Attractors of convex maps with positive Schwarzian derivative in the presence of noise

One-dimensional maps have proved to be useful models for understanding the transition to turbulence. We investigate a smooth perturbation of the well-known logistic system in order to examine numerically the change in the bifurcation behavior which is observed, when the Schwarzian derivative is allowed to become positive. We find coexistence of a fixed point attractor and a periodic or chaotic two-band-attractor. The chaotic two-band attractor can disappear by yielding a preturbulent state which will asymptotically settle down to a fixed-point. The chaotic behavior of some systems can be destroyed by arbitrarily small amounts of external noise. The concept of (ε, δ)-diffusions is used to describe the sensitivity of attractors against external noise. We also observe a direct transition from a fixed-point to a chaotic one-band attractor. This can be interpreted as type-III-intermittency of Pomeau and Manneville but with an almost linear scaling behavior of the Lyapunov exponent.     

Integrability aspects of a Schwarzian PDE

We present a reduction of the anti-self-dual Yang–Mills (ASDYM) equations to a system of partial differential equations (PDEs) introduced recently by Nijhoff et al. (Phys. Lett. A 267 (2000) 147). An auto-Bäcklund transformation of the reduced system is also presented. The system under consideration is related to a fourth-order nonlinear PDE of the Schwarzian type. The symmetry group of the latter equation is calculated and similarity reductions to the Schwarzian equation and the full Painlevé III, V and VI are presented.     

沟道效应的运动阻尼与系统走向混沌的临界特征

利用正弦平方势,把粒子运动方程化为具有运动阻尼和周期调制的摆方程.利用Melnikov方法分析了系统的全局分叉和混沌行为，指出了沟道辐射本底增强和沟道效应的无规行为与混沌之间的相关性. 关键词： 沟道效应 摆方程 分叉 混沌     

Experimental evidence for chaotic behaviour of a parametrically forced pendulum

Experimental evidence is presented for chaotic type nonperiodic motions of a parametrically forced pendulum. A bifurcation diagram is measured directly, showing successive subharmonic bifurcations to ?/4, onset of a periodic motion and the appearance of periodic motions via intermittency. The experimentally determined threshold values of the amplitude of the driving force for the first period doublings and the onset of a periodic motion are found to be in good agreement with the theoretical predictions.     

Simulation of chaotic behaviour in population dynamics

In the time evolution of populations, many attractors can be found: fixed points, limit cycles and chaotic regimes. Usually, chaotic behaviour is observed in species which have well defined breeding seasons and a high fertility rate. Different mathematical models have been used in order to simulate those regimes. In this paper, we use the bitstring model introduced to simulate the evolution of age-structured populations -- the Penna Model -- to simulate a sort of cyclic and chaotic behaviours. In comparison with the standard logistic map, our results show a time changing parameter. Received: 5 August 1997 / Revised: 8 November 1997 / Accepted: 13 November 1997     

Approximation-error-ADP-based optimal tracking control for chaotic systems with convergence proof

In this paper, an optimal tracking control scheme is proposed for a class of discrete-time chaotic systems using the approximation-error-based adaptive dynamic programming （ADP） algorithm. Via the system transformation, the optimal tracking problem is transformed into an optimal regulation problem, and then the novel optimal tracking control method is proposed. It is shown that for the iterative ADP algorithm with finite approximation error, the iterative performance index functions can converge to a finite neighborhood of the greatest lower bound of all performance index functions under some convergence conditions. Two examples are given to demonstrate the validity of the proposed optimal tracking control scheme for chaotic systems.     

The problem of chaotic behaviour in a homogeneous arbitrarily dimensional cosmology

The problem of chaotic behaviour in multidimensional mixmaster models is discussed. We show that the chaotic regime disappears in the full class of homogeneous arbitrarily dimensional cosmological models if the dimension of space is n≠3.     

Nonlinear/chaotic behaviour in thermo-acoustic instability

This paper is concerned with the dynamic system nonlinear behaviour encountered in classical thermo-acoustic instability. The Poincaré map is adopted to analyse the stability of a simple non-autonomous system considering a harmonic oscillation behaviour for the combustion environment. The bifurcation diagram of a one-mode model is obtained where the analysis reveals a variety of chaotic behaviours for some select ranges of the bifurcation parameter. The bifurcation parameter and the corresponding period of a two-mode dynamic model are calculated using both analytical and numerical methods. The results computed by different methods are in good agreement. In addition, the dependence of the bifurcation parameter and the period on all the relevant coefficients in the model is investigated in depth.     

Period multiplication and chaotic behaviour in a driven non-linear piezoelectric resonator

Experimental observations show that the frequencies of piezoelectrically active vibrational resonances of Rochelle salt can be tuned over a 2 : 1 range by an applied voltage. At high driving amplitudes, subharmonics of the driving frequency occur, accompanied by regions of chaotic behaviour, due to the non-linear nature of the resonant system.     

Competing Ising interactions and chaotic glass-like behaviour on a Cayley tree

For the Ising model on a Cayley tree with competing nearest neighbour coupling J and next nearest neighbour coupling J′, we find in addition to the expected paramagnetic, ferromagnetic and antiferromagnetic phases, an intermediate range of J′/J < 0 values where the local magnetization has chaotic oscillatory glass-like behaviour.     

Wave chaotic behaviour generated by linear systems

It is shown that regimes with dynamical chaos are inherent not only to nonlinear system but they can be generated by initially linear systems and the requirements for chaotic dynamics and characteristics need further elaboration. Three simplest physical models are considered as examples. In the first, dynamic chaos in the interaction of three linear oscillators is investigated. Analogous process is shown in the second model of electromagnetic wave scattering in a double periodical inhomogeneous medium occupying half-space. The third model is a linear parametric problem for the electromagnetic field in homogeneous dielectric medium which permittivity is modulated in time.     

Regular and chaotic behaviour of multimode gas lasers

The build-up of multimode gas laser spectra is studied on the basis of the full system of Lamb's equations of motion. Comparison is made with the results of the phase-averaged equations that constitute the so-called free-running approximation. It turns out that the range of validity of the latter is approximately given byr?0.5 wherer is the ratio of the homogeneous linewidth and the mode spacing (provided the former is mainly due to the decay of both the upper and the lower level). However, forr?2, the free-running approximation which generally predicts a steady-state spectrum to occur, breaks down. Actually, in this case multimode oscillation of a chaotic character takes place, and no steady-state regime is attained.     

On the Feigenbaum-Cvitanović equation in the theory of chaotic behaviour

We propose an analytic perturbative approach for the determination of the Feigenbaum-Cvitanović function and the universal parameterα occurring in the Feigenbaum scenario of period doubling for approach to chaotic behaviour. We apply the method to the caseZ=2 whereZ is the order of the unique local maximum of the nonlinear map. Our third order approximation givesα=2.5000 as compared to “exact” numerical valueα=2.5029 ... We also obtain a reasonably accurate value of the Feigenbaum-Cvitanović function.     

1-cocycles on the group of contactomorphisms on the supercircles S1|1 and S1|2 generalizing the Schwarzian derivative

We give an explicit construction of the $1$-cocycles of the group of contactomorphisms on the supercircle $S^{1|m}$ for $m=1,2$, with coefficients in the space of differential operators acting on $S^{1|m}$-tensor densities. We show that they satisfy properties similar to those of the super-Schwarzian derivative.     

Temporal chaotic behaviour of vortex motion in a type-II superconductors with periodically-distributed pinning centres

Temporal chaotic character of vortex motion in systems where defects are arranged in periodic arrays has been investigated by computer simulation. Due to the high nonlinearity of the vortex–defect interaction, the temporal evolution of the vortex motion is chaotic with a power spectrum similar to what have been observed in the experiments. It is found that the strength of both the vortex–vortex and vortex–defect interactions have no significant effects on the chaotic motion of the vortices, however, the mismatch between these two interactions causes attractor crisis of the system. Different from them, the Lorentz force is not the origin of the attractor crisis, but it causes a divergent motion of the vortex (i.e., the flux flow).     

The Painlevé property,integrability and chaotic behaviour of a two-coupled Duffing oscillators

Integrability and chaotic behaviour in a two-coupled Duffing oscillators are studied. The coupling is nonlinear. Painlevé test is performed to identify integrable cases of damped- and force-free system. Exact analytical solutions are given for the integrable cases. Effect of external periodic forces for (i) single well with infinite height potential, (ii) potential with a hump at the centre and (iii) single well with finite height hump potential are numerically investigated. Occurrence of multiple attractors and period doubling cascades of coexisting attractors is presented.     

Period-doubling bifurcation and chaotic behaviour in non-equilibrium superconducting film

Based on the theory of a previous letter [2], the possibility of period-doubling bifurcations and chaotic behaviour appearing in superconducting film under high quasiparticle injection and applied periodic field is argued. The complicated behaviour of system and the corresponding region of parameter are given.