Computation of Lyapunov-Type Numbers for Invariant Curves of Planar Maps

Invariant manifolds play an important role in the study of dynamical systems, and it is often crucial to know when they persist under small perturbations of the system. Sufficient conditions for persistence have been formulated in the literature in terms of so-called Lyapunov-type numbers, which measure and compare attractivity rates. The numerical evaluations of these numbers in a simple setting is the subject of this paper. We consider planar diffeomorphisms depending on a parameter; the diffeomorphisms have invariant curves which deform as the parameter changes. The Lyapunov-type numbers are then monitored as functions of parameter and are related to the dynamics on the invariant curves. As an example, we consider the delayed logistic map. We also describe how the invariant curves have been computed.

[1]  M. van Veldhuizen,et al.  Convergence results for invariant curve algorithms , 1988 .

[2]  Luca Dieci,et al.  Computation of invariant tori by the method of characteristics , 1995 .

[3]  G. Vegter,et al.  Algorithms for computing normally hyperbolic invariant manifolds , 1997 .

[4]  G. Moore,et al.  Computation and Parametrisation of Invariant Curves and Tori , 1996 .

[5]  P. J. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[6]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[7]  Robert D. Russell,et al.  Computation of invariant tori by orthogonal collocation , 2000 .

[8]  D. Aronson,et al.  Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study , 1982 .

[9]  Volker Reichelt,et al.  Computing Invariant Tori and Circles in Dynamical Systems , 2000 .

[10]  M. van Veldhuizen.  A new algorithm for the numerical approximation of an invariant curve , 1987 .

[11]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[12]  A. Denjoy,et al.  Sur les courbes définies par les équations différentielles à la surface du tore , 1932 .

[13]  Neil Fenichel.  Persistence and Smoothness of Invariant Manifolds for Flows , 1971 .

[14]  Leon O. Chua,et al.  Practical Numerical Algorithms for Chaotic Systems , 1989 .

[15]  Rutherford Aris,et al.  Numerical computation of invariant circles of maps , 1985 .

[16]  C. Caramanis.  What is ergodic theory , 1963 .

[17]  W. D. Melo,et al.  ONE-DIMENSIONAL DYNAMICS , 2013 .