Effect of bounded noise on the chaotic motion of a Duffing Van der pol oscillator in a ϕ6 potential

This paper investigates the chaotic behavior of an extended Duffing Van der pol oscillator in a ϕ⁶ potential under additive harmonic and bounded noise excitations for a specific parameter choice. From Melnikov theorem, we obtain the conditions for the existence of homoclinic or heteroclinic bifurcation in the case of the ϕ⁶ potential is bounded, which are complemented by the numerical simulations from which we illustrate the bifurcation surfaces and the fractality of the basins of attraction. The results show that the threshold amplitude of bounded noise for onset of chaos will move upwards as the noise intensity increases, which is further validated by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the less possible chaotic domain in parameter space. The effect of bounded noise on Poincare maps is also investigated.     

Effect of random noise on chaotic motion of a particle in a ϕ6 potential

The chaotic behaviors of a particle in a triple well ϕ⁶ potential possessing both homoclinic and heteroclinic orbits under harmonic and Gaussian white noise excitations are discussed in detail. Following Melnikov theory, conditions for the existence of transverse intersection on the surface of homoclinic or heteroclinic orbits for triple potential well case are derived, which are complemented by the numerical simulations from which we show the bifurcation surfaces and the fractality of the basins of attraction. The results reveal that the threshold amplitude of harmonic excitation for onset of chaos will move downwards as the noise intensity increases, which is further verified by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the more possible chaotic domain in parameter space. The effect of noise on Poincare maps is also investigated.     

Selection of windows, attractors and self-similar patterns in a coupled map lattice

It is shown that a lattice of diffusively coupled logistic maps displays self-similar period-doubling cascades to chaos with all the known stages of pattern formation. The location of the self-similar patterns is determined. The basins of attraction yielding window structures, so far believed to be negligibly small, are shown to cover virtually all initial conditions given a certain maximum amplitude to the random initial conditions. As a consequence a means for selecting attractors in a CML is obtained. A new pattern selection regime at high nonlinearity is reported and the basins of attraction of some attractors of small lattices are investigated.     

Partially nearly riddled basins in systems with chaotic saddle

We show that chaotic attractors can have partially nearly riddled basins of attraction, i.e., basins which consist both of large open sets and a set in which small open sets which belong to the basins of different attractors are intermingled. We argue that such basins are robust for systems with the chaotic saddle located between at least two attractors and in the presence of noise cause the uncertainties similar to those implied by riddled basins.     

Noise-induced basin hopping in a gearbox model

We investigate some dynamical effects of adding a certain amount of noise in a theoretical model for rattling in single-stage gearbox systems with a backlash, consisting of two wheels with a sinusoidal driving. The parameter intervals we are dealing with show an extremely involved attraction basin structure in phase space. One of the observable effects of noise is basin hopping, or the switching between basins of different attractors. We characterize this effect and its relation to the presence of chaotic transients.     

Noise-induced basin hopping in a gearbox model

We investigate some dynamical effects of adding a certain amount of noise in a theoretical model for rattling in single-stage gearbox systems with a backlash, consisting of two wheels with a sinusoidal driving. The parameter intervals we are dealing with show an extremely involved attraction basin structure in phase space. One of the observable effects of noise is basin hopping, or the switching between basins of different attractors. We characterize this effect and its relation to the presence of chaotic transients.     

Noninvertibility and the structure of basins of attraction in a model adaptive control system

Summary We present a two-dimensional, nonlinear map, arising from a simple adaptive control problem, which exhibits disconnected boundaries separating the basins of attraction of its coexisting attractors. We perform a detailed study of the relation between this phenomenon and the noninvertible nature of the map and demonstrate how the complex basin structure is caused by a change in the number of preimages of points along a stable manifold.     

Global analysis of boundary and interior crises in an elastic impact oscillator

The crisis phenomena of a Duffing–Van der Pol oscillator with a one-side elastic constraint are studied by the composite cell coordinate system method in this paper. By computing the global properties such as attractors, basins of attraction and saddles, the vivid evolutionary process of two kinds of crises: boundary crisis and interior crisis are shown. The boundary crisis is resulted by the collision of a chaotic attractor and a periodic saddle on the basin boundary. It is observed that there are two types of interior crises. One is caused by the collision of a chaotic attractor and a chaotic saddle within the interior of basin of attraction. The other one occurs because a period attractor collides with a chaotic saddle within the interior of basin of attraction. The saddles of system play an important role in the crisis process. The results show that this method is an efficient tool to perform the global analysis of elastic impact oscillators.     

Allee effects in a Ricker-type predator–prey system

We study a discrete host–parasitoid system where the host population follows the classical Ricker functional form and is also subject to Allee effects. We determine basins of attraction of the local attractors of the single population model when the host intrinsic growth rate is not large. In this situation, existence and local stability of the interior steady states for the host–parasitoid interaction are completely analysed. If the host's intrinsic growth rate is large, then the interaction may support multiple interior steady states. Linear stability of these steady states is provided.     

An extension of the Antoci-Dei-Galeotti evolutionary model for environment protection through financial instruments

This work moves from a recent paper by Antoci et al. (2009) [1] where a dynamic model is proposed to describe an innovative method for improving environmental quality based on the exchange of financial activities, promoted by a Public Administration, between firms and tourists in a given region. We extend their analysis in two directions: we first perform a global analysis of the basins of attraction to check the stability extents of the coexisting stable attractors of the model, and we show that some undesirable and sub-optimal stable equilibria always exist, whose basins may be quite intermingled with those of the optimal equilibrium; then we introduce a structural change of the model by assuming that the Public Administration, besides its action as an intermediary between visitors and polluting firms, also performs a direct action for the pollution control. We show how the cost of this direct action of the Public Administration can be balanced by proper taxes and we prove that undesired equilibria can be ruled out by a suitable balance of financial instruments and direct actions of Public Administration for environmental remediation.     

Homogenization of a stochastically forced Hamilton-Jacobi equation

We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation is deterministic, and, in general, the noise has an enhancement effect, for which we provide a quantitative estimate. As an application, we perform a noise sensitivity analysis for Hamilton-Jacobi equations forced by a noise term with small amplitude, and identify the scaling at which the macroscopic enhancement effect is felt. The results depend on new, probabilistic estimates for the large scale Hölder regularity of the solutions, which are of independent interest.     

On a method of evaluation of attraction domain for fixed point of nonlinear point mapping of arbitrary dimension

In this paper we describe the method of attraction domain evaluation for equilibrium states of nonlinear discrete dynamic system based on Lyapunov functions method. Attraction domain evaluation size is equilibrium state neighborhood where the first difference of Lyapunov function is negative. Lyapunov function is chosen as positive quadratic form for which the negativity of its first difference by virtue of linearized system is guaranteed with given supply. We propose the method of attraction domain extension.     

Multistability in a system of two coupled oscillators with delayed feedback

In this paper, nonlocal dynamics of a system of two differential equations with a compactly supported nonlinearity and delay is studied. For some set of initial conditions asymptotics of solutions of considered system is constructed. By this asymptotics we build a special mapping. Dynamics of this mapping describes dynamics of initial system in general: it is proved that stable cycles of this mapping correspond to exponentially orbitally stable relaxation periodic solutions of initial system of delay differential equations. It is shown that amplitude, period of solutions of initial system, and number of coexisting stable solutions depend crucially on coupling parameter. Algorithm for constructing many coexisting stable solutions is described.     

Nonlinear effects in a discrete-time dynamic model of a stock market

The time evolution of prices and savings in a stock market is modeled by a discrete time nonlinear dynamical system. The model proposed has a unique and unstable steady-state, so that the time evolution is determined by the nonlinear effects acting out of the equilibrium. The nonlinearities strongly influence the kind of long-run dynamics of the system. In particular, the global geometric properties of the noninvertible map of the plane, whose iteration gives the evolution of the system, are important to understand the global bifurcations which change the qualitative properties of the asymptotic dynamics. Such global bifurcations are studied by geometric and numerical methods based on the theory of critical curves, a powerful tool for the characterization of the global dynamical properties of noninvertible mappings of the plane. The model unfolds more complex chaotic and unpredictable trajectories as a consequence of increasing agents' “speculative” or “capital gain realizing” attitudes. The global analysis indicates that, for some ranges of the parameter values, the system has several coexisting attractors, and it may not be robust with respect to exogenous shocks due to the complexity of the basins of attraction.     

Inaccessible States in Time-Dependent Reaction Diffusion

Using asymptotic methods we show that the long-time dynamic behavior in certain systems of nonlinear parabolic differential equations is described by a time-dependent, spatially inhomogeneous nonlinear evolution equation. For problems with multiple stable states, the solution develops sharp fronts separating slowly varying regions. By studying the basins of attraction of Abel's nonlinear differential equation, we demonstrate that the presence of explicit time dependence in the asymptotic evolution equation creates “forbidden regions” where the existence of interfaces is excluded. Consequently, certain configurations of stable states in the nonlinear system become inaccessible and cannot be achieved from any set of real initial conditions.     

A discrete dynamical system as a model of a neural network with generalized Hebbian synapses

We study the dynamical behavior of a discrete time dynamical system which can serve as a model of a learning process. We determine fixed points of this system and basins of attraction of attracting points. This system was studied by Fernanda Botelho and James J. Jamison in [A learning rule with generalized Hebbian synapses, J. Math. Anal. Appl. 273 (2002) 529-547] but authors used its continuous counterpart to describe basins of attraction.     

Noise of quantum solitons and their quasi-coherent states

Quantum noise of optical solitons is analysed based on the exact solutions of the quantum nonlinear Schrödinger equation (QNSE) and the construction of the quantum soliton states. The noise limits are obtained for the local photon number and for the local quadrature phase amplitude. They are larger than the vacuum fluctuation. So in the fundamental soliton states the variance of the local photon number and the local quadrature phase amplitude cannot be squeezed. The soliton states with the minimum noise are quasi-coherent states, in which the quantum dispersion effects are negligible.     

Two characteristics of planar intertwined basins of attraction

In this paper, we investigate the intertwined basins of attraction for planar dynamical systems. We prove that the intertwining property is preserved by topologically equivalent systems. Two necessary and sufficient conditions for a planar system having intertwined basins are given.     

Dissipative relativistic standard map: Periodic attractors and basins of attraction

The dissipative relativistic standard map, introduced by Ciubotariu et al. [Ciubotariu C, Badelita L, Stancu V. Chaos in dissipative relativistic standard maps. Chaos, Solitons & Fractals 2002;13:1253–67.], is further studied numerically for small damping in the resonant case. We find that the attractors are all periodic; their basins of attraction have fractal boundaries and are closely interwoven. The number of attractors increases with decreasing damping. For a very small damping, there are thousands of periodic attractors, comprising mostly of the lowest-period attractors of period one or two; the basin of attraction of these lowest-period attractors is significantly larger compared to the basins of the higher-period attractors.