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.     

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.     

On the realization of the Hunt-Ott strange nonchaotic attractor in a physical system

The object of investigation is a system consisting of two coupled nonautonomous van der Pol oscillators the characteristics frequencies of which differ by a factor of 2. The system is subjected to an external action in the form of slow periodic modulation of an oscillation-controlling parameter and also to an additional action at a frequency that is in an irrational relation with the modulation frequency. It is shown that the variation of the oscillation phase over a modulation period can be approximated by a 2D map on a torus that has a robust (structurally stable) Hunt-Ott strange nonchaotic attractor. Calculations of the quantitative characteristics of the attractor corresponding to the initial set of nonautonomous coupled oscillators (such as phase sensitivity exponent, structures and scaling of rational approximations, as well as Lyapunov exponents and their parameter dependence) confirm the presence of the Hunt-Ott 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.     

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.     

Experimental observation of strange nonchaotic attractors in a driven excitable system

We report the observation of strange nonchaotic attractors in an electrochemical cell. The system parameters were chosen such that the system observable (anodic current) exhibits fixed point behavior or period one oscillations. These autonomous dynamics were thereafter subjected to external quasiperiodic forcing. Systematically varying the characteristics (frequency and amplitude) of the superimposed external signal; quasiperiodic, chaotic and strange nonchaotic behaviors in the anodic current were generated. The inception of strange nonchaotic attractors was verified using standard diagnostic techniques.     

Wada basins of strange nonchaotic attractors in a quasiperiodically forced system

Whether Wada basins of strange nonchaotic attractors (SNAs) can exist has been an open problem. Here we verify the existence of Wada basin for SNAs in a quasiperiodically forced Duffing map. We show that the SNAs? basins are full Wada for a set of parameters of positive measure. We identify two types of SNAs? Wada basins by the basin cell method. It suggests that SNAs cannot be predicted reliably for the specific initial conditions.     

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 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.     

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.     

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.     

Multifarious intertwined basin boundaries of strange nonchaotic attractors in a quasiperiodically forced system

A variety of different dynamical regimes involving strange nonchaotic attractors (SNAs) can be observed in a quasiperiodically forced delayed system. We describe some numerical experiments giving evidences of intertwined basin boundaries (smooth, non-Wada fractal and Wada property) for SNAs. In particular, we show that Wada property, fractality and smoothness can be intertwined on arbitrarily fine scales. This suggests that SNAs can exhibit the final state sensitivity and unpredictable behaviors. An interesting dynamical transition of SNAs together with associated mechanisms from non-Wada fractal to Wada intertwined basin boundaries is examined. A scaling exponent is used to characterize the intertwined basin boundaries.     

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.     

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.     

The simplest case of a strange attractor

It is shown that stochastic motion of strange attractor type may arise in a system with stable limit cycle if the perturbation of the system is periodical. Analytical and numerical analyses of the conditions for the strange attractor are developed.     

Representation of a strange attractor from an experimental study of chemical turbulence

We show that in Belousof-Zhabotinsky reaction the experimentally observed turbulence can be depicted by a three dimensional “Strange Attractor”.