Narrow Results

The ideas underpinning a technique of using nonlinear approximations to control chaotic dynamical systems are shown to be useful as a basis for the development of effective methods for targeting these systems. The inverses of the maps under consideration are not required. The methods are applied to the Hénon and Ikeda maps and also to an Ikeda-like map.

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.

A technique of using nonlinear approximations to control chaotic dynamical systems is extended so it can be used to control such systems when only data generated can be observed.

A method for controlling onto saddle-type fixed points developed by Yagasaki and Uozumi is extended so as to make the capture region many times larger than that of the original method.

The dynamics of a system defined by an endomorphism is essentially different from that of a system defined by a diffeomorphism due to interaction of invariant objects with the so-called critical locus. A planar endomorphism typically folds the phase space along curves J₀ where the Jacobian of the map is singular. The critical locus, denoted J₁, is the image of J₀. It is often only piecewise smooth due to the presence of isolated cusp points that are persistent under perturbation. We investigate what happens when the stable set W_s of a fixed point or periodic orbit interacts with J₁ near such a cusp point C₁. Our approach is in the spirit of bifurcation theory, and we classify the different unfoldings of the codimension-two singularity where the curve W_s is tangent to J₁ exactly at C₁. The analysis uses a local normal-form setup that identifies the possible local phase portraits. These local phase portraits give rise to different global manifestations of the behavior as organized by five different global bifurcation diagrams.

A method of greatly decreasing the activation time of a control method based on stable manifold information is proposed.

In the work [Chua, 1992], a deep intuition of its author gave rise to the choice of singularities corresponding to Chua's circuit. Therefore, it is the only one probably exhibiting three saddle points named Chua's singularities in this paper. One of the singularities is a saddle in forward time (dt > 0) of integration, whereas the other two are saddles in backward time (dt < 0) of integration. In the following, the term Chua's Chaos denotes chaos related to Chua's singularities. These singularities are the source of all special surfaces that are the subject of this contribution.

We named the surface to which all other surfaces are bound as the Double-Arm Stable Manifold (DASM). The beauty and multifunctionality of this surface represents the unfathomable Intelligence in the sense of [Tolle, 2003]. The presence of the DASM in the state space is a sufficient condition for the generation of Chua's chaos or corresponding periodic windows.

Since Chua's singularities are not limited by circuit morphology or the order of state equations, the research on Chua's chaos seems to be still very promising.

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.

In this paper, we consider the cooperative system

Saddle fixed points are the centerpieces of complicated dynamics in a system. The one-dimensional stable and unstable manifolds of these saddle-points are crucial to understanding the dynamics of such systems. While the problem of sketching the unstable manifold is simple, plotting the stable manifold is not as easy. Several algorithms exist to compute the stable manifold of saddle-points, but they have their limitations, especially when the system is not invertible. In this paper, we present a new algorithm to compute the stable manifold of two-dimensional systems which can also be used for noninvertible systems. After outlining the logic of the algorithm, we demonstrate the output of the algorithm on several examples.

Let ϕ_t be a Lévy process in a semisimple Lie group G of noncompact type regarded as a stochastic flow on a homogeneous space of G, called a G-flow. We will determine the Lyapunov exponents and the stable manifolds of ϕ_t, and the stationary points of an associated vector field. As examples, SL(d,R)-flows and SO(1,d)-flows on SO(d) and S^{d - 1} are discussed in details.

In the local dynamics of Newton's method, a generic double root of a holomorphic function of two variables has a Cantor family of holomorphic superstable manifolds.