The CAPD library and its applications

Transcription

1 The CAPD library and its applications Daniel Wilczak Institute of Computer Science and Computational Mathematics Jagiellonian University, Kraków, Poland June 11, 2014

2 Outline: 1 Short history of the CAPD 2 Introduction to CAPD by short examples 3 Applications periodic orbits for the Lorenz system chaotic dynamics in the Rössler system continuation of stable elliptic periodic orbits

3 Computer Assisted Proofs in Dynamics

4 Marian Mrozek First version of rigorous ODE solver M.Mrozek, K.Mischaikow Encyclopedia Britanica: paper on chaos in the Lorenz equations amog 4 best results in mathematics in CAPD publically available at continuously developed at the Jagiellonian University

5 Marian Mrozek First version of rigorous ODE solver M.Mrozek, K.Mischaikow Encyclopedia Britanica: paper on chaos in the Lorenz equations amog 4 best results in mathematics in CAPD publically available at continuously developed at the Jagiellonian University

6 Marian Mrozek First version of rigorous ODE solver M.Mrozek, K.Mischaikow Encyclopedia Britanica: paper on chaos in the Lorenz equations amog 4 best results in mathematics in CAPD publically available at continuously developed at the Jagiellonian University

7 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 Piotr Zgliczyński Tomasz Kapela Daniel Wilczak

8 The capddynsys 4.0 in 2014: C 0 C 1 C r ODE solvers Poincaré maps and their r th order derivatives Differential inclusions supports: double, long double, multiprecision, interval, mpfr-intervals Some applications: C 0 -computations; chaotic dynamics for many textbook systems, bifurcations for reversible systems C 1 -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 C 2 -computations; cocoon bifurcations, homoclinic tangencies C 3 C 5 computations; bifurcations of periodic orbits for ODEs, KAM stability of periodic solutions, invariant tori around periodic orbits

9 The capddynsys 4.0 in 2014: C 0 C 1 C r ODE solvers Poincaré maps and their r th order derivatives Differential inclusions supports: double, long double, multiprecision, interval, mpfr-intervals Some applications: C 0 -computations; chaotic dynamics for many textbook systems, bifurcations for reversible systems C 1 -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 C 2 -computations; cocoon bifurcations, homoclinic tangencies C 3 C 5 computations; bifurcations of periodic orbits for ODEs, KAM stability of periodic solutions, invariant tori around periodic orbits

10 The capddynsys 4.0 in 2014: C 0 C 1 C r ODE solvers Poincaré maps and their r th order derivatives Differential inclusions supports: double, long double, multiprecision, interval, mpfr-intervals Some applications: C 0 -computations; chaotic dynamics for many textbook systems, bifurcations for reversible systems C 1 -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 C 2 -computations; cocoon bifurcations, homoclinic tangencies C 3 C 5 computations; bifurcations of periodic orbits for ODEs, KAM stability of periodic solutions, invariant tori around periodic orbits

11 The capddynsys 4.0 in 2014: C 0 C 1 C r ODE solvers Poincaré maps and their r th order derivatives Differential inclusions supports: double, long double, multiprecision, interval, mpfr-intervals Some applications: C 0 -computations; chaotic dynamics for many textbook systems, bifurcations for reversible systems C 1 -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 C 2 -computations; cocoon bifurcations, homoclinic tangencies C 3 C 5 computations; bifurcations of periodic orbits for ODEs, KAM stability of periodic solutions, invariant tori around periodic orbits

12 The capddynsys 4.0 in 2014: C 0 C 1 C r ODE solvers Poincaré maps and their r th order derivatives Differential inclusions supports: double, long double, multiprecision, interval, mpfr-intervals Some applications: C 0 -computations; chaotic dynamics for many textbook systems, bifurcations for reversible systems C 1 -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 C 2 -computations; cocoon bifurcations, homoclinic tangencies C 3 C 5 computations; bifurcations of periodic orbits for ODEs, KAM stability of periodic solutions, invariant tori around periodic orbits

13 The capddynsys 4.0 in 2014: C 0 C 1 C r ODE solvers Poincaré maps and their r th order derivatives Differential inclusions supports: double, long double, multiprecision, interval, mpfr-intervals Some applications: C 0 -computations; chaotic dynamics for many textbook systems, bifurcations for reversible systems C 1 -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 C 2 -computations; cocoon bifurcations, homoclinic tangencies C 3 C 5 computations; bifurcations of periodic orbits for ODEs, KAM stability of periodic solutions, invariant tori around periodic orbits

14 Compilation of the library:./configure [options] make -j (takes some time, more than 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

15 Compilation of the library:./configure [options] make -j (takes some time, more than 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

16 Compilation of the library:./configure [options] make -j (takes some time, more than 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

17 Compilation of the library:./configure [options] make -j (takes some time, more than 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

18 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

19 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

20 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

21 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

22 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

23 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

24 The Rössler system ẋ = y z ẏ = x +by ż = b +z(x a) Theorem (Zgliczyński 1997) For a = 5.7 and b = 0.2 the system is Σ 2 -chaotic.

25 The Rössler system ẋ = y z ẏ = x +by ż = b +z(x a) Theorem (Zgliczyński 1997) For a = 5.7 and b = 0.2 the system is Σ 2 -chaotic.

26 Topological tool Theorem (Zgliczyński, 1997) If a map f : N M R 2 looks like this f N f M M N then it is Σ 2 chaotic. any binifinite sequence of symbols N and M is realized by a trajectory of f each periodic sequence of symbols N and M is realized by a periodic trajectory of f with the same period

27 Topological tool Theorem (Zgliczyński, 1997) If a map f : N M R 2 looks like this f N f M M N then it is Σ 2 chaotic. any binifinite sequence of symbols N and M is realized by a trajectory of f each periodic sequence of symbols N and M is realized by a periodic trajectory of f with the same period

28 Topological tool Theorem (Zgliczyński, 1997) If a map f : N M R 2 looks like this f N f M M N then it is Σ 2 chaotic. any binifinite sequence of symbols N and M is realized by a trajectory of f each periodic sequence of symbols N and M is realized by a periodic trajectory of f with the same period

29 Poincaré section Π = {(x,y,z) : x = 0, ẋ = (y +z) > 0} z 0 0 y 5 0 x 5 10 Wall time reported in the article: 50 hours ( sec) on 50MHz processor

30 Poincaré section Π = {(x,y,z) : x = 0, ẋ = (y +z) > 0} z 0 0 y 5 0 x 5 10 Wall time reported in the article: 50 hours ( sec) on 50MHz processor

31 The Lorenz system ẋ = 10( x +y) ẏ = 28x y xz ż = xy 8 3 z Poincaré section Π = {(x,y,z) : z = 27, ż < 0} Theorem (Z. Galias, W. Tucker, 2011) The Poincaré map P: Π Π has exactly 8798 periodic orbits of periods 2-16.

32 The Lorenz system ẋ = 10( x +y) ẏ = 28x y xz ż = xy 8 3 z Poincaré section Π = {(x,y,z) : z = 27, ż < 0} Theorem (Z. Galias, W. Tucker, 2011) The Poincaré map P: Π Π has exactly 8798 periodic orbits of periods 2-16.

33 The Lorenz system ẋ = 10( x +y) ẏ = 28x y xz ż = xy 8 3 z Poincaré section Π = {(x,y,z) : z = 27, ż < 0} Theorem (Z. Galias, W. Tucker, 2011) The Poincaré map P: Π Π has exactly 8798 periodic orbits of periods 2-16.

34 Π ± = {(x,y,z) : z = 27, ż R ± } P ± : Π ± Π Interval Newton Method applied to: x 1 x 1 P (x 2p ) F: R 4p x 2. x 2 P + (x 1 ). R4p x 2p P + (x 2p 1 ) x 2p

35 Validation of zeros of maps Interval Newton Operator f : R n R n smooth X R n - a convex, compact set x 0 int(x) N(f,X,x 0 ) := x 0 conv(df(x)) 1 f(x 0 ) Theorem If N(f,X,x 0 ) int(x) then f has a unique zero x int(x). Moreover x N(f,X,x 0 ). Toy example: f(x) = x(x 2 +2)+1 Approx zero of f is x x 0 = 0.5 f(x 0 ) = X = [ 1,0] Df(X) [2,5] N(f,X,x 0 ) [ 0.475, ] intx = ( 1,0)

36 Validation of zeros of maps Interval Newton Operator f : R n R n smooth X R n - a convex, compact set x 0 int(x) N(f,X,x 0 ) := x 0 conv(df(x)) 1 f(x 0 ) Theorem If N(f,X,x 0 ) int(x) then f has a unique zero x int(x). Moreover x N(f,X,x 0 ). Toy example: f(x) = x(x 2 +2)+1 Approx zero of f is x x 0 = 0.5 f(x 0 ) = X = [ 1,0] Df(X) [2,5] N(f,X,x 0 ) [ 0.475, ] intx = ( 1,0)

37 Validation of zeros of maps Interval Newton Operator f : R n R n smooth X R n - a convex, compact set x 0 int(x) N(f,X,x 0 ) := x 0 conv(df(x)) 1 f(x 0 ) Theorem If N(f,X,x 0 ) int(x) then f has a unique zero x int(x). Moreover x N(f,X,x 0 ). Toy example: f(x) = x(x 2 +2)+1 Approx zero of f is x x 0 = 0.5 f(x 0 ) = X = [ 1,0] Df(X) [2,5] N(f,X,x 0 ) [ 0.475, ] intx = ( 1,0)

38 Validation of zeros of maps Interval Newton Operator f : R n R n smooth X R n - a convex, compact set x 0 int(x) N(f,X,x 0 ) := x 0 conv(df(x)) 1 f(x 0 ) Theorem If N(f,X,x 0 ) int(x) then f has a unique zero x int(x). Moreover x N(f,X,x 0 ). Toy example: f(x) = x(x 2 +2)+1 Approx zero of f is x x 0 = 0.5 f(x 0 ) = X = [ 1,0] Df(X) [2,5] N(f,X,x 0 ) [ 0.475, ] intx = ( 1,0)

39 Validation of zeros of maps Interval Newton Operator f : R n R n smooth X R n - a convex, compact set x 0 int(x) N(f,X,x 0 ) := x 0 conv(df(x)) 1 f(x 0 ) Theorem If N(f,X,x 0 ) int(x) then f has a unique zero x int(x). Moreover x N(f,X,x 0 ). Toy example: f(x) = x(x 2 +2)+1 Approx zero of f is x x 0 = 0.5 f(x 0 ) = X = [ 1,0] Df(X) [2,5] N(f,X,x 0 ) [ 0.475, ] intx = ( 1,0)