Bifurcation structure of bubble oscillators

U. Parlitz, V. Englisch, C. Scheffczyk, andW. Lauterbom Institut ffir Angewandte Physik {L4P}, Technische Hochschule Darmstadt. Schlossgartenstrasse 7, D-6100 Darmstad6 Federal Republic of Gerrnany

(Received 10 August 1989; accepted for publication 7 March 1990)

Methods from chaos physics are applied to a model of a driven spherical gas bubble in water to determine its dynamic properties, especially its resonance behavior and bifurcation structure. The dynamic properties are described in a growing level of abstraction by radius-time curves, trajectories in state space, strange attractors in the Poincar6 plane, basins of attraction, bifurcation diagrams, winding number diagrams, and phase diagrams. A sequence of bifurcation diagrams is given, exemplifying the recurrent pattern in the bifurcation set and its relation to the resonances of the system. Period-doubling cascades to chaos and back ("period bubbling") are a prominent recurring feature connected with each resonance (demonstrated for period-1, period-2, and period-3 resonances, and observed for some higher-order resonances). The recurrent nature of the bifurcation set is most'easily seen in the phase diagrams given. A similar structure of the bifurcation set has also been found for other nonlinear oscillators (Duffing, Toda, laser, and Morse).

PACS numbers: 43.25.Yw, 43.25.Zx

INTRODUCTION

When looking for a theoretical understanding of the spectrum of acoustic cavitation noise, one is led to the prob- lem of driven nonlinear bubble oscillations. Even a short

occupation with this problem reveals that it belongs to a class of difficult problems', namely, the class of driven non- linear oscillators (see, e.g., Ref. 1 ), which for decades has defied a solution. The reason now becomes more and more

obvious, as driven nonlinear oscillators necessarily seem to be deterministically chaotic systems with all their involved and incredibly complex dynamics. • The first indications in this direction and in this context arose in numerical calcula-

tions of a simple bubble model, 3 which gave rise to an elabo- rate study yielding the surprising result that parameter re- gions exist where there are no usual steady-state solutions. n By investigating a more complex bubble model, it was shown that this behavior was not a deficiency of the simple model, but of a more general nature. s-7 Today, by applying methods from modern chaos theory, there is no longer any doubt about the character of the new solutions: s42 They are chao- tic oscillations.

Such oscillations show sensitive dependence on initial conditions. Arbitrarily small perturbations of the oscillation grow exponentially in time until they reach the same order of magnitude as the variables of the system. From then on, a prediction made of the state of the system becomes meaning- less. In a chaotic system those digits of the initial state that lie beyond the finite resolution of any measurement (or compu- tation) become important for the dynamics of the system after a finite time. Therefore, long-term predictions are im- possible, although the dynamics itself is completely deter- ministic. This is one of the main results of the growing field of nonlinear dynamics. In a previous paper, 12 we gave a re- view of some of the most important notions and methods of this new theory with special attention to acoustics and exam- ples from cavitation theory. In this paper, the new tools are

applied to investigate the dynamical properties of a single spherical cavitation bubble that is subjected to a periodic external sound field. In Sec. I, we introduce the mathemat- ical model for the driven spherical bubble. To improve the accuracy of the numerical results and to reduce the CPU- time consumption of the simulations, all computations have been carried out for an equivalent system that is given in Appendix B. By means of the diffeomorphism that relates the original equations of motion and the equivalent dynami- cal system, all results are translated back and expressed in term• of the variables and parameters of the bubble model.

The driven radial oscillations of the bubble may be peri- odic as well as chaotic. When looking for the parameter val- ues where the oscillations undergo bifurcations and become chaotic, one finds that these phenomena are closely connect- ed with the resonances of the system. The whole bifurcafion set, i.e., the set of parameter values for which bifureations of the oscillation occur, is ordered in a spedfie recurring way. This superstructure of the bifurcation set is typical for peri- odically driven nonlinear oscillators.

In this paper, essentially two methods are used to visual- ize the parameter dependence of the dynamics of the bubble. In Sec. II, bifurcation diagrams are given with one coordi- nate of the attractor (the radius of the bubble) plotted versus a control parameter (mostly the driving frequency). They make visible the changes of the attractor at the bifurcation points in a clear and simple way. The bifurcation diagrams are used to give an overview on the occurrence of resonances and bifureations, and to illustrate some typical bifurcation scenarios. Additionally, plots of the basins of some attrac- ton are presented. They are intertwined in a complicated way even for coexisting periodic solutions.

A disadvantage of bifurcation diagrams is that they can show the dependence of the dynamical system on a single parameter only. A method to summarize the dependence on two (or more) parameters in a single diagram is the con-

1061 J. Acoust. Soc. Am. 88 (2), August 1990 0001-4966/90/081061-17500.80 ¸ 1990 Acoustical Society of America 1061

struction of phase diagrams. A phase diagram is a chart of the parameter space showing parameter values where bifur- cations take place. When the parameter space is two-dimen- sional, these bifurcation points in general constitute bifurca- tion curves in the parameter plane. The set of all bifurcation points is called the bifurcation set of the dynamical system. If a system depends on more than two parameters, any two- dimensional phase diagram is, of course, only a cross section of the actual bifurcation set. In Sec. III phase diagrams of the bubble model are presented that show the parameter de, pendence and the superstructure of the bifurcation set in a very condensed way.

I. THE BUBBLE MODEL

The bubble model used for the numerical simulations is

given by a nonautonomous ordinary differential equation ( 1 ) of second order:

velocity, and •c = • is the polytropic exponent of the gas in the bubble. The numerical values are for a gas bubble in water undergoing adiabatic oscillations.

The model describes the radial oscillations of a spherical gas bubble in a liquid driven by a sound field, and includes (to some approximation) sound radiation, the main damp- ing mechanism at higher amplitudes of the oscillation. Equa- tion (1) is equivalent to the bubble model of Keller and Miksis •6 to first order in !/c. In the formulation of Prosper- etti, •? the model of Keller and Miksis reads as

P

+ • d•' (2) with

(1 ' J• •'R• 'Jl- 31• 2(1 -- R"•' • -- (Pstat P "• 20' •{Rn •3tc 20' c / 2 3c/ . - k - - P+• dP, p pc at k • = P(R,•,t) + P, sin(2•vt) + Pstat --

(1) with

• -- Pstat -Jr- Pv -- P, sin(2•rvt), - where R, is the equilibrium radius of the bubble, v is the frequency of the driving sound field, Ps is the amplitude of the driving sound field, P•t,, =' 100 kPa is the static pressure, Pv = 2.33 kPa is the vapor pressure, tr = 0.0725 N/m is the surface tension, p = 998 kg/m 3 is the density of the liquid, /• = 0.001 N s/m 3 is the viscosity, c = 1500 m/s is the sound

where we have explicitly noted a constant vapor pressure term P• as we use it in our equations. This amounts to a slight modification of the constant pressure term Pst•, in the case of cold water and does not affect significantly the bubble mod- el.

The main modification concerns the driving term P• Xsin [2•rv(t + R/c) ] in Eq. (2) where the retarding term R/c appears. Our model has the advantage over the Keller and Miksis model in that the radius R of the bubble does not occur in the argument of the sinusoidal driving terms. Thus the state space Mofthe equivalent autonomous system has a cyclic variable and is given by M=R+XRXS • = {(R,U,O)}, where U= • andO = vtmod 1. The auton-

omous system can be written in the form

[ U2 (3---•) + ( 1+(1 3/c,q) (-Pstat• Pø b= _ 20' •(R• • •" 20' 41.t U

( •) Pstat-Pv+P•sin(2•) -R• -- 1 + 2•rvPs p pc

For the toruslike state space M= R + XRXS •, a global Poincar6 cross section (see Ref.. 12) can easily be construct- ed as will be shown below. Thus the neglect of R/c in the driving term enables us to define a global Poincar6 map in a straightforward way. To demonstrate that, the removal of R/c does not affect the physical validity of the model, we show in Appendix A that Eqs. ( 1 ) and (2) are equivalent up to terms of order O(c-2). Since these higher-order terms

(3)

have already been neglected during the derivation of Eq. (2), both models ( 1 ) and (2) are supposed to describe the physical reality with the Same accuracy.

Equation (3) defines a vector field v: (R,U,O)-• (•, •J,O) = v(R,U,O) on the state space M. This vector field v generates a flow map <b': M-•M on M that gives the time evolution of (initial) states x = (R,U,O)•I, i.e., •'(x) = x (t) = {R (t), U( t ), O (t) } are the solutions of the differ-

1062 J. Acoust. Soc. Am., Vol. 88, No. 2, August 1990 Parlitz eta/.: Bifurcation structure 1062

ential equations (3) at time t with initial conditions (R(0),U(0),O(0)} = (R,U,O)•tI. The set Yx = {•'(x):t•R) of all images •'(x) ofx is called trajectory (or orbit) through x. Periodic solutions of the differential equation correspond to closed trajectories. When a periodic orbit is stable, it is a simple example of an attractor. An attractor •4 is a subset of the state space M that attracts all trajectories starting in a larger subset B of M called its basin. Attractors may be very complicated fractal sets, then called strange attractors, Chaotic trajectories are, in general, part of such a strange attractor and belong to a periodic oscilla- tions. Basins usually are complicated and highly intertwined sets even for periodic coexisting attractors.

An elegant way to visualize the interesting geometrical structure of (strange) attractors and their basins is given by the so-called Poincar6 map P. To define this map, a hyper- plane X C Min the state space M has to be chosen that fulfills transversality and recurrence conditions for trajectories. In general, the definition of a suitable Poincar6 cross section • is a difficult task and can only be done locally. But for driven oscillators like the bubble model ( 1 ) or (3), a natural way to define Y. is to cut the toruslike state space M transversely to the cyclic O direction at a fixed value Oo of(} (see Fig. 1). The Poincar6 map P is then given by the flow map q•' re- stricted to X, i.e., P (Rp,Up): =•]•(Rp, Up): = q•r(Rp, Up,O o) with T = 1/w. The labelp of the variables Rp and Up is used to indicate that these are coordinates of points (Rp,Up) in the cro•s section Y.. Every time a trajec- tory crosses X, it leaves a point of intersection (Rp,Up). These points {(R •,, U•, )}, n•Z, constitute an orbit of the Poincar6 map that is, in this case, easily visualized by a set of points in the Rp- Up plane because the Poincar6 cross section Y is a two-dimensional plane. Figure 2 shows a strange at- tractor of the Poincar6 map of the bubble model, i.e., a two- dimensional transversal cut through the attractor in the three-dimensional state space M.

In the following the bubble model ( I ) or (3) is investi- gated numerically for its properties, especially its bifurca- tiens and chaotic regions for a bubble of equilibrium radius R, = 10pro. In all diagrams the radius variable R or Rp is normalized by R,. Figure 3 shows two typical periodic oscil-

FIG. 1. Visualization of the solid toruslike State space M = R + X R X S • of the bubble model (3}. The shaded R-U plane at •o defines the surface of section of a global Poincar6 map. Period- I trajectories like that shown in the figure possess one point of intersection with the Poincar6 surface of section. This point is a fixed point of the associated Poincar6 map.

Up [m/•]

Up [m/•]

45.0

27.0

9.0-

-9.0-

-27.0

-45.0

16.0

14.4

12.8

11.2

9.6

•.0

Ps = 300 kPa •= 600 kHz

0.30 0.•75 1.45 2.025 2.60

1.5 1'.6 1•.7 1•.6 1.9 Rp/Rn

FIG. 2. (a) Poincar6 cross section of a strange attractor of the bubble mod- el. The box indicates the location of the enlargement shown in (b} in order to visualize the complicated fraetal and self-similar local slructure of the attractor.

lations that have been plotted after the transients decayed. In Fig. 3(a), the period of the oscillation equals the period of the sinusoidal driving. Therefore, the dots plotted whenever a period T = l/w of the driving has elapsed always occur for the same value of the radius R. Figure 3 (b) shows a period-2

2.Ofi -

1.64

0.76

0.32

1.42850 1.43424

(a) 1.43998 1.4457•

t [ms] 1.45146 1.4572

1.60 -

R-• L0•- 0.73-

0.44- I 0.83330 0.638:32 0.84334 0.54638

t [msl 0.8,5336 0.6684

FIG. 3. Radius-time curves of two periodic bubble oscillations for R, = 10/.tin and P, = 80kPa. (a) Period-I oscillation occurring for v = 70 kHz and (b) period-2 oscillation occurring for v = 120 kHz. The black dots on the solution curves are plotted at times t• = k l/v (k = 1,2,3,...), i.e., whenever a period ofthe sinusoidal driving has elapsed.

1063 J. Acoust. Sec. Am., Vol. 88, No. 2, August 1990 Parlitz eta/.: Bifurcation structure 1063

225,0 -

120.0 -

U l•-O- (m/•1-90.0-

--10•.0-

-300.0

3O.O-

U tO.O- [m/.1-10.0-

-30.0-

-õ0.0 0.2

100.0

60.0 -

20.0 - u

[•/'1-20.0-

-60.0-

-ioo.o

18.o-

-6.0-

-18.0-

-30.0

0.00 2:2 3.00

,50.0 -

0.7 1.2 .7

0.0 0.6 ,'.2 ;., 2.,

30.0 (d)

o., o.? ;.o ;.3

FIG. 4. Projections of four periodic trajectories into the R-Uplanc with R normalized with R.. The black dots give the locations of the projected state points [R(t)/R., U(t) ] when the trajectories are illuminated stroboscopi- cally at times t k = k i/v (k = i,2,3,.:.) (compare Fig. 3}. The periodic os- cillations shown have been computed for P• = 80 kPa, R. ---- 10/ma, and different driving frequencies v belonging to different resonances R..,• of the system. The classification of the resonances is explained in detail in the text. ( a ) v = 200 kHz, period- 1, resonance R L •; (b) v = 70 kHz, period- 1, reso- nance R4j; (c) v----490 kHz, period-2, resonance R,,2; and (d) v = 120 kHz, period-2, resonance Rs. 2 .

oscillation; i.e., in this case, the period of the oscillation equals two periods Tofthe driving. In both diagrams, sharp peaks in the direction of small values of the radius occur. They result from the collapse of the bubble and become more pronounced for increasing amplitudes of the driving sound field. For the numerical treatment, the system is therefore transformed with the help of a suitable diffeomorphism to get less steep bubble collapses and thus relaxed conditions for the accuracy of the calculations and for computer time. This transformation is given in detail in Appendix B.

z FIG. 5. The torsion number n of a

periodid orbit y gives the average number of windings of a neighboring trajectory V' around yduring one pe- riod of y. It may be computed by counting the rotations of the differ- ence vector .V(t) pointing from r to r'.

Another way to visualize the solutions of the bubble model is by using plots of projections of the trajectory into the R- U plane as can be seen in Fig. 4. As in the case of the radius-time curve in Fig. 3, a dot has been plotted after each period of the driving. The trajectories in Fig. 4(a) and (b) belong to period-1 solutions, whereas those in Fig. 4(c) and (d) show the orbits of period-2 oscillations. The dots can also be interpreted as the points of intersection of the trajec- tory in state space with the Poincar• cross section X at Oo = 0. They represent the orbits of the associated Poincar• map: a fixed point in the case of Fig. 4(a) and (b) and a period-2 cycle in Fig. 4(c) and (d). As illustrated in Ref. 12, these simple periodic attractors of the Poincar• map may undergo period-doubling (PD) cascades where the number of attractor points doubles at each step. Beyond the accumu- lation point of the cascade, the attractors become complicat- ed self-similar sets as exemplified in Fig. 2. The trajectories shown in Fig. 4 belong to different resonances of the bubble model. This can be seen from the number of loops of the orbits (or from the number of local maxima or minima in the radius time curves in Fig. 3). This loop number and the period of the oscillation may be used to classify the reson- ances of the bubble model. Unfortunately, the loop number is not invariant under diffeomorphic transformations of the dynamical system; i.e., its value depends on the choice of the coordinate system. Therefore, the torsion number •sj9 that counts the average number of rotations of a neighboring or- bit about the given periodic trajectory during one period of the oscillation (see Fig. 5 ) is used to characterize resonances and to label the bifurcation curves of periodic orbits in the phase diagrams.

In the following, we give a brief introduction into the new concept of torsion numbers (and generalized winding numbers) of a given orbit. Let ?, be an orbit of the bubble oscillator associated with the solution x(t) = (R(t),U(t), O(t)} of Eqs. (3) and let y' be a neighboring orbit ofy given by the solution z(t) = x(t) + y(t) of (3). The perturbation y(t) = (y•(t),y:(t),y•(t)} is assumed to be infinitesimally small. Then, the local torsion of the flow is described by the rotations of thc difference vector y (t) about y (compare Fig. 5). The time evolution of y(t) is given by the variational equation (4) of the differential equation (3), where Dv(x(t)} is the Jacobi matrix of the vector field v:

•(t) = Dv(x(t)).y(t) = Ov2 ø•2 ø•v2 o•/. j, y(t). (4) 0

Equation (4) immediately follows from a Taylor series ex- pansion ( 5 ) of v at the (state) point x (t)•14:

t 064 J. Acoust. Soc. Am., Vol. 88, No. 2, August 1990 Parlitz eta/.: Bifurcation structure 1064

:9(0 = •.(t) - •(t) = v(z(t)) - v(x(t))

= v(x(t) + y(t)) - v(x(t))

= Dv(x(t)).¾(t) + higher order terms. (5)

Sincej;• (t) vanishes in Eq. (4) [compare Eq. (3)],y3(t) is constant in time, and thus only the first two equations of (4) are of importance to describe the local (i.e., linearized) flow about x(t) or y. Using polar coordinates (.V•,Y2)= (r X cos a,r sin a) we obtain the (reduced) variational equa- tions in polar coordinates:

OR/ •U

( c•v2 c•V2sina)cosa. (7 ) •t= --sin 2a+ a•Rcosa +3U The mean angular velocity • of the difference vector y(t) is given by

l•(y) = lira --1 f &(s)ds =' lira a(t) -- a(0) ,-,0 t Jo ,-•o '• (8) We call f• (y) the torsion frequency of the orbit y. It is the nonlinear analog to the eigenfrequency of linear systems. Since the difference vector y(t) always rotates dockwise [because.O, = Y2 in Eq. (4) ], the torsion frequency f• is neg- ative for all orbits of the bubble oscillator. We use the torsion

frequency fth = - 2zrvofthe driving harmonic oscillator to define the (generalized) winding number w = w(y) of the orbit y:

w(y) = fl(y)/•h. (9)

In contrast to the (ordinary) winding number, z"9 the defini- tion (9) of to(y) is not restricted to orbits that lie on an invariant torus in phase space. Since the bubble model is a strictly dissipative system (i.e., its Poincar6 map is every- where contracting), the existence of Hopf bifurcations and invariant tori can be excluded rigorously. Thus quasiperio- dic motion cannot occur and only the (generalized) winding number defined above can be used to describe resonant prop- erties of the oscillations. Definition (9) also includes the case of chaotic attractors with certain ergodic properties. 2ø The winding numbers at the bifurcation points of period-

-doubling cascades constitute a converging alternating se- quence. We come back to this point in the next section.

For periodic oscillations with period (number) m = re(y) (where period rn means that the oscillation re- peats after m periods of the driving frequency), the quantity

n = n(y) = mw(y) = m•(y)/( -- 2try) (10)

is called the torsion number of the closed orbit y. Around saddle-node (SN) and period-doubling (PD) bifurcation points in parameter space, the Jacobi matrix DP(R•,, U•, ) of the Poincar6 map P possesses real eigenvalues. For negative eigenvalues, the torsion number is an odd multiple of «; for positive eigenvalues, it equals an integer. •s Since the eigen- values of DP depend continuously on the parameters of the system, the torsion number n is a fixed integer ns• around SN bifurcation points and undergoes (as does the period m) a doubling n-,2n = nen•N at PD bifurcation points with n•,o (and n) also being constants in a whole neighborhood of

the bifurcation point. These integer values usually equal the number of loops of the trajectory and are used in Sec. III to label the bifurcation curves in the phase diagrams. The minxr- ing number w = n/m is also a constant around SN as well as PD bifurcation curves in parameter space (see Sec. II for examples).

Some other methods to investigate especially chaotic bubble oscillations as, for instance, attractor maps and the determination of various fractal dimensions are not treated

in this paper in order to limit the material being presented. The reader who is interested in these topics may consult Ref. 12 where these techniques are applied to chaotic attractors of the bubble model and further references are given.

II. BIFURCATION DIAGRAMS AND BASINS OF ATTRACTION

In a bifurcation diagram, one coordinate value of the attractor in the Poincar6 plane of section is plotted versus a parameter of the system, called a "control parameter." In the diagrams to follow, we take as dependent coordinate the normalized radius Rp/R• in the Poincar• plane Oo = 0, and as control parameter the frequency v of the driving sound field. We call these frequency bifurcation diagrams or v-bi- furcation diagrams. In this form the bifurcation diagram ap- pears as a slight but important modification of the usual amplitude resonance curves where the maximum of the re- sulting steady-state oscillation amplitude is plotted versus the driving frequency. Bifurcation diagrams are more apt to show the nature of the attractors for different parameter val- ues. For instance, when an attractor (a steady-state solu- tion) is of period 2, i.e., repeats only after two oscillations of the driving sound field, two points are plotted at that value of the control parameter (compare Figs. 3 and 4). For a strange attractor like that in Fig. 2, the projection yields a large number of points which appear scattered along a verti- cal line bounded by the size of the attractor with respect to the coordinate considered in the bifurcation diagram. In a series of bifurcation diagrams, the parameter dependence of the attractors of the bubble oscillator will now be demon-

strated. Figure 6(a) is a frequency bifurcation diagram (R•/R, vs v) for a sound-pressure amplitude of P• = 20 kPa and equilbrium radius R,= 10/zm. The diagram has been obtained in the following way. Starting with v = 30 kHz, the attractor has been calculated by solving the differ- ential equation given above until (after 100 periods of the. driving) a steady-state oscillation is reached. Then, up to 100 points of the orbit in the Poincar• plane are plotted. If the solution is of period 1, only a single point occurs in the diagram. After the projection of the attractor has been plot- ted in this way, the control parameter v is increased by small step Av. Starting with the last point from the previous attrac- tor as initial value, the new attractor is calculated for the frequency value v + Av. This procedure is repeated up to v = 720 kHz. To cover the hysteresis of the large resonance R•.•, the procedure is reversed from v = 720 kHz; i.e., the frequency is lowered in steps down to v = 30 kHz. For P• = 20 kPa only period-1 oscillations are present. In the re-

gions where there are two points above one frequency value, there are coexisting attractors of period 1. They appear be-

1065 J. Acoust. Soc. Am., Vol. 88, No. 2, Augpst 1990 Parlitz eta/.: Bifurcation structure 1065

2.10

1.91 -

1.72-

1.34.-

1.15 -

0.96

3

Ps-- 20 kPa R,,= 10 /•m

(b) 0

30 145 260 375 490 605

v

FiG. 6. (a) A frequency bifurcation diagram sho•n• the no•aHzed radi- • •rdinate R•/R,in the Poin•r• plane venus the d•ving fr•uency v. Tbe dia;ram h• •en compu/• for incr•sing and d•rcasing v in order to v•ua]ize the hyster•}s ph•om•a •oc{at• w{/h coexisting attractor. For •ch new v value, tbe data of tbe p•ing v value have new initial •nd{tions and the r•ult• •e ploU• a•er the t•nsicnts have d•ay•. For ihc •lativdy small dHvin z amplitude of P, = 20 kP• all o•il- lations are •d• i and only two (•ddle-node) bffurcations in conn•fion •}th the r•nan• R•., •ur. Th• bifu•utions I•d to zransitions onto c•x{stin; att•to• that a• indicted by two a•ows. B•id• the r•o- •n• Rm. m t•t is well known from the ha•onic •i!lator. •me higher- order •d•-I r•onan•s (like R:., ) are •dy visible. (b) To.ion num- ar diagram •rr•nding to (a). The to.ion numar n takes integer valu• at the r•onan• and bifurcation •ints. and is a well-suit• quantity for cl•sifying th• phenomena.

under parameter variations until the orbit vanishes or be- comes stable.•8

The basins of attraction in the Poincar6 plane of the coexisting attractors are shown in Fig. 7. The black basins in Fig. 7(a)-(c) belong to the lower branch of the resonance R •,• in Fig. 6 (a), whereas the white areas in the plots give the basins of the attractors at the upper branch. The three fig- ures show the basins for v = 280, 290, and 295 kHz. It can be

seen that the black basin shrinks when the control parameter v approaches the critical value v 2 ----- 297 kHz where the at- tractor of the lower branch v vanishes. For this example the loss of stability of the attractor goes together with a decrease of the area (volume) of its basin. It should be noted that this

connection between a local property of the attractor (its sta- bility) and a global feature (its basin) is not necessarily the

When the driving amplitude Ps is increased more and more, secondary, nonlinear resonances of the type R,.,, n = 2,3 ..... become visible in the bifurcation diagram. Figure 8 shows the diagram for P, = 40 kPa. On the left-hand side of the diagram, the resonances have become more pro- nounced, and two saddle-node bifurcations occur in connec- tion with the resonance R2. • , giving rise to an additional hysteresis loop similar to that of R,.•. For driving frequen- cies above the main resonance R,.•, a different kind of a resonance occurs that consists of a period-2 oscillation. As the torsion number of this period-2 attractor is one, this reso- nance is labeled R,.2. The period-2 resonance is created by period-doubling bifurcations with an additional saddle-node bifurcation within the resonance region where the period-2 solution loses its stability. This bifurcation scenario is given schematically in the pictogram in Fig. 9, where the unstable periodic orbits that are created at the bifurcation points are also included for clarity. The period-I resonances emerge from a smooth curve as peaks that become sharper and high- er when P, is increased. In contrast, the period-2 resonances

cause the diagram has been calculated up and down as ex- plained above to catch the hysteresis regions. The arrows mark the saddle-node bifurcation points v, = 260 kHz and v2 = 297 kHz where jumps occur to the other attractor. Thus two period-1 attractors coexist in the parameter inter- val

Furthermore, secondary resonances are present in Fig. 6 (a) which have been labeled by R with indices. The second index gives the period (number) m of the attractor. The first index of R is the torsion number which counts the number of

windings of a neighboring orbit around the periodic attrac- tor. •8 Its dependence on v is shown in Fig. 6(b). At the resonances R .... the torsion number takes on an integer val- ue. The corresponding plateaus in the torsion number dia- grams become wider when the driving amplitude P5 is in- creased and mark the hysteresis region of the resonance. ]'he dashed line at the resonance R •.• give the integer-valued tor- sion number of the unstable period- 1 orbit that is created and destroyed via SN bifurcations at v, and v:. The value n = 1 that occurs at the bifurcation points remains constant be- cause in strictly dissipative driven oscillators, torsion numbers of unstable orbits are integers and do not change

Up [m•l

-10

-20 -

-30

0

-lB

-zo -30 ,

O.I 0.7 I 3 1.9

Rp/Rn

FIG. 7. Poincar6 sections of the ba-

sins of the coexisting period- I attrac- tors of the resonance R•.• shown in Fig. 6(a). The basins have been com- puted forP, = 20kPa, R• = 10/zm, and {a) v = 280 kHz, {b) v--- 290 kHz, and (c} v= 295 kHz. The black and white dots inside the ba-

sins are the corresponding attractors (fixed points of the Poinear6 map). The black basin belonging to the lower branch in Fig. 6{a) shrinks and eventually vanishes when v reaches the bifurcation point v_, = 297 kHz.

1066 J. Acoust. Soc. Am., VoL 88. No. 2, August 1990 Parlitz otaL: Bifurcation structure 1066

Ps = 80 kPa R n = 10 3.00

2.24-

1.96 -

1.68- 1.40 -

1.12 -

0.64

Ps = 40 kPa Rn = 10 /•m

R•.• • R•,2

30 145 280 375 490 605 720

HG. 8. Frequency bifurcation diagram for P, = 40 kPa [compare Fig. 6(a) ]. Pronounced higher-order resonances of period-I and the first peri- od-2 resonance R•a occur. The arrows indicate transitions onto coexisting attractors at saddle-node bifurcation points.

are suddenly "born" by the bifurcations shown in Fig. 9 when the driving amplitude Ps exceeds a critical value. Be- fore the bifurcation takes place, the period-2 resonances are "invisible" in bifurcation diagrams [see Fig. 6(a) ] as well as in ordinary amplitude resonance curves. The creation of pe- riod-2 resonances described above has also been observed for

many other nonlinear oscillators. 14.1• If the system possesses additional symmetry properties like Dufiing's equation, •3.,s this type of a resonance does not occur, and is in some sense replaced by a symmetry-breaking resonance. Period-2 re- sonances typically occur first for high driving frequencies as shown in Fig. 8. But as in the case of the period- 1 resonances, further bifurcations leading to resonances R,.: with increas- ing torsion numbers occur when the driving is increased. Figure 10 shows two bifurcation diagrams for Ps = 80 kPa. Between the period-I resonances R.•, period-2 resonances R,.2 with odd torsion numbers n = 3,5,7 occur. The period number rn and the torsion number n of the period-2 reso- nances R,m are always given as the sum of the correspond- ing numbers of the neighboring period-1 resonances (e.g., R2a-,R3. 2 ,-Ru• ). This ordering principle is very useful when searching for resonances that do not occur via bifurca-

sn%••i• Rp period 1 pd period 1 R-•

FIG. 9. Pictogram of the bifurcation scenario of the resonance R,a in Fig. 8. This symbolic diagram shows the branches of the stable (connected curves } as well as the unstable period- 1 (dashed curve) and period-2 {dotted curve) orbits created at the period doubling (PD) and saddle node (SN) bifurca- tion points. The same scenario occurs in connection with all (higher-order) period-2 resonances R.a with odd torsion number n (see. e.g., Fig. 10}.

2.6t -

2.22 -

Rp

1.44 -

1.05 -

0.66 30 14•5 t 260 • 490 605 720

V [kHz]

2.64,

2.31

1.08

Rp Rn 1.65-

1.32

0.99

0.06

•'X R2a (b) R$,i •

3O 60 90 120 150 180 210

FIG. 10. Bifurcation diagrams for the driving amplitude P, = 80 kPa (com- pare Figs. 6 and 8). More higher-order resonances including the period-2 resonances R•a, Raa, and R,.• occur as can best be seen in diagram (b) showing only the resonances to the left of the main resonance R u . Arrows indicating transitions at saddle-node bifurcation points have been omitted for clarity in these figures and the following bifurcation diagrams. At those v values where more than two attractors coexist, only two of them are plot- ted due to the technique of increasing and decreasing the driving frequency only once. For this reason the branches of the period-2 resonances R,.: and Rs.: are incomplete.

tions of the basic period-1 solution. The period-3 resonances that are discussed below are candidates for the application of this searching strategy.

Period-doubling cascades and strange attractors occur after only a slight increase of P, above 80 kPa. Figure 11

Ps = 85 kPa Rn = 10 •m

Rs.,

z.aa 1 1.06- Rs., •. Rs• •

1.22- .

30 60 90 lgO 150 180 210

FIG. 11. Bifureation diagram for P• = 85 kPa showing the same frequency interval as Fig. 10(b). Now the resonances R3.: and Rs.: possess complete PD cascades. In the resonance R,.: only a second PD from period 2 to peri- od 4 is visible, but the branches of this resonance are plotted incompletely due to the technique used giving at most two coexisting attractors. The ori- gin of the additional PD bifurcation at the !eft-hand sides of Rn.:, R,.:, and Rs.: will be discussed in Sec. III.

1067 J. Acoust. Soc. Am., VoL 88, No. 2, August 1990 Parlitz otal.: Bifurcation structure 1067

shows the bifurcation diagram at P, = 85 kPa for 30 kHz< v< 210 kHz. The frequency scale has been expanded to better visualize the attractors and their bifurcations. Thus

only the region below the main resonance R •.1 is shown. The resonance R3. 2 has already developed a complete period- doubling cascade that is reached by lowering the driving fre- quency from above. The strange attractors obtained beyond the cascade growjn size toward lower frequencies and sud- denly disappear, leaving a period-1 oscillation. The reso- nance Rs. 2 also has developed a period-doubling cascade from higher frequencies downward. When its strange attrac- tors disappear, they jump to a period-2 attractor which per- sists over a large interval and almost totally covers the reso- nance R3, t . The resonance R7, 2 is in a way similar to the resonance R5.2, except that the period-doubling cascade at this value of Ps stops just after the first further bifurcation to a period-4 attractor. It then drops back to a period-2 attrac- tor which, in turn, covers most of the resonance R4,1 . The resonance R9, 2 is even less developed, but also covers a large range of frequencies leaving only a narrow band for the peri- od- 1 resonance R 5,• ß

A slight increase in the driving pressure amplitude to P, = 90 kPa greatly expands the regions with strange attrac- tors and adds period-doubling cascades (Fig. 12). The re- gions of period-1 oscillations become markedly smaller and mainly only coexist with higher-order periodic or chaotic attractors. The resonance n3. 2 now consists of two parts, a newly grown "bubble" of period 2 (marked by V) and the

3.12

2.66 -

2.24 -

Rp R'-• 1.80-

1.38 -

0.92 -

0.48

Ps = 90 kPa Rn = 10 /•m

R2,, •,, (a)

30 14,5 2•0 3?õ 440 685 '7'20 l/ [kHz]

2.82 I '

243 / R2,, 2.04 • ß

•nn 1.65

0.4õ] } • • , 30 80 00 1•0 m•0 1•0 •10

' • [kHz]

FIO. 12. Bifur•ation diagrams for • = 90 kPa: (a) ove•iew <d (b) en- largement for s•all driving fr•uenci• [•mpare Figs. 10(b) and 11 ]. Many PU •cades and v intelams with chaotic attractors •cur. Even a 3.2LPU •cade (k = 1,2,3,..) with basic •fi• three (denoted by •) is visible. The "bubble" denoted by discuss• in S<. III.

Ps =90 kPa v= 117.8 kHz -1.1

Up [m/al

-1.9

-2.7 -

-3.5 -

-4.3 -

-5.1

1.200 I

1.4,50

Rp/Rn 1.700

FIG. 13. Poincar• cross section of a strange attractor of the bubble model created by a PD cascade of the resonance Rs.:.

modified previous part with a period-doubling cascade and a strange-attractor section which this time drops back to peri- od 2, as already encountered with the resonance Rs,: in Fig. 11. Figure 13 shows the Poincar• cross section of a chaotic solution that is created by the PD cascade of the resonance Rs.: . The shape of this attractor is very similar to that of a corresponding chaotic attractor in the resonance R3, 2 (see Ref. 12, Fig. l 1 ).

In Fig. 12(b) some small intervals with attractors of period 3.2 • are visible (marked by A). They coexist in a large parameter range, but are seldom reached by the tech- nique used to compute the bifurcation diagrams. There are several ways to overcome this limitation. If one knows an initial condition belonging to the basin of a period-3 attrac- tor, one can trace the period-3 attractor by increasing or decreasing the control parameter as it is done for the reso- nance R,3 at the end of this section. Suitable initial condi- tions may be found with a different kind of bifurcation dia- gram. Instead of a single point of the attractor belonging to the previous parameter value being used as the new initial condition, many points on a grid in the Poincar• plane are taken. Color coding the different attractors may then be- come necessary to be able to distinguish them in the plot. Another method to search for period-3 solutions or other attractors of interest consists in the computation of basins of attraction like those shown in Fig. 7. Figure 14 shows two basin plots for P• = 170 kPa and v = 430 kHz where four coexisting attractors occur. The white, gray, and black areas in the diagram of Fig. 14 (a) belong to the period- 1, period-2, and period-3 attractor encountered, respectively. The fourth basin belongs to a period-5 attractor. Its basin is very small in size for the given parameter values and is marked by the arrow in Fig. 14(b). The grainy areas in both plots are due to the finite resolution of the grid of initial values taken. The zoom {Fig. 14(b)] into the small rectangle of Fig. 14(a) gives an impression of the yet-unresolved complex structure of the basins.

As can be seen from the bifurcation diagrams in Figs. 10-12 some specific bifurcation scenarios recur in conjunc- tion with the nonlinear resonances of the driven bubble. This repeated structure of the bifurcation set will become more clearly visible in the phase diagrams to follow in the next section. In this section we are going on to describe the devel-

1068 J. Acoust. Soc. Am., Vol. 88, No. 2, August 1990 Parlitz ot aL: Bifurcation structure 1068

(a) 30.0

20.0

IO.O

U'p 0.0

-1o.o

-20.0

-30.0

0.1 0.7 1.3 1.9 2.5

(b) 5.0

4.O

3•)

zo

LO

0.0

-I.O 1.o 1.2 L4 1.6

/ R.

1.8 2.0

FIG. 14. (a) Basins of attraction of four coexisting periodic attractors for the parameter values P, = 170 kPa, R, = 10pro, and v = 430 kHz. The white area belongs to a period-I attractor with the coordinates {(R•,/R,,, Up}}• ={(1.30,8.91}}, the grey area to a period-2 attractor with {(Ri,/R,,Ut,)}2 = {(2.24,9.29),(2.39, -- i.72)}, and the black area to a period-3 anraetor with {(RffR,,U,,)}• ={(!.64,9,42), (!.04, -- 7.51 ), (1.68,4.58) }. The basin of the period-5 attractor with the coordi- nates {(Rp/R,,,Up)}s = {(1.91,8.51),(1.52, - 5.64), (2.14,5.48),1.06, -- 13.81 ),( 1.63,3.25)} can hardly be seen in this diagram, but in the en-

largement in Fig. 14(b). (b} Enlargement of the frame indicated in (a). The curved grey layer pointed at by the arrow belongs to the basin of the period-5 attractor that is to be distinguished from the lower right grey basin of the period-2 attractor.

opment of the inner bifurcation structure of the resonance pair R •.• -R ,.z when the driving amplitude is increased. This pair may serve as a prototype or kind of elementary cell for alltheotherhigher-orderresonancepairsR,.•-R2, ,a. Fig- ure 15 shows the development of the resonances R•.• and R•.= when P, is increased drastically. The four diagrams in the left column show the occurrence of PD bifurcations

along the upper branch of the resonance R•.•. The period-2 attractors belonging to the upper branch of the resonance

R•. 2 remain almost unchanged in this series. In Fig. 15(a) two PD bifureations at the upper branch of the R•,• res• nance ereate a period-2 "bubble." The period-2 oscillation at the lower branch of the resonance R•.: is stable for a small v interval until it vanishes due to a saddle-node bifurcation.

When P• is increased further, two of the PD bifurcations merge, resulting in the diagram shown in Fig. 15(b). The next qualitative changes of the bifurcation diagram in this parameter region consist of further period doublings, creat- ing bubbles of period 4, period-8, and so on. Figure 15(e) shows the diagram after the generation of the four period-8 "bubbles." This "period-bubbling" cascade accumulates, and a parameter interval with chaotic attractors occurs. Fig- ure 15 { d) shows an enlargement of this scenario.

The right column of Fig. 15 illustrates what happens to the upper branch of the resonance R•.: when P• takes on large values. The sequence of four bifurcation diagrams shows another period-bubbling cascade, resulting again in chaos and strange attractors. The scattered points in Fig. 15 (h) denoted by •7 are not due to transients, but the result of a global bifurcation • of the attractor. To illustrate this phenomenon, Fig. 16 shows the Poinear6 sections of two typical attractors. The attractor in Fig. 16(a) was computed for v = 50 kHz and contains no "outliers." When the driving frequency is increased, the attractor suddenly is drastically enlarged due to a global bifureation, i.e., the interaction of certain invariant manifolds of the Poincar6 map. The new parts of the attractor consist of points on these invariant manifolds. They are reached (immediately after the bifurea- tion) only with a low probability. Therefore, these parts oc- cur only with a low density in Fig. 16(b).

In Fig. 17(a), a blowup of one of the period-doubling cascades shown in Fig. 15(h) is given. Figure 17(b) and (c) shows the corresponding largest Lyapunov exponent/[,,• and the winding number w vs v. While the/[.•,• vanishes at the bifurcation points of the cascade, the winding number is constant near the critical v values (see Sec. I). The height of the resulting steps in the winding number diagram is given by a simple formula similar to the result for the torsion num- ber in a period-doubling easead•. • l.•: At the k th PD bifurca- tion point of the cascade shown in Fig. 17(a), the winding number w takes the value

w• -----w• + ( -- 1)•/3mo2 k, (11) where w= is the winding number at the accumulation point of the PD cascade. Here, w• is given by the basic winding number Wo and the basic period m o of the resonance where the PD cascade takes place:

tooo = too -- l/3rno. (12)

For the example shown in Fig. 17, the values mo= 2, to o = •, and tooo = « hold. There also exist PD cascades where the ordering of "up" and "down" steps in the corresponding winding number diagrams is reversed. 14'19 These types of scenarios are described by the winding number formulas ( 13 ) and (14) which differ from Eqs. ( 11 ) and (12) only by some signs:

to• = w• -- ( -- 1)•/3mo 2•, (13) w• = too + l/3mo. (14)

1069 J. Acoust. Soc. Am., Vol. 88, No. 2, August 1990 Parlitz et aL: Bifurcation structure 1069

Pa = 170 kPa Rn = 10 •m 390

a.aa- • (•) 1.30 -

O.T8 1•o a•o • & • ?•o 81o

Ps = 190 kPa Rn= 10 /•m

(b)

2.04.

Ps = 205 kPa R.= 10 tam

2.•1- 1.72-

!.13-

0.54

2.88 - Ps = 206.3 kPa 1• = 10 tam

152'

1.18 -

2.20-

l.SO-

Rp

a8o & do

282

260 kPa R•= 10 tam

(e) I • do •8 81o

Ps = 270 kPa Rn = 10 /.•m

(f); 2.51 -

1.89 -

158-

127-

0 96 t a•o & `%;0 •`% 60o ?35 81o

231-

220-

Rp •n 13o-

2 88

2.46-

2.04-

1.52- !.20 -

0.78 -

0.36 - 380

290 kPa R. = 10 /•m

FIG. 15. Bifurcation diagrams showing the evolution of the resonances R u and R•.2 for increasing P•. In the diagrams giv- en in the left column, the resonance R u undergoes several PD bifurcation until some chaotic attractors occur. Here, R•a remains almost unchanged in this series. The right column is devoted to the reso- nance R.z that bifurcates to chaos, too, when P• is suflieiently high.

The deeper systematics behind the winding number se- quences that may be encountered in dynamical systems has still to be explored. For this problem, the study of the logistic map turns out to be the appropriate starting point. The wind- ing number is in this case given as the relative number ofR's in the R-L string of the symbolic description of the dynam- ics, and the formulas (11)-(14) are immediate conse- quences of the construction law upon period doubling? In the case of driven oscillators, where in contrast to the logistic parabola attractors can coexist and nonlinear resonances oc- cur, a scheme of all possible winding number sequences should be a helpful guide when searching for coexisting at- tractors and their bifurcations.

Even if one considers the very large driving amplitudes Ps used to compute the diagrams in Figs. 15-17 as physically irrelevant or beyond the scope of a model describing the

oscillation of a spherical bubble, the bifurcation scenarios shown are of great importance, because they also take place for more moderate values of P, at other resonances (com- pare Figs. 10-12). The period-bubbling cascades in particu- lar may be found in all resonances.

Figure 18 gives an impression of the rich bifurcation structure at high driving amplitudes P•. Many period-dou- bling cascades and jumps onto coexisting attractors take place. The period-3-2"cascade occurring between v, = 565 kHz and v• = 695 kHz belongs to the period-3 resonance R•.3. In Fig. 19, this basic period-3 solution has been traced to high and low frequencies until it vanishes due to saddle- node bifurcations. For this diagram, a smaller value of P, has been chosen than in Fig. 18 to obtain finite cascades. Again, the typical period-bubbling scenario is found, now starting with basic period 3.

1070 d. Acoust. Soc. Am., VoL 88, No. 2, August 1990 Parlitz ot at: Bifurcation structure 1070

40.0

24.0-

-8.0-

-24.0 -

-40.0

0.30

Ps =290 kPa u=550 kHz

(a)

0,875 1.450 2.025 2.õ0

Rp/Rn

Up

Ps =290 kPa v=585 kHz 40.0

24.0 -

8.0-

-8.0 -

-24.0 -

-40.0 0.30 0.875

--'"'"'. (b) ! :< ">,• ...

.

1.4,50 2.025 2.60

Rp/Rn FIG. 16. Two Poincar& cross sections of a strange attractor (a) before and (b) after the attractor is "exploded" due to a global bifurcation.

This section is closed with an example of a P,-bifurca- tion diagram shown in Fig. 20. Jumps onto coexisting attrac- tors and period-doubling cascades 2'2" and 3.2"occur. More diagrams like those of Fig. 20 (P,-bifurcation dia- grams) in combination with v-bifurcation diagrams would give more insight into the global bifurcation properties of the driven bubble oscillator. But the material is best combined in

an even higher kind of abstraction than bifurcation dia- grams, i.e., phase diagrams, where only the bifurcation points in parameter space are considered and plotted. This is the topic of the next section.

III. PHASE DIAGRAMS

In this section phase diagrams of the v-P,.-parameter plane are given that have been computed for R, = 10/zm. The numerical methods that were used are briefly explained in Appendix D. For clarity, only certain bifurcation curves of particular interest are shown in the following figures. The curves are labeled by (n,m), where n is the torsion number and m is the period number as introduced above. In the case of the SN curves, both numbers are integers in a small strip around each curve. For PD curves this strip is divided into two parts by the bifurcation curve. In both substrips the number ni and rn i are constant, and the relations n2 = 2n• and rn 2 = 2rn, hold. The larger values ofn and m are used to label the PD cuves. Figure 21 shows some bifurcation curves belonging to the resonances R,l and R,2. The solid curves give the locations of PD bifurcations and the dashed curves those of SN bifurcations. To facilitate the interpretation of the curves, cuts through the diagram denoted by (a)-(f) are

P$ •

2.8• ---'--•' 2.54 -

290. kPa R n = 10.

2.26 -

Rp t.98 t • L701 t 14' (a) 0.25

0.00

-0.25

-0.50 -

-0.75

0.40 w

0.35 -

0.30 -

0.25

(c) 0.20

390

(b)

11

8

16

4 0 430 450 4 0 490 510

•' [kHz]

FIG. 17. (a) Enlargement of the PD cascade shown in Fig. 15(h). In this diagram, v has only been increased and not decreased again. (b) Corre- sponding largest Lyapunov exponent d. ma x of the Poincar6 map. Here, 2•,• vanishes at the bifurcation points and becomes positive for chaotic attrac- tors. (c) Corresponding winding number diagram with plateaus around the bifurcation points. The first numbers of the cascade are w• = •, w: = •, w3 = •6, and w4 = ,•. The winding number w• at the accumulation point of the PD cascade in this case equals «.

introduced and discussed. The line labled (a) cuts and reso- nance horn ( 1,1 ) and just encounters the hysteresis that oc- curs when v is varied for fixed P, above a critical value [com- pare Fig. 6(a) ]. The SN bifurcation curve ( 1,1 ) gives the

3.30

2.79

2.28 -

Rp •n 1.77-

1.26 -

0.75 -

0.24 3O0

Ps = 350 kPa R n = 10 /am

385 470 555 640 725 810

v

FIG. 18. Bifurcation diagram for a relatively high driving amplitude. Many (coexisting) periodic and chaotic attractors occur, in particular a period- 3' 2" cascade. To compute this diagram v has only been increased.

1071 J. A'coust. Soc. Am., Vol. 88, No. 2, August 1990 Parlitz oral.: Bifurcation structure 1071

2.88 -

2.45 -

2.02:

Rp •nn 1,õ9-

1.16-

P•= 320 kPa. Rn = 10 /•m

0.73 -

0.30,

570 655 74•0 825 995 1080

FIG. 19. Bifurcation diagram showing a period-3 attractor created by SN bifurcations and the first two period doublings to period 6 and period 12. These period-3' 2" oscillations constitute the period-3 resonance R •.•.

boundaries of the hysteresis region. Inside this region, two period- 1 attractors coexist, at least near the boundaries. Cut (b) is more complicated. The PD and SN bifurcations en- countered when following this line are best explained by means of the pictogram in Fig. 9 [see also the right part of the bifurcation diagram in Fig. 12(a) ]. Each bifurcation in the pictogram corresponds to a point of intersection of the SN and PD bifurcation curves (1,2), and the line (b) in Fig. 21. The pictogram of Fig. 9 is a good example for learning how to interpret the corresponding bifurcation curves, since it shows the connection of the curves via the unstable orbits.

When comparing phase diagrams with bifurcation diagrams at higher driving pressures Ps as given in Sec. II, the situation becomes even more involved. Line (c) in Fig. 21 denotes a cut at P• = 170 kPa. It corresponds to the sequence ofbifur- cations shown in Fig. 15(a). The "bubble" in Fig. 15(a) is the result of a twofold intersection of the line (c) with the PD curve (1,2). Another difference between lines (b) and (c) is the cut of the SN curve (1,2) emanating from the PD curve (1,2) at about (v,P•) = (578 kHz, 136 kPa). At this point the PD bifurcation changes from subcritical to super- critical. t

Cut (d) corresponds to Fig. 15 (b). The "bubble" oc- curring in Fig. 15(a) has now burst open since line (d) is located above the local maximum of the bifurcation curve

3.24

2.75 -

2.26 -

Rp

1.28 -

0.79 -

0.30

3O

600 kHz Rn = 10 k6rn

12'5 220 315 410 505 600 Ps

FIG. 20. A Pt-bifurcation diagram showing, among other things, the peri- od-2.2" cascade of the resonance Rt.2 and the period-3.2" cascade ofRh3. To compute this diagram, Ps has only been increased.

30O

200 -

Ps [kPa]

100 -

(f)

(d)

(b)

(-) 0 i i

0 300 600 900

V [kHz]

FIG. 21. Phase diagram of the bubble model ( 1 ) for R,= 10/•m. The solid and dashed curves give the locations of PD and SN bifurcations, respective- ly. The curves are labeled as (n,m) by the torsion number n and the period number m. The different bifurcation scenarios associated with the cuts (a)- (f) through the resonance horns ( 1,1 ) and (1,2) are discussed in the text. The dots indicate those bifurcations that occur in Figs. 6 (a), 9, and 15.

(1,2). In the case of the resonances R9,2, R7,2, and R•,a, this situation already occurs for Pt about 85 kPa as can be seen in Fig. 11. Line (e) additionally intersects the two PD bifurca- tion curves (3,4) and (5,8), giving the transition to period-4 and period-8 oscillations shown in Fig. 15 (c). The other two PD curves (1,4) and (3,8) belong to the right branch of the resonance R1,2 (compare Fig. 15). Cut (f) corresponds to the bifurcation diagram in Fig. 15 (f). In Fig. 22, the first five period- 1 and the first four period-2 resonance horns are plot- ted to give an impression of the superstructure of the bifurca- tion set. The pattern of recurring resonance horns shown in the diagram is continued by the bifurcation curves belonging

600

(a)

lOO

Ps [k•]

10.

20 100 350 • [•]

2000

I000-_

Ps [kl•]

100:

20 80

• ...., ,... (b)

ß ........ ........ -,', 100 1000

V [kHz]

FIG. 22. Phase diagrams of the bubble model ( 1 ) for R,, = 10/•m. (a) Saddle-node bifurcation curves for the first five harmonic resonances of pe- riod 1. (b) PD (solid) and $N (dotted) bifurcation curves for the first four resonances of period 2. The curves with higher torsi(•n numbers are distort- ed copies of the ( 1,1 ) and (1,2) resonances given in Fig. 21, respectively.

1072 d. Acoust. Sec. Am., Vol. 88, No. 2, August 1990 Parlitz et al.: Bifurcation structure 1072

tooo

100 t• ' (13.3) (iO.3) (7.3) (4.3)

50 . . , . . , • , , , , , , , , • 30 100 1000

V [kHz]

FIG. 23. Ph•e diagram of the bubble model ( 1 ) for R. = 10/zm. •ddle- node bifu•ati{m curves with period 3 and Iorsion number up to 14. The period-3 solutions exist in the interior of the enclosed areas. The acute inner areas mark regions of coexisting period-3 solutions brought about by the overturning.of the corresponding period-3 resonances (see also Fig. 24 for a better visualization }.

to the resonances R,., and R2,,.2 with n = 6,7,8 .... (com- pare Fig. 12). A very similar superstructure has been found to be valid for the Toda oscillator TM and a simple laser mod- el.'5 Symmetric systems like Duffing's equation also possess recurring structures in their bifurcation sets, but of a differ- ent type.'3 We conjecture that the bifurcation sets of all driv-

Ps [va•]

7 78 149 220

42O

• •,•) '73 •

240

(1•)

{3fi-

• [•]

Ps

135

80

•5 100 1115 170

71 83 95 107

57 64 71 78

Ual•' • (14,•

48 53 ,68 63

FIG. 24. Saddle-node bifurcation curves of period 3 from Fig. 23, shown enlarged on individual scales. Note the similarities between the shapes in each column.

½4.1) (3,1) (2,1) (1,1) (O,l)

{11,3) (10,3) (8,3) (7,3) (S,3) (4,3) (2,3) (1,3)

FIG. 25. Farey ordering of nonlinear resonances by torsion number and period number.

en nonlinear oscillators possess a superstructure and are, within a few large classes of oscillators, very similar. The fact that the SN bifurcation curves are closed should not be mis-

interpreted as meaning that, for very large Pt, only a single period-1 solution exists, as in the case of very small driving amplitudes. On the contrary, for large P, the bifurcation set is very complicated due to many PD cascades to chaos and coexisting attractors, whose bifurcation curves are not given here. The only additionally coexisting attractors considered here are given in the phase diagrams of period-3 resonances shown in Figs. 23 and 24. The bifurcation curves (1,3), (2,3), (4,3), (5,3), (7,3), (8,3), (10,3), (11,3), { 13,3),and (14,3) give the location in parameter space where period-3 attractors are "born" due to SN bifurcations (compare Fig. 19). Period-3 bifurcation curves whose torsion equals an in- teger multiple of 3 have not been observed. Inside the period- 3 islands, further PD bifurcations take place. The different appearance of the islands is the result of the choice of the v- P, surface of section in the v-P,-R,-... parameter space.

As can be seen from the period- 1, period-2, and period-3 bifurcation curves given in Figs. 22 and 23, the resonant creation of periodic orbits obeys a Farey ordering. Between two given periodic orbits characterized by the tupels (n,,m,) and (n2,m2), another orbit with (n3,m3) = (n, + n2,rn, + ma) exists. Thus, for the onset of the re-

sonanees, the (abstract) scheme shown in Fig. 25 holds. The Farey ordering of periodic orbits is well known from dynam- ical systems where the trajectories lie on an invariant torus in state space. '2 These systems are usually modeled by the sine circle map, and the Farey ordering is valid as long as the torus exists, i.e., below the so-called critical curve in param- eter space. The present system, however, does not have in- variant tori in state space as quasiperiodic orbits do not exist. Indeed, investigations of similar systems indicate a much richer structure than given by the Farey ordering? 3

IV. CONCLUSION

In this article, the dynamics of a single spherical gas bubble of radius R, = 1011m in water that is subjected to a sinusoidally varying sound field is described. Depending on the amplitude and the frequency of the external excitation, the oscillations of the bubble undergo qualitative changes and settle down to periodic oscillations (of different period), or stay fluctuating, yielding chaotic oscillations as a final state. The parameter dependence of the bifurcation points, i.e., the parameter values for which qualitative changes of the dynamics occur (e.g., a change in period) is shown by means of different types of diagrams. The best illustration of the bifurcation set of the system, i.e., the union of all bifurea-

1073 J. Acoust. Soc. Am., Vol. 88, No. 2, August 1990 Parlitz otal: Bifurcation structure 1073

tion points, is given by phase diagrams showing the loca- tion (s) of all points in parameter space where given bifurca- tions take place. Although a certain experience is necessary to read and interpret this kind of diagram, it is, in some sense, the key for a deeper understanding of the bifurcation proper- ties of the bubble model. Phase diagrams show most clearly how the bifurcations one is interested in are organized.

When the bifurcation set is restricted to a parameter plane, e.g., the v-P, plane, the restricted bifurcation set con- sists of a set of curves that enclose areas of qualitatively simi- lar behavior. The curves can be classified by two numbers derived from the period of the oscillation and the torsion of the local flow of the system. Sets of bifurcation curves recur with the nonlinear resonances.

One main building block of the recurring pattern stems from the harmonic resonances R,.,, n = 1,2 ..... where the period is always that of the driving, but the torsion number increases one by one. The bifurcation scenario repeats in shape in the region of each harmonic resonance. Obviously, even global bifurcations leading to a breakup of the "bub- bling" structure recur in a similar way. As the finer details are very complex, the organization behind the development of the scenario cannot yet be given and will be difficult to obtain.

The structure of the bifurcation set seems to be universal

for a large class of driven nonlinear oscillators, among them the Toda oscillator, t4 a simple laser model,'5 the Morse os- cillator, 23 other bubble oscillators, 4-12 and (with some modi- fications due to its symmetric potential) the Duffing oscilla- tor. ]3'24-26 It can be expected that further bubble models incorporating refinements of various kinds will show the same gross structure of the bifurcation set, with just shifted lines in parameter space. It is thus conjectured that the es- sential response of a spherical bubble to a sinusoidal excita- tion has been found.

ACKNOWLEDGMENTS

We thank the members of the Nonlinear Dynamics Group at the Drittes Physikalisches Institut, University of G6ttingen, now at the Institut f'dr Angewandte Physik, Technical University Darmstadt, for many valuable discus- sions, and Ray Glynn Holt for improving the readability of the manuscript. The computation has been done on the Sperry 1100/82, the IBM 3090/200, and the VAX 8650 of the Gesellschaft f'fir Wissenschaftliche Datenverarbeitung, G6ttingen, the CRAY X-MP of the Konrad Zuse-Zentrum ftir Informationstechnik, Berlin, the IBM 3090/200 of the Hochsehulrechenzentrum, Darmstadt, and the Gould Pow- er Node 6040 of the Institut ffir Angewandte Physik, Darm- stadt.

APPENDIX A: EQUIVALENCE OF BUBBLE MODELS

To show the mathematical equivalence of the bubble models (1) and (2) up to terms of order O(c-Z), Eq. (A1) has to be verified:

--(1 +•-) -Ps sin[2rrv(t + R/c)] P

= _(1+•) Pssin(2rrvt) R d [p, sin (2•rvt)] +O(c-2). (A1) pc dt

Relation (A1) is equivalent to Eq. (A2):

(1 +R/c) sin[2rrvt + 2•rv(R /c) l = ( 1 + •/c) sin(2rrvt)

+ (R/c)2•rv cos(2rrvt) + O(c-•), (A2)

which can be proved by substituting the Taylor series expan- sion in Eq. ( A3 ) into the corresponding expression in (A2), resulting in Eq. (A4):

sin [2•rvt + 2•v(R/c) ]

= sin (2rrvt) + 2rrv(R/c) cos (2•rvt) + 0(c-2), (A3)

( 1 + •/c) sin [2rrvt + 2rrv(R/c) ] = ( I + •/c) sin (2rrvt) + 2•rv(R/c) cos (2rrvt)

+ 2rrv(•R/c 2) cos(2rrvt) + O(c-2). (A4) The bubble model (2) and thus (1) can also be derived from the Gilmore model 8'17 by linearizing the enthalpy H around P•t,• at infinity and neglecting terms of order O(c-2).

APPENDIX B: TRANSFORMATION OF THE BUBBLE MODEL INTO A C • -EQUIVALENT SYSTEM

Figure Bl(a) shows a typical periodic bubble oscilla- tion with sharp (downward oriented) collapse peaks. The singular behavior of the bubble during the collapse phase leads to serious numerical problems that can only partly be solved by sophisticated integration algorithms and small time steps during the integration. Therefore, the simulation of bubble oscillations becomes very CPU-time consuming

3.40

2.58 •

0.94 -

0.12

1.0000 1.0030'• !.00604 1.00900 1.01200 1.0151

t

03•o ..... (b)

xi4•0.o4 \ ]' • •, \

0.0

10 -3 * t

FIG. BI. Period-I oscillation of a bubble for R• = 10/tm, P• = 150 kPa, and v = 200 kHz. (a) Radius-time curve for the original system and (b) transformed radius versus transformed time.

1074 J. Acoust. Soc. Am., Vol. 88, No. 2, August 1990 Parlitz otal: Bifurcation structure 1074

and at times even prohibitive. The problem may be overcome by a nonlinear change of the coordinate system to get. smoother oscillations in the new system. In order to preserve all qualitative features of the dynamics, the transformation from the original bubble model S to the equivalent system S' is done by means ofa C © diffeomorphism. A C © diffeomor- phism is an invertible mapf wheref and its inverse f- i are infinitely many times differentiable. A suitable diffeomor- phic transformation of the state coordinates R, U, and O of the bubble model [see Eq. (3) in Sec. I ] into the coordinates x, x:, and x3 of the equivalent system S' is given in Eq. (B 1 ):

x• = a exp(,8R/R• ), x• = yUexp(,8R/R, ), x 3 = O.

(B1)

The inverse map reads as

R R• ln(•), U= a x• ,8 7 x• O = x3. (B2)

The parameters a, ,8, and y of the transformation are to be chosen so that the oscillation in the (x,x2,x 3) coordinates becomes as smooth as possible. Suitable values for a, ,8, and 7' are given below. Besides the transformation of the state- space coordinates, the time t is also rescaled. Let t' be the time in the equivalent system S'. Then, the time scaling used for the simulations is given by Eq. (B3):

t' = rot, (B3) with

Vo = a,8 / rR..

Substitution of the variables R, U, and O in Eq. (3) in Sec. I by the new coordinates x•, x2, and x 3 yields the vector field (B4) of the new equivalent system for k •- 9:

X• : X2,

X• = Xi{y 2 "• [(t03 +p4y)y 2 + (P5 --PAY) z-4

-- (p? + pay)z - • -- (1 + P9Y) (P•o + P2 sin x3)

--p•zcosx3]/[ (1 --poY)Z + P•2]}, (B4) x;

with

Y = x__•2, z = 1 In x..•, x• ,8 a

where

101=2/r¾, t02 = Ps 3 Yo tS'-•p' P3 -- .2,8'

g 1 (p 2a•, 3• P4=•' P5=p• sta, --P• +•) P6 = P5' c

2a 4• • = , P8=•, P9=•, P• 82•pR, 8•pR, c

P• -- P• 2•vP•R, 4• Pro= p62• ' P•= •pc ' P•=pcR,

and

• = a/y.

The prime at x•, x•, and x• denotes differentimion with re- spect to t '.

Dynamical systems [ or vector fields ( 3 ) and (B4) ] that are related by a C © diffeomorphism and a reparametriza- tion of time are called C © equivalent because they possess the same qualitative properties. •

Figure B1 (b) shows the oscillation given in Fig. B1 (a) in the coordinates of the new system $'. This smooth radius- time curve has been computed with Eq. (B4) without any numerical problems during the collapse phase. Therefore, all simulations of bubble oscillations presented in this paper and Refs. 10 and 12 have been carried out for the equivalent systems S '. The results are then translated back by means of the inverse of the diffeomorphism and are expressed in terms of the original variables R and U. This technique reduces the CPU-time consumption. For the results given in Sees. I and II, the parameters chosen for the transformation are a = 1, ,8 = 2, and y = 0.001. The phase diagrams have been com- puted with a = 0.01, ,8 = 2, and y = 0.001. These values have been determined empirically by a number of tests (with moderate amplitudes of the oscillation). A more systematic approach for searching optimal values of a, ,8, and 7 (or alternative transformations) is an open problem for future research.

APPENDIX C: THE JACOBI MATRIX OF THE VECTOR FIELD

For investigating the stability or other properties of an oscillation, one is often interested in the time evolution of an arbitrarily small perturbation that is given by the variational equations of the system (see Sec. I or Ref. 18). In the follow- ing, we give the variational equations of the equivalent sys- tem S', which are used in this paper to compute torsion and winding numbers as well as Lyapunov exponents of bubble oscillations. In the case of the Lyapunov exponents, the re- sults obtained with Eq. (B4) are multiplied by the time scal- ing factor v o [see Eq. (B3) ] to determine the values of the Lyapunov exponents with respect to the time t of the original bubble equations. To simplify the notation, we use the fol- lowing abbreviations for the vector field (B4) of the equiva- lent system (C1):

jC• = X2, Jc2=yx2+A(x•,x2,x3)/B(x•,x2,x3) • (C1) with

A = X I [ (P3 + P4 y)y2 + (P5 -- P6 Y) z-4

- (p7+psy)z -•- (1 +poy)C--zD], B= (1 --poy)z+p,2,

C =P,o +P: sin x•,

D = Pl I COS

where

__ 1 y=X2, z=__ln x•. Xl ,8

The third equation 5% = P l is omitted in this context since its inclusion only leads to a trivial extension of the system of variational equations [compare Eqs. (4)-(7) in Sec. I and Ref. 18 for details ]. Therefore, only the partial derivatives of the vector field (C 1 ) are considered in the following that are

1075 J. Acoust. Sec. Am., Vol. 88, No. 2, August 1990 Parlitz ot at: Bifurcation structure 1075

necessary to formulate the reduced, i.e., two dimensional, set of variational equations.

These partial derivatives of the (reduced) vector field (el) are

05ct = O, 05c, = 1, 8x• •x 2

O• e____ __y•+ Al B-- 8x• B: '

with

Ox,

A• = 0A = (2p3 + 3p4y)y __p•-4 __ P• __P9C ' Jxe z

P9

B, •Xl -- +7 x,, 8B z

Be -- -- -- P9 l•x 2 x !

(C2)

,I

Dv,(x) 0

= ( _yZ + (A•B -- B•A)/B e 1 ) 2y+ (A2B-- Bv4)/B • '

where A and B as well as the partial derivatives A ,, Ae, Bj, andB e are given by Eqs. (CI) and {C2) [or (C3}].

APPENDIX D: COMPUTATION OF BIFURCATION .CURVES

The bifurcation curves shown in See. III are computed by solving the three-dimensional fixed-point equations (D1):

fF•(u) = 0, =0 ,v

LO.(.) =0, (D1)

where u, with n--• (Ul,U2,tks,U4)ER +xRXR>øXR >ø, ---- x,p, u e = Xiv, ua = P,, and u 4 = v. Here, x•v and x2v are

the coordinates of the Poincar6 cross section of the trans-

formed system (CI). Thus F = (FvF •) is the fixed-point condition function for a periodic point in the Poincar6 cross section:

F(U) = (U•,Ue) -- Pt'")(u•,u2)

[Fl(ll ) = It I -- = Ue

where P •"') denotes the/th component of the m-fold Poin- ear6 map.

Here, G.• denotes the characteristic polynomial X(/•,u) = det [pld -- D P(")(u•,u e) ] of the linearized Poin-

The expressions for the partial derivatives may be further simplified by introducing a new set of normalized param- eters:

A• = -- (q• + qzp)y • + [q•- (qn- q•)/z]-4

-- [% -- (q, + qsy)/z]/z -- C -- (z + qo)D,

A2 = (qlo q- q•Y)Y -- ql2 2-n -- q!3 2-1 -- q14 C,

B• = [ (qv•z -• q,•)y + qo]/x•, B e = _ qln(Z/Xl),

with

qi=pz, qe=2p4, qa=pz, _ 4p• p; ps

qs--T' q•=P" q'=7' q*=•' qo=l/•, q•o=2p> q,•=3pn, q•==P6,

q•=P8, q•=P*, q•=P•.

(C3)

Thus the Jacobian Dr, (x) of the reduced variational equa- tions reads as

(CA)

I

car6 map D pt-o (u 1,Ue) for fixed eigenvaluesp = pu = + 1 or]J b = -- 1:

G.•(u) =X(P =P• = • 1,u)

Ou,

- 8u• (u,,uO• (u,,uO. Ch•sing•o = + 1 lmds to •ddle-n•e bifu•tion •ints, and • = -- I givm •fi•-doubling bifur•tion •ints.

The valu• of the line• Poincar• map

DPt•)(u•,u2),

DP•(u•,uz) = (u,,u•), i,j= 1,2,

e• • obtain• by solving simultan•usly •. (C 1 ) with the initial •ndition x(0) = (u•,ue) tog•h• with the r•u• two-dimensional matrix v•ational •uation (D2):

•,(t)=•,(x(t)}Y,(t), Y,(t=0) = I• (D2) where Y, is a 2X2-matfix •d •e/a•bian •,{x(t)) is given in •. (•). The •lution Y, (t) of (D2) at fix• tim• t = roT, where m •uals the • of the fix• •int and Tis the forcing •fi•, yields the four v•u• of 8P•m•/•i, i,j = L2 (Ref. 25).

1076 J. Acoust. Soc. Am., Vol. 88, No. 2, August 1990 Parlitz et at: Bifurcation structure 1076

The system (D1) was solved by a Newton-like meth- od, :7 which computes the Jacobian of the fixed-point equa- tions (D1) by a difference method, so that the Jacobian of (D 1 ) need not explicitly be given.

Upon solving the system (D1), one of the four fixed- point coordinates (u,,u2, u3,u 4) = (x,p,x2p,P,,v) has to be fixed. This leads to just one bifurcation point in the P,-v parameter plane. The bifureation curves are obtained by starting at an arbitrary initial bifurcation point and then tracing the curve by varying one of the four fixed-point co- ordinates, using a continuation algorithm with linear predic- tion. 28

•J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Sys- tems, and Bifurcations of Vector Fields (Springer, Berlin, 1983).

2H. G. Schuster, Deterministic Chaos (Physik, Weinheim, 1984), 2nd ed. (1987).

3W. Lauterborn, "Subharmonic oscillations of gas bubbles in water," Acustica 22, 238-239 (1969/70) (in German).

4W. Lauterborn, "Numerical investigation of nonlinear oscillations of gas bubbles in liquids," J. Aeoust. Soc. Am. 59, 283-293 (1976).

•E. Cramer, in Cavitation and Inhornogeneities in Underwater Acoustics, edited by W. Lauterborn (Springer, Berlin, 1980), pp. 54-63.

6E. Cramer and W. Lauterborn, "On the dynamics and acoustic emission of spherical cavitation bubbles in a sound field," Acustica 49, 226-238 (1981) (in German).

?W. Lauterborn, "Cavitation bubble dynamics--new tools for an intricate problem," Appl. Sei. Res. 38, 165-178 (1982).

xW. Lauterboru and E. Suchla, "Bifurcation superstructure in a model of acoustic turbulence," Phys. Rev. Lett. 53, 2304-2307 (1984).

9U. Parlitz and W. Lauterborn, "Periodic and chaotic bubble oscillations," in Proceedings of the 12th International Congress on Acoustics (ICA12), Toronto, 1986, edited by T. F. W. Embleton, G. A. Daigle, M. R. Stinson, and A. C. C. Warhock (Beauregard, Canada, 1986), Vol. Ill, paper 14-7.

•øW. Lauterborn and U. Parlitz, "On the bifurcation structure of bubble

oscillators," in Proceedings of the Xllth International Symposium on Non- linear Acoustics, edited. by V. Kedrinskii (Institute of Hydrodynamics, Novosibirsk, 1987), pp. 71-80.

•'P. Smereka, B. Birnir, and S. Banerjee, "Regular and chaotic bubble oscil-

lations in periodically driven pressure fields," Phys. Fluids 30, 3342-3350 (1987).

•:W. Lauterborn and U. Parlitz, "Methods of chaos phs'sics and their cation to acoustics," J. Acoust. Soe. Am. 84, 1975-1993 (1988).

•3U. Parlitz and W. Lauterborn, "Superstructure in the bifurcation set of the Du0ting equation • + d-• + x + x 3 =leos(cot)," Phys. Lett. 107A, 351-355 (1985).

14T. Kurz and W. Lauterborn, "Bifurcation structure of the Toda oscilla- tor," Phys. Rev. A 37, 1029-1031 (1988).

•sW. Lauterborn and R. Steinhoff, "Bifurcation structure of a laser with pump modulation," J. Opt. Soc. Am. B 5, 1097-1104 (1988).

•6j. B. Keller and M. Miksis, "Bubble oscillations of large amplitude," J. Acoust. Soc. Am. 68, 628-633 (1980).

•A. Prosperetti, "Bubble phenomena in sound fields: part one," Ultrason- its 22, 69 (1984).

•U. Parlitz and W. Lauterborn, "Resonances and torsion numbers of driv- en dissipative nonlinear oscillators," Z. Natufforsch. 41a, 605-614 (1986).

•gU. Parlitz and W. Lauterborn, "Period-doubling cascades and devil's staircases of the driven van der Pol oscillator," Phys. Rev. A 36, 1428- 1434 (1987).

:øD. Ruelie, "Rotation numbers for diffeomorphisms and flows," Ann. Inst. Henri Poincar6 42, 109-115 (1985). •P. Beiersdorfer, "Universality of the topology of period doubling dynami- cal systems," Phys. Lett. 100A, 379-382 (1984).

::T. Uezu, "Topology in dynamical systems," Phys. Lett. 93A, 161-166 (1983).

2aW. Knop and W. Lauterborn, "Bifurcation structure of the classical Morse oscillator," J. Chem. Phys. (1990) (to appear).

•4S. Sato, M. Sano, and Y. Sawada, "Universal scaling property in bifurca- tion structure of Du•ng's and generalized Dufiing's equations," Phys. Rev. A 28, 1654-1658 (1983).

2sH. Kawakami, "Bifurcation of periodic responses in forced dynamic non- linear circuits: Computation of bifurcation values of the system param- eters," IEEE Trans. Circuits Syst. CAS-31,248-260 (1984).

_,6j. M. T. Thompson, "Chaotic phenomena triggering the escape from a potential well," Proc. R. Soc. London A 421, 195-225 (1989).

:?M. J. D. Powell, "A hybrid method for nonlinear equations," in Numeri- cal Methods for Nonlinear Algebraic Equations, edited by P. Rabinowitz (Gordon and Breach, New York, 1970), pp. 87-114.

:•R. Seydel, "A continuation algorithm with step control," International Series of Numerical Mathematics ( Birkhiiuser, Basel, 1984), Vol. 70, pp. 480-494.

1077 J. Acoust. Soc. Am., Vol. 88, No. 2, August 1990 Parlitz et aL: Bifurcation structure 1077