Journal ArticleDOI
Richness of dynamics and global bifurcations in systems with a homoclinic figure-eight*
TLDR
In this article, the authors consider 2D flows with a homoclinic figure-eight to a dissipative saddle and derive the bifurcation diagram using topological techniques.Abstract:
We consider 2D flows with a homoclinic figure-eight to a dissipative saddle. We study the rich dynamics that such a system exhibits under a periodic forcing. First, we derive the bifurcation diagram using topological techniques. In particular, there is a homoclinic zone in the parameter space with a non-smooth boundary. We provide a complete explanation of this phenomenon relating it to primary quadratic homoclinic tangency curves which end up at some cubic tangency (cusp) points. We also describe the possible attractors that exist (and may coexist) in the system. A main goal of this work is to show how the previous qualitative description can be complemented with quantitative global information. To this end, we introduce a return map model which can be seen as the simplest one which is 'universal' in some sense. We carry out several numerical experiments on the model, to check that all the objects predicted to exist by the theory are found in the model, and also to investigate new properties of the system.read more
Citations
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Journal ArticleDOI
Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps
A. S. Gonchenko,Sergey Gonchenko +1 more
TL;DR: In this paper, the authors focus on the existence of pseudohyperbolic attractors for 3D diffeomorphisms and show that three-dimensional maps may have only five different types of such attractors, which they call the discrete Lorenz, figure-8, double-figure-8 and super-Lorenz.
Journal ArticleDOI
Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane
TL;DR: In this article, the authors consider the dynamics of an unbalanced rubber ball rolling on a rough plane and demonstrate the existence of complex chaotic dynamics such as strange attractors and mixed dynamics.
Book ChapterDOI
Theory and Applications of the Mean Exponential Growth Factor of Nearby Orbits (MEGNO) Method
TL;DR: The Mean Exponential Growth Factor of Nearby Orbits (MEGNO) as discussed by the authors is a fast indicator that provides a clear picture of the resonance structure, the location of stable and unstable periodic orbits as well as a measure of hyperbolicity in chaotic domains which coincides with that given by the maximum Lyapunov characteristic exponent.
Journal ArticleDOI
Hyperchaos and multistability in the model of two interacting microbubble contrast agents
TL;DR: The model of two coupled gas bubbles provides a new example of physically relevant system with multistable hyperchaos, and it is demonstrated that the dynamics of two bubbles can be essentiallyMultistable.
Journal ArticleDOI
Hyperchaos and Multistability in Nonlinear Dynamics of Two Interacting Microbubble Contrast Agents
TL;DR: In this paper, the authors consider a model of two coupled contrast agents interacting via the Bjerknes force and exposed to an external ultrasound field, and demonstrate that in this five-dimensional nonlinear dynamical system various types of complex dynamics can occur, namely, periodic, quasi-periodic, chaotic and hypechaotic oscillations of bubbles.
References
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Book
Introduction to Applied Nonlinear Dynamical Systems and Chaos
TL;DR: The Poincare-Bendixson Theorem as mentioned in this paper describes the existence, uniqueness, differentiability, and flow properties of vector fields, and is used to prove that a dynamical system is Chaotic.
Journal ArticleDOI
A universal instability of many-dimensional oscillator systems
TL;DR: In this article, the authors demonstrate the mechanism for a universal instability, the Arnold diffusion, which occurs in the oscillating systems having more than two degrees of freedom, which results in an irregular, or stochastic, motion of the system as if the latter were influenced by a random perturbation even though, in fact, the motion is governed by purely dynamical equations.
Journal ArticleDOI
A Two-dimensional Mapping with a Strange Attractor
TL;DR: In this article, the same properties can be observed in a simple mapping of the plane defined by: \({x i + 1}} = {y_i} + 1 - ax_i^2,{y i+ 1} = b{x_i}\).
Journal ArticleDOI
The Dynamics of the Henon Map
TL;DR: In this paper, the Henon map with expansion combined with strong contraction is modeled on the treatment of the one-dimensional system x→1-ax 2 and the perturbation of a from the value a=2 and b small.