The Many Facets of Chaos

2015 ◽  
Vol 25 (04) ◽  
pp. 1530011 ◽  
Author(s):  
Evelyn Sander ◽  
James A. Yorke
Keyword(s):  
Dynamical Behavior ◽  
Basin Of Attraction ◽  
Specific Property ◽  
Invariant Sets ◽  
Basic Invariant ◽  
Stable And Unstable Manifolds ◽  
Time Period ◽  
Fractal Attractor ◽  
Unstable Manifolds ◽  
The Many

There are many ways that a person can encounter chaos, such as through a time series from a lab experiment, a basin of attraction with fractal boundaries, a map with a crossing of stable and unstable manifolds, a fractal attractor, or in a system for which uncertainty doubles after some time period. These encounters appear so diverse, but the chaos is the same in all of the underlying systems; it is just observed in different ways. We describe these different types of chaos. We then give two conjectures about the types of dynamical behavior that is observable if one randomly picks out a dynamical system without searching for a specific property. In particular, we conjecture that from picking a system at random, one observes (1) only three types of basic invariant sets: periodic orbits, quasiperiodic orbits, and chaotic sets; and (2) that all the definitions of chaos are in agreement.

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.

COMPUTING INVARIANT SETS

2003 ◽  
Vol 13 (02) ◽  
pp. 497-504 ◽  
Author(s):  
ALE JAN HOMBURG ◽  
ROBIN DE VILDER ◽  
DUNCAN SANDS
Keyword(s):  
Finite Dimension ◽  
Periodic Points ◽  
Invariant Sets ◽  
Stable And Unstable Manifolds ◽  
Unstable Manifolds

We describe algorithms for computing hyperbolic invariant sets of diffeomorphisms and their stable and unstable manifolds. This includes the calculation of Smale horseshoes and the stable and unstable manifolds of periodic points in any finite dimension.

scholarly journals An invariant of basic sets of Smale flows

1997 ◽  
Vol 17 (6) ◽  
pp. 1437-1448 ◽  
Author(s):  
MICHAEL C. SULLIVAN
Keyword(s):  
Topological Invariant ◽  
Invariant Sets ◽  
Stable And Unstable Manifolds ◽  
One Dimensional ◽  
Smooth Flow ◽  
Unstable Manifolds ◽  
Smale Flows ◽  
Basic Sets

We consider one-dimensional flows which arise as hyperbolic invariant sets of a smooth flow on a manifold. Included in our data is the twisting in the local stable and unstable manifolds. A topological invariant sensitive to this twisting is obtained.

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

2018 ◽  
Vol 28 (14) ◽  
pp. 1850169
Author(s):  
Lingli Xie
Keyword(s):  
Vector Field ◽  
Saddle Point ◽  
Equilibrium Point ◽  
Necessary Conditions ◽  
Continuous System ◽  
Homoclinic Bifurcation ◽  
Stable And Unstable Manifolds ◽  
Geometrical Properties ◽  
Unstable Manifolds

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.

scholarly journals Hadamard–Perron theorems and effective hyperbolicity

2014 ◽  
Vol 36 (1) ◽  
pp. 23-63 ◽  
Author(s):  
VAUGHN CLIMENHAGA ◽  
YAKOV PESIN
Keyword(s):  
Closing Lemma ◽  
Stable And Unstable Manifolds ◽  
Local Dynamics ◽  
Local Diffeomorphisms ◽  
Finite Amount ◽  
Amount Of Information ◽  
Unstable Manifolds

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.

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.

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

1999 ◽  
Author(s):  
Stefano Lenci ◽  
Giuseppe Rega
Keyword(s):  
Local Level ◽  
Basins Of Attraction ◽  
Dynamical Regime ◽  
Impact Oscillator ◽  
Stable And Unstable Manifolds ◽  
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.

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