Global Invariant Manifolds Near Homoclinic Orbits to a Real Saddle: (Non)Orientability and Flip Bifurcation

2013 ◽  
Vol 12 (4) ◽  
pp. 1803-1846 ◽  
Author(s):  
Pablo Aguirre ◽  
Bernd Krauskopf ◽  
Hinke M. Osinga
Keyword(s):  
Invariant Manifolds ◽  
Homoclinic Orbits ◽  
Flip Bifurcation ◽  
Global Invariant

scholarly journals GEOMETRY OF HOMOCLINIC CONNECTIONS IN A PLANAR CIRCULAR RESTRICTED THREE-BODY PROBLEM

2007 ◽  
Vol 17 (04) ◽  
pp. 1151-1169 ◽  
Author(s):  
MARIAN GIDEA ◽  
JOSEP J. MASDEMONT
Keyword(s):  
Symbolic Dynamics ◽  
Body Problem ◽  
Invariant Manifolds ◽  
Libration Point ◽  
Homoclinic Orbits ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Lyapunov Orbits ◽  
Homoclinic Connections ◽  
Three Body

The stable and unstable invariant manifolds associated with Lyapunov orbits about the libration point L1between the primaries in the planar circular restricted three-body problem with equal masses are considered. The behavior of the intersections of these invariant manifolds for values of the energy between that of L1and the other collinear libration points L2, L3is studied using symbolic dynamics. Homoclinic orbits are classified according to the number of turns about the primaries.

scholarly journals COMPUTING TWO-DIMENSIONAL GLOBAL INVARIANT MANIFOLDS IN SLOW–FAST SYSTEMS

2007 ◽  
Vol 17 (03) ◽  
pp. 805-822 ◽  
Author(s):  
J. P. ENGLAND ◽  
B. KRAUSKOPF ◽  
H. M. OSINGA
Keyword(s):  
Invariant Manifolds ◽  
Multiple Time Scales ◽  
Test Case ◽  
Multiple Time ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Boundary Problems ◽  
Case Examples ◽  
Unstable Manifolds ◽  
Global Invariant

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.

Amplitude Nonlinear Normal Modes of Piecewise Linear Systems

2001 ◽  
Author(s):  
Dongying Jiang ◽  
Vincent Soumier ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Invariant Manifold ◽  
Invariant Manifolds ◽  
Piecewise Linear ◽  
Autonomous Systems ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Flip Bifurcation ◽  
Strong Nonlinearity ◽  
Nonlinear Modes ◽  
Nonlinear Normal

Abstract A numerical method for constructing nonlinear normal modes for piecewise linear autonomous systems is presented. Based on the concept of invariant manifolds, a Galerkin based approach is applied here to obtain nonlinear normal modes numerically. The accuracy of the constructed nonlinear modes is checked by the comparison of the motion on the invariant manifold to the exact solution, in both time and frequency domains. It is found that the Galerkin based construction approach can represent the invariant manifold accurately over strong nonlinearity regions. Several interesting dynamic characteristics of the nonlinear modal motion are found and compared to those of linear modes. The stability of the nonlinear normal modes of a two-degree of freedom system is investigated using characteristic multipliers and Poincaré maps, and a flip bifurcation is found for both nonlinear modes.

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

2005 ◽  
Vol 15 (03) ◽  
pp. 763-791 ◽  
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.

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

2019 ◽  
Vol 29 (08) ◽  
pp. 1950105 ◽  
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.

THE SLOW INVARIANT MANIFOLD OF A CONSERVATIVE PENDULUM-OSCILLATOR SYSTEM

1996 ◽  
Vol 06 (04) ◽  
pp. 673-692 ◽  
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.

Homoclinic and Heteroclinic Bifurcations Close to a Twisted Heteroclinic Cycle

1998 ◽  
Vol 08 (02) ◽  
pp. 359-375 ◽  
Author(s):  
Martín G. Zimmermann ◽  
Mario A. Natiello
Keyword(s):  
Reaction Diffusion ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Reaction Diffusion Equation ◽  
Heteroclinic Cycle ◽  
Flip Bifurcation ◽  
Double Loop ◽  
Orbit Flip ◽  
Inclination Flip ◽  
Two Parameters

We study the interaction of a transcritical (or saddle-node) bifurcation with a codimension-0/codimension-2 heteroclinic cycle close to (but away from) the local bifurcation point. The study is motivated by numerical observations on the traveling wave ODE of a reaction–diffusion equation. The manifold organization is such that two branches of homoclinic orbits to each fixed point are created when varying the two parameters controlling the codimension-2 loop. It is shown that the homoclinic orbits may become degenerate in an orbit-flip bifurcation. We establish the occurrence of multi-loop homoclinic and heteroclinic orbits in this system. The double-loop homoclinic orbits are shown to bifurcate in an inclination-flip bifurcation, where a Smale's horseshoe is found.

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