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

1994 ◽  
Vol 04 (04) ◽  
pp. 933-941 ◽  
Author(s):  
ANDREW L. KATZ ◽  
EARL H. DOWELL
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Chaotic Attractor ◽  
Duffing Oscillator ◽  
Stable And Unstable Manifolds ◽  
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.

Splitting of Separatrices in the Rapidly Forced Duffing Oscillator

1993 ◽  
Author(s):  
Alexander F. Vakakis
Keyword(s):  
Asymptotic Approximation ◽  
Duffing Oscillator ◽  
Perturbation Parameter ◽  
Negative Stiffness ◽  
Stable And Unstable Manifolds ◽  
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.

scholarly journals Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

2010 ◽  
Vol 17 (1) ◽  
pp. 1-36 ◽  
Author(s):  
M. Branicki ◽  
S. Wiggins
Keyword(s):  
Dynamical Systems ◽  
Finite Time ◽  
Vector Fields ◽  
Transient Flow ◽  
Time Interval ◽  
Stable And Unstable Manifolds ◽  
Lagrangian Transport ◽  
Transport Barriers ◽  
Unstable Manifolds ◽  
Hyperbolic Trajectories

Abstract. We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of transport barriers in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterising transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate in a concrete manner the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of flow transition which occurs when finite-time hyperbolicity is lost or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing and important area of dynamical systems theory.

Homoclinic motions and chaos in the quasiperiodically forced van der Pol-Duffing oscillator with single well potential

1994 ◽  
Vol 445 (1925) ◽  
pp. 597-617 ◽  
Keyword(s):  
Periodic Orbits ◽  
Invariant Tori ◽  
Duffing Oscillator ◽  
Double Resonance ◽  
Unperturbed System ◽  
Van Der Pol ◽  
Stable And Unstable Manifolds ◽  
Special Cases ◽  
Forced Oscillator ◽  
Single Well

We study a two-frequency quasiperiodically forced oscillator with single well po­tential which reduces to the van der Pol and Duffing oscillators in certain special cases. The unperturbed system without damping and forcing terms has a one-parameter family of periodic orbits. We concentrate on the dynamics near the unperturbed resonant periodic orbits. Using the second-order averaging method and a version of Melnikov’s method, we show that when double resonance occurs, the stable and unstable manifolds of normally hyperbolic invariant tori intersect transversely, i. e. transverse homoclinic motions exist, near the unperturbed reso­nant periodic orbits in certain parameter regions. Such homoclinic motions yield chaotic dynamics characterized by a generalization of the Bernoulli shift. Numer­ical simulation results are also given to demonstrate the theoretical results.

A GEOMETRIC MODEL FOR THE DUFFING OSCILLATOR

1993 ◽  
Vol 03 (03) ◽  
pp. 685-691 ◽  
Author(s):  
J.W.L. McCALLUM ◽  
R. GILMORE
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Attractor ◽  
Geometric Model ◽  
Duffing Oscillator ◽  
Chaotic Attractors ◽  
Unstable Periodic Orbits ◽  
Linking Numbers ◽  
Full Period ◽  
Parameter Values

A geometric model for the Duffing oscillator is constructed by analyzing the unstable periodic orbits underlying the chaotic attractors present at particular parameter values. A template is constructed from observations of the motion of the chaotic attractor in a Poincaré section as the section is swept for one full period. The periodic orbits underlying the chaotic attractor are found and their linking numbers are computed. These are compared with the linking numbers from the template and the symbolic dynamics of the orbits are identified. This comparison is used to validate the template identification and label the orbits by their symbolic dynamics.

Nonlinear Dynamical Systems, Their Stability, and Chaos

2014 ◽  
Vol 66 (2) ◽  
Author(s):  
Amol Marathe ◽  
Rama Govindarajan
Keyword(s):  
Fluid Flow ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Fluid Mechanics ◽  
Fixed Points ◽  
Chaotic Systems ◽  
Nonlinear Dynamical Systems ◽  
Not Given ◽  
Nonlinear Dynamical ◽  
Detailed Explanation

This introduction to nonlinear systems is written for students of fluid mechanics, so connections are made throughout the text to familiar fluid flow systems. The aim is to present how nonlinear systems are qualitatively different from linear and to outline some simple procedures by which an understanding of nonlinear systems may be attempted. Considerable attention is paid to linear systems in the vicinity of fixed points, and it is discussed why this is relevant for nonlinear systems. A detailed explanation of chaos is not given, but a flavor of chaotic systems is presented. The focus is on physical understanding and not on mathematical rigor.

CHAOS AND CHAOTIC TRANSIENTS IN A FORCED MODEL OF THE ECONOMIC LONG WAVE: THE ROLE OF HOMOCLINIC BIFURCATION TO INFINITY

1995 ◽  
Vol 05 (03) ◽  
pp. 741-749 ◽  
Author(s):  
JEPPE STURIS ◽  
MORTEN BRØNS
Keyword(s):  
Limit Cycle ◽  
Basin Of Attraction ◽  
Homoclinic Bifurcation ◽  
Periodic Forcing ◽  
Stable And Unstable Manifolds ◽  
Long Wave ◽  
Unstable Manifolds ◽  
Chaotic Transients ◽  
The Basin Of Attraction

When an autonomous system of ordinary differential equations exhibits limit cycle behavior but is close in parameter space to a homoclinic bifurcation to infinity in which the limit cycle blows up to infinite amplitude and disappears, periodic forcing of the system may result in the appearance of both chaos and chaotic transients. In this paper, we use numerical techniques to map out Arnol’d tongues of a forced model of the economic long wave and illustrate how the system becomes chaotic and also exhibits chaotic transients for certain parameter combinations. Based on linearizations at infinity, we argue that infinity acts like a saddle with stable and unstable manifolds. By numerical computation, we show that chaotic transients occur when the manifolds intersect. Depending on parameters, two types of bifurcations have been identified: A chaotic attractor blows up to infinite size and disappears or the boundary of the basin of attraction of a periodic solution becomes fractal.

Periodic Flows to Chaos Based on Discrete Implicit Mappings of Continuous Nonlinear Systems

2015 ◽  
Vol 25 (03) ◽  
pp. 1550044 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Time Delay ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Single Step ◽  
Nonlinear Dynamical ◽  
Periodic Flows ◽  
Mapping Structures ◽  
Stability And Bifurcations

This paper presents a semi-analytical method for periodic flows in continuous nonlinear dynamical systems. For the semi-analytical approach, differential equations of nonlinear dynamical systems are discretized to obtain implicit maps, and a mapping structure based on the implicit maps is employed for a periodic flow. From mapping structures, periodic flows in nonlinear dynamical systems are predicted analytically and the corresponding stability and bifurcations of the periodic flows are determined through the eigenvalue analysis. The periodic flows predicted by the single-step implicit maps are discussed first, and the periodic flows predicted by the multistep implicit maps are also presented. Periodic flows in time-delay nonlinear dynamical systems are discussed by the single-step and multistep implicit maps. The time-delay nodes in discretization of time-delay nonlinear systems were treated by both an interpolation and a direct integration. Based on the discrete nodes of periodic flows in nonlinear dynamical systems with/without time-delay, the discrete Fourier series responses of periodic flows are presented. To demonstrate the methodology, the bifurcation tree of period-1 motion to chaos in a Duffing oscillator is presented as a sampled problem. The method presented in this paper can be applied to nonlinear dynamical systems, which cannot be solved directly by analytical methods.

scholarly journals Shooting methods and topological transversality

1999 ◽  
Vol 129 (6) ◽  
pp. 1137-1155 ◽  
Author(s):  
B. Buffoni
Keyword(s):  
Dynamical Systems ◽  
Chaotic Dynamics ◽  
Shooting Method ◽  
Heteroclinic Orbits ◽  
Kolmogorov Equation ◽  
Shooting Methods ◽  
Stable And Unstable Manifolds ◽  
Heteroclinic Solutions ◽  
Unstable Manifolds

We show that shooting methods for homoclinic or heteroclinic orbits in dynamical systems may automatically guarantee the topological transversality of the stable and unstable manifolds. The interest of such results is twofold. First, these orbits persist under perturbations which destroy the structure allowing the shooting method and, second, topological transversality is often sufficient when some kind of transversality is required to obtain chaotic dynamics. We shall focus on heteroclinic solutions in the extended Fisher–Kolmogorov equation.

Sign in / Sign up

Export Citation Format

Share Document