The structure of a strange attractor

A few examples are given of an iterative map of the horseshoe type for which everything can be expressed in elementary functions. The strange attractor can be an analytic curve or a set of parallel lines with a Cantor set as its cross-section.     

Revisiting a magneto-elastic strange attractor

We revisit an early example of a nonlinear oscillator that exhibits chaotic motions when subjected to periodic excitation: the magneto-elastically buckled beam. In the paper of Moons and Holmes (1980) [1] magnetic field calculations were outlined but not carried through; instead the nonlinear forces responsible for creation of a two-well potential and buckling were fitted to a polynomial function after reduction to a single mode model. In the present paper we compute the full magnetic field and use it to approximate the forces acting on the beam, also using a single mode reduction. This provides a complete model that accurately predicts equilibria, bifurcations, and free oscillation frequencies of an experimental device. We also compare some periodic, transient and chaotic motions with those obtained by numerical simulations of the single mode model, further illustrating the rich dynamical behavior of this simple electromechanical system.     

Asymptotic form of a strange attractor

A new model is proposed which has a strange attractor as a stationary state for small parameter values. The asymptotic form of the strange attractor is discussed by using the method of nonlinear scales.     

Turbulence without strange attractor

It is shown that pipe-flow turbulence consists of transients. The fractal dimensions of the dynamical process are thus all zero. Nevertheless, this is compatible with Grassberger-Procaccia analyses suggesting the existence of a high-dimensional strange attractor. The usefulness of the Grassberger-Procaccia method to detect the aging of transients is demonstrated.     

An enzyme reaction with a strange attractor

A simple model of the oscillating peroxidase-oxidase reaction is presented. The model is capable of reproducing all previously observed dynamic patterns exhibited by the experimental system including its chaotic behaviour.     

A two-dimensional mapping with a strange attractor

Lorenz (1963) has investigated a system of three first-order differential equations, whose solutions tend toward a strange attractor. We show that 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=bx _i . Numerical experiments are carried out fora=1.4,b=0.3. Depending on the initial point (x ₀,y ₀), the sequence of points obtained by iteration of the mapping either diverges to infinity or tends to a strange attractor, which appears to be the product of a one-dimensional manifold by a Cantor set.     

Scaling of intermittent behaviour of a strange nonchaotic attractor

We describe a type of intermittency present in a strange nonchaotic attractor of a quasiperiodically forced system. This has a similar scaling behaviour to the intermittency found in an attractor-merging crisis of chaotic attractors. By studying rational approximations to the irrational forcing we present a reasoning behind this scaling, which also provides insight into the mechanism which creates the strange nonchaotic attractor.     

Correlation functions of a time-continuous dissipative system with a strange attractor

We determine the exact decay of time correlation functions of a continuous-time chaotic system. In contrast to discrete-time chaotic systems where these correlations decay as a rule exponentially fast we find in our continuous-time system long-time tails well known from many-particle systems.     

A strange attractor in stationary coherent states

An example of the realization of a strange attractor in stationary coherent states is given.     

A dynamical system with a strange attractor and invariant tori

This paper describes a simple three-dimensional time-reversible system of ODEs with quadratic nonlinearities and the unusual property that it is exhibits conservative behavior for some initial conditions and dissipative behavior for others. The conservative regime has quasi-periodic orbits whose amplitude depend on the initial conditions, while the dissipative regime is chaotic. Thus a strange attractor coexists with an infinite set of nested invariant tori in the state space.     

Renormalization group analysis of the global properties of a strange attractor

This paper considers the circle map at the special point: the one at which there is a trajectory with a golden mean winding number and at which the map just fails to be invertable at one point on the circle. The invariant density of this trajectory has fractal properties. Previous work has suggested that the global behavior of this fractal can be effectively analyzed using a kind of partition function formalism to generate anf versus curve. In this paper the partition function is obtained by using a renormalization group approach.     

Conversion of a chaotic attractor into a strange nonchaotic attractor in an one dimensional map and BVP oscillator

In this paper we investigate numerically the possibility of conversion of a chaotic attractor into a nonchaotic but strange attractor in both a discrete system (an one dimensional map) and in a continuous dynamical system — Bonhoeffer—van der Pol oscillator. In these systems we show suppression of chaotic property, namely, the sensitive dependence on initial states, by adding appropriate i) chaotic signal and ii) Gaussian white noise. The controlled orbit is found to be strange but nonchaotic with largest Lyapunov exponent negative and noninteger correlation dimension. Return map and power spectrum are also used to characterize the strange nonchaotic attractor.     

Study of a dynamical system with a strange attractor and invariant tori

A simple three-dimensional time-reversible system of ODEs with quadratic nonlinearities is considered in a recent paper by Sprott (2014). The author finds in this system, that has no equilibria, the coexistence of a strange attractor and invariant tori. The goal of this letter is to justify theoretically the existence of infinite invariant tori and chaotic attractors. For this purpose we embed the original system in a one-parameter family of reversible systems. This allows to demonstrate the presence of a Hopf-zero bifurcation that implies the birth of an elliptic periodic orbit. Thus, the application of the KAM theory guarantees the existence of an extremely complex dynamics with periodic, quasiperiodic and chaotic motions. Our theoretical study is complemented with some numerical results. Several bifurcation diagrams make clear the rich dynamics organized around a so-called noose bifurcation where, among other scenarios, cascades of period-doubling bifurcations also originate chaotic attractors. Moreover, a cross section and other numerical simulations are also presented to illustrate the KAM dynamics exhibited by this system.     

Bifurcations and observation of strange attractor in He-Ne lasers

We report the experimental observation of deterministic chaos in an He-Ne laserat 0.6328μm by varying the discharge current.Strange attractor of laser system in the recon-structed phase-space picture has been obtained.The mechanism of the phenomena was dis-cussed.