GEODESIC PARAMETRIZATION OF GLOBAL INVARIANT MANIFOLDS OR WHAT DOES THE EQUADIFF 2003 POSTER SHOW?

We present the GLOBALIZEBVP algorithm for the computation of two-dimensional stable and unstable manifolds of a vector field. Specifically, we use the collocation routines of AUTO to solve boundary problems that are used during the computation to find the next approximate geodesic level set on the manifold. The resulting implementation is numerically very stable and well suited for systems with multiple time scales. This is illustrated with the test-case examples of the Lorenz and Chua systems, and with a slow–fast model of a somatotroph cell.

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A SURVEY OF METHODS FOR COMPUTING (UN)STABLE MANIFOLDS OF VECTOR FIELDS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127405012533 ◽

2005 ◽

Vol 15 (03) ◽

pp. 763-791 ◽

Cited By ~ 165

Author(s):

B. KRAUSKOPF ◽

H. M. OSINGA ◽

E. J. DOEDEL ◽

M. E. HENDERSON ◽

J. GUCKENHEIMER ◽

...

Keyword(s):

Vector Field ◽

Unstable Manifold ◽

Stable Manifold ◽

Vector Fields ◽

Invariant Manifolds ◽

Lorenz System ◽

Dimensional Manifold ◽

Two Dimensional ◽

The Lorenz System ◽

Global Invariant

The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.

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Global Invariant Manifolds of Dynamical Systems with the Compatible Cell Mapping Method

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419501050 ◽

2019 ◽

Vol 29 (08) ◽

pp. 1950105 ◽

Cited By ~ 2

Author(s):

Xiao-Le Yue ◽

Yong Xu ◽

Wei Xu ◽

Jian-Qiao Sun

Keyword(s):

Dynamical Systems ◽

Periodic Solutions ◽

Invariant Manifolds ◽

High Efficiency ◽

Search Algorithm ◽

Mapping Method ◽

Cell Mapping ◽

Stable And Unstable Manifolds ◽

Generalized Cell Mapping ◽

Global Invariant

An iterative compatible cell mapping (CCM) method with the digraph theory is presented in this paper to compute the global invariant manifolds of dynamical systems with high precision and high efficiency. The accurate attractors and saddles can be simultaneously obtained. The simple cell mapping (SCM) method is first used to obtain the periodic solutions. The results obtained by the generalized cell mapping (GCM) method are treated as a database. The SCM and GCM are compatible in the sense that the SCM is a subset of the GCM. The depth-first search algorithm is utilized to find the coarse coverings of global stable and unstable manifolds based on this database. The digraph GCM method is used if the saddle-like periodic solutions cannot be obtained with the SCM method. By taking this coarse covering as a new cell state space, an efficient iterative procedure of the CCM method is proposed by combining sort, search and digraph algorithms. To demonstrate the effectiveness of the proposed method, the classical Hénon map with periodic or chaotic saddles is studied in far more depth than reported in the literature. Not only the global invariant manifolds, but also the attractors and saddles are computed. The computational efficiency can be improved by up to 200 times compared to the traditional GCM method.

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THE SLOW INVARIANT MANIFOLD OF A CONSERVATIVE PENDULUM-OSCILLATOR SYSTEM

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127496000345 ◽

1996 ◽

Vol 06 (04) ◽

pp. 673-692 ◽

Cited By ~ 12

Author(s):

IOANNIS T. GEORGIOU ◽

IRA B. SCHWARTZ

Keyword(s):

Invariant Manifold ◽

Invariant Manifolds ◽

Slow Manifold ◽

Equilibrium States ◽

Periodic Motions ◽

Nonlinear Normal Mode ◽

Oscillator System ◽

Slow Invariant Manifold ◽

Nonlinear Normal ◽

Global Invariant

We analyze the motions of a conservative pendulum-oscillator system in the context of invariant manifolds of motion. Using the singular perturbation methodology, we show that whenever the natural frequency of the oscillator is sufficiently larger than that of the pendulum, there exists a global invariant manifold passing through all static equilibrium states and tangent to the linear eigenspaces at these equilibrium states. The invariant manifold, called slow, carries a continuum of slow periodic motions, both oscillatory and rotational. Computations to various orders of approximation to the slow invariant manifold allow analysis of motions on the slow manifold, which are verified with numerical experiments. Motion on the slow invariant manifold is identified with a slow nonlinear normal mode.

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