Splitting of Separatrices in the Rapidly Forced Duffing Oscillator

Analytic Approximations ◽

Melnikov Integral ◽

Unstable Manifolds ◽

Splitting Of Separatrices ◽

Splitting Distance

Abstract The splitting of the stable and unstable manifolds of the rapidly forced Duffing oscillator with negative stiffness is investigated. The method used relies on the computation of analytic approximations for the orbits on the perturbed manifolds, and the asymptotic approximation of these orbits by successive integrations by parts. It is shown, that the splitting of the manifolds becomes exponentially small as the perturbation parameter tends to zero, and that the estimate for the splitting distance given by the Melnikov Integral dominates over high order corrections.

FROM SINGLE WELL CHAOS TO CROSS WELL CHAOS: A DETAILED EXPLANATION IN TERMS OF MANIFOLD INTERSECTIONS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127494000666 ◽

1994 ◽

Vol 04 (04) ◽

pp. 933-941 ◽

Cited By ~ 16

Author(s):

ANDREW L. KATZ ◽

EARL H. DOWELL

Keyword(s):

Dynamical Systems ◽

Nonlinear Systems ◽

Chaotic Attractor ◽

Duffing Oscillator ◽

Detailed Explanation ◽

Single Well ◽

Unstable Manifolds ◽

Parameter Values

The study of stable and unstable manifolds, and their intersections with each other, is a powerful technique for interpreting complex bifurcations of nonlinear systems. The escape phenomenon in the twin-well Duffing oscillator is one such bifurcation that is elucidated through the analysis of manifold intersections. In this paper, two escape scenarios in the twin-well Duffing oscillator are presented. In each scenario, the relevant manifold structures are examined for parameter values on either side of the escape bifurcation. Included is a description of the role of the hilltop saddle stable manifolds, which are known to separate the single well basins (should single well attractors exist). In each of the two bifurcation scenarios, it is shown through a detailed analysis of Poincaré maps that a homoclinic intersection of the manifolds of a specific period-3 saddle implies the destruction of the single well chaotic attractor. Although the Duffing oscillator is used to illustrate the ideas advanced here, it is thought that the approach will be useful for a variety of dynamical systems.

CONSTRUCTION OF WHISKERS FOR THE QUASIPERIODICALLY FORCED PENDULUM

Reviews in Mathematical Physics ◽

10.1142/s0129055x07003127 ◽

2007 ◽

Vol 19 (08) ◽

pp. 823-877 ◽

Cited By ~ 3

Author(s):

MIKKO STENLUND

Keyword(s):

Equations Of Motion ◽

Perturbation Parameter ◽

Simple Construction ◽

Time Axis ◽

Kam Tori ◽

Unstable Manifolds ◽

Forced Pendulum

We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, giving a simple construction of unstable KAM tori and their stable and unstable manifolds for analytic perturbations. We extend analytically the solutions of the equations of motion, order by order in the perturbation parameter, to a uniform neighborhood of the time axis.

A Numerical Scheme for Computing Stable and Unstable Manifolds in Nonautonomous Flows

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741630041x ◽

2016 ◽

Vol 26 (14) ◽

pp. 1630041 ◽

Cited By ~ 3

Author(s):

Sanjeeva Balasuriya

Keyword(s):

Lyapunov Exponent ◽

High Resolution ◽

Rossby Wave ◽

Time Variation ◽

Numerical Scheme ◽

Numerical Approximation ◽

Duffing Oscillator ◽

Finite Time Lyapunov Exponent ◽

There are many methods for computing stable and unstable manifolds in autonomous flows. When the flow is nonautonomous, however, difficulties arise since the hyperbolic trajectory to which these manifolds are anchored, and the local manifold emanation directions, are changing with time. This article utilizes recent results which approximate the time-variation of both these quantities to design a numerical algorithm which can obtain high resolution in global nonautonomous stable and unstable manifolds. In particular, good numerical approximation is possible locally near the anchor trajectory. Nonautonomous manifolds are computed for two examples: a Rossby wave situation which is highly chaotic, and a nonautonomus (time-aperiodic) Duffing oscillator model in which the manifold emanation directions are rapidly changing. The numerical method is validated and analyzed in these cases using finite-time Lyapunov exponent fields and exactly known nonautonomous manifolds.

Study on the Geometrical Properties and Existence of Orbit Homoclinic to a Saddle Point in n-Dimensional Autonomous Vector Field with Polynomials

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418501699 ◽

2018 ◽

Vol 28 (14) ◽

pp. 1850169

Author(s):

Lingli Xie

Keyword(s):

Vector Field ◽

Saddle Point ◽

Equilibrium Point ◽

Necessary Conditions ◽

Continuous System ◽

Homoclinic Bifurcation ◽

Geometrical Properties ◽

According to the theory of stable and unstable manifolds of an equilibrium point, we firstly find out some geometrical properties of orbits on the stable and unstable manifolds of a saddle point under some brief conditions of nonlinear terms composed of polynomials for [Formula: see text]-dimensional time continuous system. These properties show that the orbits on stable and unstable manifolds of the saddle point will stay on the corresponding stable and unstable subspaces in the [Formula: see text]-neighborhood of the saddle point. Furthermore, the necessary conditions of existence for orbit homoclinic to a saddle point are exposed. Some examples including homoclinic bifurcation are given to indicate the application of the results. Finally, the conclusions are presented.

Hadamard–Perron theorems and effective hyperbolicity

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2014.49 ◽

2014 ◽

Vol 36 (1) ◽

pp. 23-63 ◽

Cited By ~ 3

Author(s):

VAUGHN CLIMENHAGA ◽

YAKOV PESIN

Keyword(s):

Closing Lemma ◽

Local Dynamics ◽

Local Diffeomorphisms ◽

Finite Amount ◽

Amount Of Information ◽

We prove several new versions of the Hadamard–Perron theorem, which relates infinitesimal dynamics to local dynamics for a sequence of local diffeomorphisms, and in particular establishes the existence of local stable and unstable manifolds. Our results imply the classical Hadamard–Perron theorem in both its uniform and non-uniform versions, but also apply much more generally. We introduce a notion of ‘effective hyperbolicity’ and show that if the rate of effective hyperbolicity is asymptotically positive, then the local manifolds are well behaved with positive asymptotic frequency. By applying effective hyperbolicity to finite-orbit segments, we prove a closing lemma whose conditions can be verified with a finite amount of information.

Stable and Unstable Manifolds

Computational Ergodic Theory - Algorithms and Computation in Mathematics ◽

10.1007/3-540-27305-0_11 ◽

2005 ◽

pp. 333-362

Keyword(s):

Linear response theory for diffeomorphisms with tangencies of stable and unstable manifolds—a contribution to the Gallavotti–Cohen chaotic hypothesis

Nonlinearity ◽

10.1088/1361-6544/aae740 ◽

2018 ◽

Vol 31 (12) ◽

pp. 5683-5691 ◽

Cited By ~ 2

Author(s):

David Ruelle

Keyword(s):

Linear Response ◽

Linear Response Theory ◽

Response Theory ◽

Chaotic Hypothesis ◽

Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories

Nelineinaya Dinamika ◽

10.20537/nd1601008 ◽

2016 ◽

pp. 121-143

Author(s):

S. P. Kuznetsov ◽

Keyword(s):

Phase Trajectories ◽

Oscillating Systems ◽

Unstable Manifolds ◽

Hyperbolic Chaos

Non-Smooth Dynamics and Non-Classical Bifurcations in Impulse-Impact Oscillators

Volume 7B: 17th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc99/vib-8054 ◽

1999 ◽

Author(s):

Stefano Lenci ◽

Giuseppe Rega

Keyword(s):

Local Level ◽

Basins Of Attraction ◽

Dynamical Regime ◽

Impact Oscillator ◽

Impact Oscillators ◽

Unstable Manifolds ◽

Local And Global Bifurcations ◽

Initial Description ◽

Smooth Dynamics

Abstract Some aspects of the nonlinear dynamics of an impulse-impact oscillator are investigated. After an initial description of the prototype mechanical model used to illustrate the results, attention is paid to the classical local and global bifurcations which are at the base of the changes of dynamical regime. Some non-classical phenomena due to the particular nature of the investigated system are then considered. At a local level, it is shown that periodic solutions may appear (or disappear) through a non-classical bifurcation which involves synchronization of impulses and impacts. Similarities and differences with the classical bifurcations are discussed. At a global level, the effects of the non-continuity of the orbits in the phase space on the basins of attraction topology are investigated. It is shown how this property is at the base of a non-classical homoclinic bifurcation where the homoclinic points disappear after the first touch between the stable and unstable manifolds.

Random invariant manifolds for ill-posed stochastic evolution equations

Stochastics and Dynamics ◽

10.1142/s0219493720500136 ◽

2019 ◽

Vol 20 (02) ◽

pp. 2050013

Author(s):

Alexandra Neamţu

Keyword(s):

Invariant Manifolds ◽

Evolution Equations ◽

Linear Part ◽

Time Behavior ◽

Stochastic Evolution ◽

Stochastic Evolution Equations ◽

Long Time ◽

Ill Posed ◽

We establish the existence of random stable and unstable manifolds for ill-posed stochastic partial differential equations (SPDEs). Namely, we assume that the linear part does not generate a [Formula: see text]-semigroup. Using the theory of integrated semigroups, we are able to analyze the long-time behavior of random dynamical systems generated by such SPDEs.