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List of Symbols

For each item the major uses throughout the book are given first. Important special uses within a chapter or a section are indicated by the chapter or section number in parentheses. Minor uses are not noted. Scalars appear in italic type; vectors, tensors, and matrices appear in boldface type.

A

Ad(z,i)

Ag(j,h)

BsA. C C" C" Cor C(z,i) d d(A,A') deg(C) dt'n dh

d(j)(SN) d(SN) d(z,z')

coefficient matrix of a linear system; basic absorption proba­bility matrix (=:=TN) (10.3); (with various superscripts and subscripts attached) various special absorption probability matrices (10.3) and (10.4) domicile absorption probability of cell z into its ith domicile (11.10) group absorption probability of transient cellj into persistent group Bh (11.10) covering set oCa strange attractor determined by GCM (13.2) simple cell mapping k-cube (8.1) cell mapping C applied k times core of singularities (5) ith image cell of z (11.1) period of a periodic persistent group (10) distance between cell set A and cell set A' degree of a map C (2.8) jth basic F-determinant (8.1) period of the hth persistent group (10.4) determinant of .(j)(x) (5.3) determinant of.+ (x) (5.3) distance between cell z and cell Z

List of Symbols

Db(z,z') DG Dm(z) DmG(z, i) DSA

ej

E(A, tx), E(tx) Epc

J;j

f\n) JiJ

FL

F(x, t, p) F(z, C) F(z, Ck )

G G(z)

Gr(z) hi iz

int(x)

I(P,F) I(S,F) I(z) I(z*;F) J(z) k

K(g) Kr(h) jji

m' N Nc Nci Ndm

N(g) N(g, i)

343

determinant of matrix A domain of attraction of the kth periodic solution of SCM (11.3) cell doublet of z and z' Jacobian matrix of mapping G number of domiciles of cell z (11.1) persistent group which is the ith domicile of cell z (11.1) covering set of a strange attractor (13.1) jth unit cell vector in ZN extended attractor (7.3) set of periodic cells of a SCM (11.3) probability of being in cell i at least once, starting from cell j (10.3) probability of being in cell i at the nth step for the first time, starting from cell j (10.3) affine function: X N -+ yN (SA) vector field of a differential system cell mapping increment function (4.2) k-step cell mapping increment function (4.2) point mapping group number of the persistent group to which the cell z belongs (11.1) group number of cell z in SCM algorithm (8.2) cell size in the Xi direction inclusion function of ZN into X N (5) largest integer, positive or negative, which is less than or equal to x index of a singular point P with respect to vector field F (2.8) index of surface S with respect to vector field F (2.8) number of image cells of z (11.1) index of singular cell z* with respect to cell function F total number of pre-images of cell z (11.2) spring modulus or system parameter (3.3); number of persis­tent groups (10) period of the gth persistent group (11.1) period of the hth periodic solution of SCM (11.3) central moment (13) cell r-multiplet (SA) a matrix (10.3) number of cells in the cell set of interest number of cells in the Xi direction number of multiple-domicile cells (lOA) member number of the gth persistent group (11.1) number of cells in the ith subgroup of the gth persistent group (11.6)

344

Nh,r(h)

Np NpIJ

Nps Nsd N.g

Nt Nt" {N} {N+} pj(n) pjj Pl'!)

IJ

p(n)

p

P-dPG P(z) P(z,i) Q

~N

R(z, i) sr

SN

S St(z) t, tj

t*, tt

T

'Jr(z, z', zIt) U(A,L), UL

v;, V(i)

9

List of Symbols

number of cells in the r(h)th subgroup of the hth persistent group number of cells in subgroup Bj (10.3); number of cells in the xj-direction number of persistent cells (10.3) number of discovered persistent groups (11.1) total number of periodic solutions in a SCM (11.3) number of the single-domicile cells (10.4) total number of persistent subgroups (10.4) number of transient cells (10.3) number oftransient groups (11.1) set of positive integers from 1 to N set of nonnegative integers from 0 to N probability of the state being in cell i at step n (10.1) transition probability of cellj mapped into cell i probability of being in cell i after n steps, starting from cellj (10.3) cell probability vector at step n (10); (with various super­scripts attached) various special cell probability vectors (10.4) transition probability matrix (10.3); (with various super­scripts and subscripts attached) various special transition probability matrices (10.3) and (10.4) a persistent group of period d (11.3) periodicity number of cell z in SCM algorithm (8.2) transition probability from cell z to its ith image cell (11.1) block transition probability matrix (10.3); (with various superscripts and subscripts attached) various special block transition probability matrices (10.3) and (10.4) N-dimensional Euclidean space ith pre-image of cell z (11.2) r-simplex in X N (5.3)

closure of SN

set of cells; hypersurface; cell singular entity (7.3) step number of cell z in the SCM algorithm (8.2) barycentric coordinate vector and its ith component (5.3) and (5.4) barycentric coordinate vector for the zero x* of the affine function FL and its ith component (5.4) block transit probability transition matrix (10.3); (with vari­ous superscripts and subscripts attached) various special transit probability matrices (10.3) and (10.4) cell triplet of z, z' and zIt L-neighborhood of a set A (7.2) gth basin of attraction of the zeroth level (12.1) gth basin of attraction ofthe ith level (12.1)

List of Symbols

v" Vo v,(i) o W· WU

x, x(t) x(d)(n)

Xi

Xj

Xij X+

X*, x*(i) XN

Z, z(n) Zi

7L ZN 7L+ OC

ocd(z, i)

OCij

r

r(P) t5( . ) .1 A Ai A(A,L), AL A(P)

vertex set in XN

boundary set of the zeroth level (12.1) boundary set of the ith level (12.1) stable manifolds unstable manifolds state vector, dimension N center point of cell z(n) ith state variable, ith component of x jth element of a set ith component of point Xj

augmented x-vector fixed point, ith periodic point of a periodic solution state space, dimension N state cell or cell vector, dimension N ith cell state variable, ith component of z the set of integers cell state space, dimension N the set of nonnegative integers system parameter (3)

345

domicile absorption probability of cell z into its ith domicile (11.1 ) absorption probability from transient cellj to persistent cell i (to.3); (with various superscripts attached) components of various special absorption probability matrices (10.4) (i,j)th components of r and expected absorption time of transient cellj into persistent cell i (10.4) expected absorption time matrix (10.4); (with various super­scripts attached) various special expected absorption time matrices (10.4) convex hull of a set of point P (5.3) Dirac's delta function defining symbol for a simplex (5.3) or a cell multiplet (5.4) eigenvalue or system parameter (3.3) eigenvalue Lth layer surrounding set A hyperplane of the smallest dimension containing a set of points P (5.3) parameter vector of a system J1 value at a bifurcation point system parameter; mean values in the Xi direction (12.3) J1. value at the period doubling accumulation point conditional expected absorption time of transient cell j into persistent cell i expected absorption time of transient cellj (10.3) largest Liapunov exponent (13.4)

346

r fIl fils fIl: $(x) $+

fIl+(x) $(J1(x)

W, wo, w" !l(Z) aS N

{ ... y o

List of Symbols

Liapunov exponents (2.7); standard deviation in the Xi

direction (12.3) N-simplex in yN (5.4) hypersurface (2.8) a time interval, period of excitation fundamental matrix (3); assembling matrix (10.4) assembling matrix (10.4) augmented assembling matrix (10.4) a matrix array of a set of vectors (5.3) augmented assembling matrix (10.4) augmented fIl(x) (5.3) fIl(x) withjth column deleted (5.3) circular frequencies (3.3) limit set of cell z (7) boundary of SN

complement of a set empty set

Index

Primary references are italicized.

Absorbing cell, 13, 252 Absorption into

persistent cells, 222-224 (Section 10.4.1)

persistent groups, 224-227 (Section 10.4.2)

persistent subgroups, 227-230 Section 10.4.3)

Absorption probability, 13, 220, 263-264

Absorption probability matrix basic, 220 domicile, 247 group, 221 subgroup, 230

Acyclic group, 13,216 Affine functions, 109-110 Algorithms for generalized cell mapping,

13,244-269 (Chapter 11) Algorithms for simple cell mapping, 10,

139-152 (Chapter 8) for global analysis, 10,146-151

(Section 8.2) for locating singular multiplets, 10,

139 -146 (Section 8.1) Area contracting mapping, 53

Associated vector field, 8, 120 (Section 6.1)

Attracting layer range, 134 Attractor, 12

extended, 134 of fractal dimension, 12 invariant, 135, 169 positively invariant, 135

Autonomous point mapping, 19 Autonomous systems, 17, 154 Averaging method

state space, 15,306 time, 15, 306

Barycentric coordinates, 10, 107-108 (Section 5.3.2)

Basin of attraction, 282 of the first level, 282 of the zeroth level, 282

Bifurcation in point mapping, 3, 21-25, 36-37

period doubling, 26 period doubling accumulation point,

26 P-K solution into P-K solutions, 22-23

348

Bifurcation in point mapping (cont.) P-K solution into P-MK solutions

23-25 Bifurcation in simple cell mapping, 90-

91 Birth of new periodic solutions, 25,

36-37 Boundary of basins of attraction, 12

of the first level, 282 fractal, 12 refinement of, 281 of the zeroth level, 282

Cell dynamical system, 334 Cell function, 87 Cell mapping increment function, 7, 89

k-step, 7,89 Cell multiplets; see Multiplets Cell probability vector; see Probability

vector Cells, 2, 85

regular, 87, 147 full-,99 sub-,99

singular, 87, 102 core of, 88 first order, 98 second order, 98 solitary, 87

Cell state space, cell space, 6, 85-88 (Section 4.1)

first level refined, 279 zeroth level, 278

Cells under processing, 148,253 Cell-to-cell mapping, 2 Cell vectors, dimension N, 86

adjoining, 86 contiguous, 86 linearly independent, 86

Center point method, 153-154 (Section 9.1)

Central correlation function matrix, 310 Central moments, 310 Chaotic motion, 6, 42-43, 306 Chaotic rounding error, 333 Characteristics of cell singularities, 9,

127-138 (Chapter 7) bounded,9,129,131

homebounded, 9, 130, 133 invariant, 9, 130, 133 positively invariant, 9, 130, 133 rooted, 9, 129-130, 132 unbounded,9, 128-129, 131

Index

Compactification, 155-157 (Section 9.2) Compatible simple and generalized

mappings, 14,248-251 (Section 11.3)

Computing time, 170, 191-192,206,313 Conditional expected absorption time,

223 domicile, 247 group, 226

Cores of periodic cells, 89 of P-K cells, 89 of singular cells, 88 of singular multiplets, 123

Degree of a map, 4, 43-44 Digital control systems, 15,333 (Section

14.3) Discrete optimal control, 15, 334 Discrete trajectory, 3, 19

even branch of, 19 Jth branch ofa Kth order, 19 Kth order, 19 odd branch of, 19

Distance between two cells, 86 Domain of attraction, 3

for point mapping, 38-42 (Section 2.6) for simple cell mapping, 91 (Section

4.4) Domicile, 230, 264-266

multiple-domicile, 230 single-domicile, 230

Domicile number, 246 Dominant SCM, 251 Duffing systems under forcing, 289-299

(Section 12.5) involving strange attractors, 299-304

(Section 12.6), 321-323 (Section 13.3),328

Dynamical equivalence, 155

Entropy of a mapping, 15

Index

metric, 15 topological, 15

Equilibrium cell, 88 Equilibrium state, 3 Examined cell, 258 Expected absorption time, 13, 220, 263-

264 group, 226

Expected return time, 214

First image track, 255 Fixed point, 19 Fractals, 12,207 Fundamental matrix, 49

GCM,247 first level refined, 280 zeroth level, 278

Generalized cell mapping, 2, 208-233 (Chapter 10)

stationary, 209 Generating map, 15 Global domain of attraction; see

Domain of attraction Group

acyclic; see Acyclic group periodic; see Periodic group persistent; see Persistent group transient; see Transient group

Group number, 148,245

Henon-Pomeau map, 313-318,328 Hinged bar with oscillating support

by point mapping, 5, 68-77 (Section 3.4)

by simple cell mapping, 11, 171-177 (Section 9.6)

Hinged bar under periodic impact load

elastically restrained, 66-68 (Section 3.3.4)

by GCM, 318-321 (Section 13.3.3) by point mapping, 5, 51-68 (Section

3.3) by simple cell mapping, 11, 167-171

(Section 9.5)

349

two coupled bars, 5, 77-84 (Section 3.5)

Homoclinic point, 40 Hypersurface section map, 17

Impulsive parametric excitation, 4, 48-84 (Chapter 3)

oflinear systems, 49-51 (Section 3.2) Inclusion functions, 98, 109 Increments

backward, 87 forward, 87 second order, 87

Index, Poincare's theory of global results, 45, 47, 123-124 of a Jordan cell surface, 121 for N-dimensional vector field, 43-

45 (Section 2.8) of a periodic point, 46-47 for point mappings, 3, 46-47 (Section

2.9) for simple cell mapping, 8, 120-126

(Chapter 6) of a singular multiplet, 122-124

(Section 6.3) of a singular point, 45 of a surface, 44, 46-47 for two-dimensional systems, 4, 43

In-phase mode, 78, 196 Invariant extension, 135, 165 Invariantly connected, 128 Invariant set, 19, 128 Iterative method, 277-285 (Chapter 12)

Jacobian matrix, 20 Jordan cell surface, 121

admissible, 121 Jordan hypersurface, Jordan surface,

8,43,121 admissible 8, 121

Jth basic F-determinant, 144

Liapunov exponents, 3, 42 the largest, 42, 323-326 (Section 13.4)

Liapunov function, 181 for simple cell mapping, 332

350

Liapunov's direct method, 177 Limit cell, 128 Limit cycles, 164, 195-203 Limiting probability, 13,216,260-262

(Section 11.7),308-309 Limit point, 19 Limit set, 19, 128 Linear oscillator under harmonic

forcing, 286-287 (Section 12.3) Logistic map, 25-28 (Section 2.4.1)

by generalized cell mapping, 210-211, 234-242,269-271

with a small flat top, 30-32 Looping cell, 255 Lth layer, 131

Manifolds stable, 38, 53 unstable, 38, 53

Markov chains, 11,211-221 (Section 10.3)

Means, 286, 309 Member number, 246 Method of backward expansion, 40, 63 Minimum jump variation, 90 Mixed mapping systems, 15,333 Mixed mode P-2 solutions, 82 Mixed state space, 333 Multiplets, r-multiplets, 8, 108-109

(Section 5.4.1) cores of singular, 9 degenerate, 8, 113 (Section 5.4.4) isolable singular, 9, 122 nondegenerate, 8, 111-113 (Section

5.4.3) nonisolable singular, 123 regular, 8, 111-113 semisingular, 113 singular, 8, 111-113, 143-146

Negatively invariant see, 19 Neighborhood mapping properties,

130-135 (Section 7.2) Neighborhood of a cell set, 130-135

One-dimensional cell functions, 98-100 (Section 5.1)

Index

One-dimensional point maps, 25-32 (Section 2.4)

Orientation of gradient vectors, 23, 36-37

Out-of-phase mode, 78, 196

Periodic cells, 89, 216 advancing-type, 169

Periodic group, 13,216,217-219 (Section 10.3.2)

Periodicity number, 148, 246 Periodic motions, periodic solutions

advancing-type, 58-59 enduring, 249 nonenduring, 250 for point mappings, 3, 19 for simple cell mapping, 88-89

(Section 4.2) Periodic point, 20

asymptotically stable 20 stable, 20 unstable, 20

Periodic system, 18, 154 Period of periodic group, 213, 257-

260 (Section 11.6) Period T map, 18 Persistent cell, 13,213-214

candidate, 253 Persistent group, 13,214,252-257

(Section 11.5) enlarged, 233 irreducible, 215, 245 refinement of, 280 zeroth level, 279

Persistent subgroup, 217-219 PG,247 P-K cells, 89 P-K point, 20 P-K solution, 20 Poincare map; see Point mapping Point mapping, 3, 16-47 (Chapter 2) Population dynamics problem; see

Logistic map Positively invariant set, 19, 128 Pre-image array, 247-248 (Section 11.2) Probability vector, 12,209

grouped, 225 subgrouped, 228

Processed cells, 149,254

Index

Processing sequence, 149,254 Pseudo-orbits, 3

Random vibration analysis, 15,331-332 (Section 14.1)

Recursive cell, 128

Sampled data systems, 21 Sampling method, 13,268-269

interior, 278-279 interior-and-boundary, 283-285

(Section 12.2) SCM,247

first level refined, 279 zeroth level, 278

Second order point mapping, 32-37 (Section 2.5)

domain of attraction, 38-42 (Section 2.6)

Selected SCM, 251 Sensitivity to initial conditions, 40, 69 Sets of cells

adjoining, 86 nonadjoining, 86

Simple cell mapping, 2, 85-97 (Chapter 4)

stationary, 7 Simple point mapping example

by generalized cell mapping, 271-276, 326

by point mapping, 38-42 (Section 2.6)

by simple cell mapping, 157-164 (Section 9.3)

Simplex, N-simplex, 10, 106-108 (Section 5.3)

elementary, 108 (Section 5.3.3) Simplicial mapping, 109-111 Single mode P-2 solutions, 79 Singular cell; see Cells, singular Singular doublet, 99, 102 Singularities of a cell function, 98-119

(Chapter 5) Singular points of second order point

mapping, 33-37 (Section 2.5.1) center, 34 of the first kind, 33-34 node, 33-34

for nonlinear mapping, 35-37 (Section 2.5.2)

saddle point, 33 ofthe second kind, 33-34 spiral point, 33

Singular points of vector fields, 43 nondegenerate, 44

Singular square, 102 Singular triplet, 102

351

Singular 1-multiplets; see Cells, singular Singular 2-multiplet; see Singular

doublet Sink cell, 10, 147 Stability of periodic solutions, 3 Standard deviations, 286 State cell; see Cells Step number, 148 Stochastic coefficients, 14 Stochastic excitation, 14 Strange attractor, 3,307-330 (Chapter

13) covering set of cells by GCM, 308 covering set of cells for, 307

Stretch-contraction-reposition map, 310-313 (Section 13.3.1),326

Subharmonic response, 58, 280, 299 Substochastic matrix, 215 Symmetrical tent map, 28-32 (Section

2.4.2) with a small flat top, 29-30

Synchronous generators, 11, 177-194 (Section 9.7)

computer algorithm, 183-185 short-circuit fault clearing time, 193-

194 (Section 9.7.4)

Theory of index; see Index Transient cell, 13,213-214 Transient group, 13,214,262-263

(Section 11.8) Transition matrix; see Transition

probability matrix Transition probability, 12,209

n-step, 212 Transition probability matrix, 12, 209

group, 226 normal form of, 215, 230 subgrouped, 229 without mixing, 223

352

Transit matrix, 215 group transit matrix, 224 subgroup transit matrix, 228

Triangulation into simplexes, 114, 117, 120,139-143 (Section 8.1.1)

Two-dimensional cell functions, 100-105 (Section 5.2)

singular entities, 102-105 (Section 5.2.2)

Two-dimensional linear simple cell mappings, 93-97 (Section 4.6)

Unassigned cell, 258 Unexamined cell, 258 Unit cell vectors, 86 Unit vector field, 323

average unit flow vector, 324

Index

van der Pol oscillators, 11 by generalized cell mapping, 287-289,

326 by simple cell mapping, 164-167

(Section 9.4) two coupled oscillators, 11,195-

207 (Section 9.8) Vector triplets, 100-101

fan, 100 star, 100

Virgin cell, 148,253

Wordlength,333

Zaslavskii map, 53-54, 318-321 Zeros of a vector function, 9

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