Growing 1D and Quasi-2D Unstable Manifolds of Maps

We present a new 1D algorithm for computing the global one-dimensional unstable manifold of a saddle point of a map. Our method can be generalized to compute two-dimensional unstable manifolds of maps with three-dimensional state spaces. This is shown here with a quasi-2D (Q2D) algorithm for the special case of a quasiperiodically forced map, which allows for a substantial simplification of the general case described in Krauskopf and Osinga (1998). The key idea is to “grow” the manifold in steps, which consist of finding a new point on the manifold at a prescribed distance from the last point. The speed of growth is determined only by the curvature of the manifold, and not by the dynamics.The performance of the 1D algorithm is demonstrated with a constructed test example, and it is then used to compute one-dimensional manifolds of a map modeling mixing in a stirring tank. With the Q2D algorithm we compute two-dimensional unstable manifolds in the quasiperiodically forced Henon map.

[1]  James A. Yorke,et al.  CALCULATING STABLE AND UNSTABLE MANIFOLDS , 1991 .

[2]  Dana D. Hobson,et al.  An efficient method for computing invariant manifolds of planar maps , 1993 .

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

[4]  John Guckenheimer,et al.  Dstool: Computer assisted exploration of dynamical systems , 1992 .

[5]  H. Aref.  Stirring by chaotic advection , 1984, Journal of Fluid Mechanics.

[6]  James A. Yorke,et al.  A procedure for finding numerical trajectories on chaotic saddles , 1989 .

[7]  J. Yorke,et al.  Crises, sudden changes in chaotic attractors, and transient chaos , 1983 .

[8]  Hinke M. Osinga,et al.  Computing invariant manifolds , 1996 .

[9]  Bernd Krauskopf,et al.  Globalizing Two-Dimensional Unstable Manifolds of Maps , 1998 .

[10]  O. Junge,et al.  Exploring invariant sets and invariant measures. , 1997, Chaos.

[11]  Lawrence Sirovich,et al.  A simple model of chaotic advection and scattering. , 1995, Chaos.

[12]  R. Fateman,et al.  A System for Doing Mathematics by Computer. , 1992 .

[13]  Juergen Kurths,et al.  Multiband strange nonchaotic attractors in quasiperiodically forced systems , 1996 .

[14]  Stephen Wiggins,et al.  Chaotic transport in dynamical systems , 1991 .

[15]  K. Ikeda,et al.  Optical Turbulence: Chaotic Behavior of Transmitted Light from a Ring Cavity , 1980 .

[16]  Michael Dellnitz,et al.  The Computation of Unstable Manifolds Using Subdivision and Continuation , 1996 .

[17]  Julio M. Ottino,et al.  Analysis of chaotic mixing in two model systems , 1986, Journal of Fluid Mechanics.

[18]  Peter C. Jurs,et al.  Mathematica , 2019, J. Chem. Inf. Comput. Sci..

[19]  J. Palis,et al.  Geometric theory of dynamical systems , 1982 .

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

[21]  Roger L. Kraft.  Hyperbolicity & Sensitive Chaotic Dynamics at Homoclinic Bifurcations (J. Palis and F. Taken) , 1996, SIAM Rev..

[22]  J. Yorke,et al.  Strange attractors that are not chaotic , 1984 .

[23]  M. Hénon,et al.  A two-dimensional mapping with a strange attractor , 1976 .

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

[25]  G. Vegter,et al.  On the computation of invariant manifolds of fixed points , 1995 .

[26]  Floris Takens,et al.  Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations : fractal dimensions and infinitely many attractors , 1993 .

[27]  Bernd Krauskopf,et al.  Investigating torus bifurcations in the forced Van der Pol oscillator , 2000 .

[28]  Christopher K. R. T. Jones,et al.  Global dynamical behavior of the optical field in a ring cavity , 1985 .

[29]  Bernd Krauskopf,et al.  Growing Unstable Manifolds of Planar Maps , 1997 .

[30]  Gert Vegter,et al.  On the computation of normally hyperbolic invariant manifolds , 1996 .

[31]  M. Hénon.  A two-dimensional mapping with a strange attractor , 1976 .

[32]  M. Dellnitz,et al.  A subdivision algorithm for the computation of unstable manifolds and global attractors , 1997 .