TheCAPDlibraryanditsapplications · Outline: 1 Short history of the CAPD 2 Introduction to CAPD by...

The CAPD library and its applications

Daniel Wilczak

Institute of Computer Science and Computational MathematicsJagiellonian University, Krakow, Poland

June 11, 2014

Outline:

1 Short history of the CAPD

2 Introduction to CAPD by short examples3 Applications

periodic orbits for the Lorenz systemchaotic dynamics in the Rossler systemcontinuation of stable elliptic periodic orbits

http://capd.ii.uj.edu.pl

Computer Assisted Proofs in Dynamics

1992 Marian MrozekFirst version of rigorous ODE solver.

1995 M.Mrozek, K.MischaikowEncyclopedia Britanica:paper on chaos in the Lorenz equations amog4 best results in mathematics in 1995

2001CAPD publically available at http://capd.ii.uj.edu.pl

1992-2014continuously developed at the Jagiellonian University

The CAPD 4.0 in 2014:

core CAPD: (Multiprecision) IA, linear algebra (dense)

capdRedHom: (co)-homology software

Pawe l Pilarczyk Pawe l D lotko Mateusz Juda

capdDynSys: validated numerics for dynamical systems

Daniel WilczakPiotr Zgliczynski Tomasz Kapela

The capdDynSys 4.0 in 2014:

C0 − C1 − Cr ODE solvers

Poincare maps and their r−th order derivatives

Differential inclusions

supports: double, long double, multiprecision, interval,mpfr-intervals

Some applications:

C0-computations;chaotic dynamics for many textbook systems, bifurcations for reversible systems

C1-computations;periodic orbits (in quite high dimensions, like 300 for the N-body problem),

hyperbolicity, homoclinic and heteroclinic solutions for ODEs both to equilibria

and periodic solutions

C2-computations;cocoon bifurcations, homoclinic tangencies

C3 − C5 computations;bifurcations of periodic orbits for ODEs, KAM stability of periodic solutions,

invariant tori around periodic orbits

Some applications:

Compilation of the library:

./configure [options]

make -j (takes some time, more than 120 000 lines of code)

Basic options:

--with-mpfr

--with-wx-config (internal graphics kernel)

Building own programs:

g++ main.cpp -o main ‘capd-config --cflags --libs‘

Optional scripts:

‘capd-gui-config --cflags --libs‘

‘mpcapd-config --cflags --libs‘

Basic options:

--with-mpfr

Optional scripts:

Basic options:

--with-mpfr

Optional scripts:

Basic options:

--with-mpfr

Optional scripts:

Main header files

#include "capd/capdlib.h"

#include "capd/mpcapdlib.h" // for multiprecision

#include "capd/krak/krak.h" // for graphics kernel

Defined types:

interval, MpFloat, MpInterval

Algebra:

DVector, LDVector, IVector, MpVector, MpIVector[Prefix]Matrix[Prefix]Hessian[Prefix]Jet

Automatic differentiation:

[Prefix]Map

ODE solvers:

[Prefix]OdeSolver, [Prefix]CnOdeSolver[Prefix]TimeMap, [Prefix]CnTimeMap[Prefix]PoincareMap, [Prefix]CnPoincareMap

Main header files

Defined types:

Algebra:

[Prefix]Map

ODE solvers:

Main header files

Defined types:

Algebra:

[Prefix]Map

ODE solvers:

Main header files

Defined types:

Algebra:

[Prefix]Map

ODE solvers:

Main header files

Defined types:

Algebra:

[Prefix]Map

ODE solvers:

Main header files

Defined types:

Algebra:

[Prefix]Map

ODE solvers:

The Rossler system

x = −y − z

y = x + by

z = b + z(x − a)

Theorem (Zgliczynski 1997)

For a = 5.7 and b = 0.2 the system is Σ2-chaotic.

The Rossler system

x = −y − z

y = x + by

z = b + z(x − a)

Theorem (Zgliczynski 1997)

For a = 5.7 and b = 0.2 the system is Σ2-chaotic.

Topological tool

Theorem (Zgliczynski, 1997)

If a map f : N ∪M → R2 looks like this

M NfHNL

then it is Σ2 chaotic.

any binifinite sequence of symbols N and M is realized by atrajectory of f

each periodic sequence of symbols N and M is realized by aperiodic trajectory of f with the same period

Topological tool

M NfHNL

Topological tool

M NfHNL

Poincare section

Π = {(x , y , z) : x = 0, x = −(y + z) > 0}

Wall time reported in the article:

50 hours (180 000 sec) on 50MHz processor

Poincare section

Π = {(x , y , z) : x = 0, x = −(y + z) > 0}

Wall time reported in the article:

50 hours (180 000 sec) on 50MHz processor

The Lorenz system

x = 10(−x + y)y = 28x − y − xz

z = xy − 83z

Poincare section

Π = {(x , y , z) : z = 27, z < 0}

Theorem (Z. Galias, W. Tucker, 2011)

The Poincare map P : Π → Π has exactly 8798 periodic orbits of

periods 2-16.

The Lorenz system

x = 10(−x + y)y = 28x − y − xz

z = xy − 83z

Poincare section

Π = {(x , y , z) : z = 27, z < 0}

periods 2-16.

The Lorenz system

x = 10(−x + y)y = 28x − y − xz

z = xy − 83z

Poincare section

Π = {(x , y , z) : z = 27, z < 0}

periods 2-16.

Π± = {(x , y , z) : z = 27, z ∈ R±}

P± : Π± 7→ Π∓

Interval Newton Method applied to:

F : R4p ∋

x1x2...

x1 − P−(x2p)x2 − P+(x1)

...x2p − P+(x2p−1)

∈ R4p

Validation of zeros of maps

Interval Newton Operatorf : Rn → R

n – smoothX ⊂ R

n - a convex, compact setx0 ∈ int(X )

N(f ,X , x0) := x0 − conv(Df (X ))−1f (x0)

Theorem

N(f ,X , x0) ⊂ int(X )

then f has a unique zero x∗ ∈ int(X ). Moreover x∗ ∈ N(f ,X , x0).

Toy example: f (x) = x(x2 + 2) + 1Approx zero of f is x∗ ≈ −0.453398.....

x0 = −0.5 f (x0) = −0.125

X = [−1, 0] Df (X ) ⊂ [2, 5]

N(f ,X , x0) ⊂ [−0.475,−0.4375] ⊂ intX = (−1, 0)

n – smoothX ⊂ R

N(f ,X , x0) := x0 − conv(Df (X ))−1f (x0)

Theorem

N(f ,X , x0) ⊂ int(X )

x0 = −0.5 f (x0) = −0.125

X = [−1, 0] Df (X ) ⊂ [2, 5]

N(f ,X , x0) ⊂ [−0.475,−0.4375] ⊂ intX = (−1, 0)

n – smoothX ⊂ R

N(f ,X , x0) := x0 − conv(Df (X ))−1f (x0)

Theorem

N(f ,X , x0) ⊂ int(X )

x0 = −0.5 f (x0) = −0.125

X = [−1, 0] Df (X ) ⊂ [2, 5]

N(f ,X , x0) ⊂ [−0.475,−0.4375] ⊂ intX = (−1, 0)

n – smoothX ⊂ R

N(f ,X , x0) := x0 − conv(Df (X ))−1f (x0)

Theorem

N(f ,X , x0) ⊂ int(X )

x0 = −0.5 f (x0) = −0.125

X = [−1, 0] Df (X ) ⊂ [2, 5]

N(f ,X , x0) ⊂ [−0.475,−0.4375] ⊂ intX = (−1, 0)

n – smoothX ⊂ R

N(f ,X , x0) := x0 − conv(Df (X ))−1f (x0)

Theorem

N(f ,X , x0) ⊂ int(X )

x0 = −0.5 f (x0) = −0.125

X = [−1, 0] Df (X ) ⊂ [2, 5]

N(f ,X , x0) ⊂ [−0.475,−0.4375] ⊂ intX = (−1, 0)

TheCAPDlibraryanditsapplications · Outline: 1 Short history of the CAPD 2 Introduction to CAPD by...

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