Attractors Of Dynamical Systems Research Articles

СКРЫТЫЕ АТТРАКТОРЫ НЕКОТОРЫХ МУЛЬТИСТАБИЛЬНЫХ СИСТЕМ С БЕСКОНЕЧНЫМ ЧИСЛОМ СОСТОЯНИЙ РАВНОВЕСИЯ

It is well known that mathematically simple systems of nonlinear differential equations can exhibit chaotic behavior. Detection of attractors of chaotic systems is an important problem of nonlinear dynamics. Results of recent researches have made it possible to introduce the following classification of periodic and chaotic attractors depending on the presence of neighborhood of equilibrium into their basin of attraction – self-excited and hidden attractors. The presence of hidden attractors in dynamical systems has received considerable attention to both theoretical and applied research of this phenomenon. Revealing of hidden attractors in real engineering systems is extremely important, because it allows predicting the unexpected and potentially dangerous system response to perturbations in its structure. In the past three years after discovering by S. Jafari and J. C. Sprott chaotic system with a line and a plane of equilibrium with hidden attractors there has been much attention to systems with uncountable or infinite equilibria. In this paper it is offered new models of control systems with an infinite number of equilibrium possessing hidden chaotic attractors: a piecewise-linear system with a locally stable segment of equilibrium and a system with periodic nonlinearity and infinite number of equilibrium points. The original analytical-numerical method developed by the author is applied to search hidden attractors in investigated systems.

Assessing causal links in fish stock–recruitment relationships

Abstract Understanding whether recruitment fluctuations in fish stock arise from stochastic forcing (e.g. environmental variations) rather than deterministic forces (e.g. intrinsic dynamics) is a long standing question with important applied consequences for fisheries ecology. In particular, the relationship between recruitment, spawning stock biomass and environmental factors is still poorly understood, even though this aspect is crucial for fisheries management. Fisheries data are often short, but arise from complex dynamical systems with a high degree of stochastic forcing, which are difficult to capture through classic modelling approaches. In the present study, recent statistical approaches based on the approximation of the attractors of dynamical systems are applied on a large dataset of time series to assess (i) the directionality of potential causal relationships between recruitment and spawning stock biomass and potential influence of sea-surface temperature on recruitment and (ii) their performance to forecast recruitment. Our study shows that (i) whereas spawning stock biomass and sea surface temperature influence the recruitment to a lesser extent, recruitment causes also parental stock size and (ii) that non-linear forecasting methods performed well for the short-term predictions of recruitment time series. Our results underline that the complex and stochastic nature of the processes characterizing recruitment are unlikely to be captured by classical stock–recruitment relationships, but that non-linear forecasting methods provide interesting perspectives in that respect.

Perpetual Points: New Tool for Localization of Coexisting Attractors in Dynamical Systems

Perpetual points (PPs) are special critical points for which the magnitude of acceleration describing the dynamics drops to zero, while the motion is still possible (stationary points are excluded), e.g. considering the motion of the particle in the potential field, at perpetual point, it has zero acceleration and nonzero velocity. We show that using PPs we can trace all the stable fixed points in the system, and that the structure of trajectories leading from former points to stable equilibria may be similar to orbits obtained from unstable stationary points. Moreover, we argue that the concept of perpetual points may be useful in tracing unexpected attractors (hidden or rare attractors with small basins of attraction). We show potential applicability of this approach by analyzing several representative systems of physical significance, including the damped oscillator, pendula, and the Henon map. We suggest that perpetual points may be a useful tool for localizing coexisting attractors in dynamical systems.

A Chaotic System with Different Families of Hidden Attractors

The presence of hidden attractors in dynamical systems has received considerable attention recently both in theory and applications. A novel three-dimensional autonomous chaotic system with hidden attractors is introduced in this paper. It is exciting that this chaotic system can exhibit two different families of hidden attractors: hidden attractors with an infinite number of equilibrium points and hidden attractors without equilibrium. Dynamical behaviors of such system are discovered through mathematical analysis, numerical simulations and circuit implementation.

Quantifying Noisy Attractors: From Heteroclinic to Excitable Networks

Attractors of dynamical systems may be networks in phase space that can be heteroclinic (where there are dynamical connections between simple invariant sets) or excitable (where a perturbation threshold needs to be crossed to a dynamical connection between “nodes''). Such network attractors can display a high degree of sensitivity to noise both in terms of the regions of phase space visited and in terms of the sequence of transitions around the network. The two types of network are intimately related---one can directly bifurcate to the other. In this paper we attempt to quantify the effect of additive noise on such network attractors. Noise increases the average rate at which the networks are explored and can result in “macroscopic” random motion around the network. We perform an asymptotic analysis of local behavior of an escape model near heteroclinic/excitable nodes in the limit of noise $\eta\rightarrow 0^+$ as a model for the mean residence time $T$ near equilibria. The heteroclinic network case has $T$ proportional to $-\ln\eta$ while the excitable network has $T$ given by a Kramers' law, proportional to $\exp (B/\eta^2)$. There is singular scaling behavior (where $T$ is proportional to $1/\eta$) at the bifurcation between the two types of network. We also explore transition probabilities between nodes of the network in the presence of anisotropic noise. For low levels of noise, numerical results suggest that a (heteroclinic or excitable) network can approximately realize any set of transition probabilities and any sufficiently large mean residence times at the given nodes. We show that this can be well modeled in our example network by multiple independent escape processes, where the direction of first escape determines the transition. This suggests that it is feasible to design noisy network attractors with arbitrary Markov transition probabilities and residence times.

Classifying and quantifying basins of attraction.

A scheme is proposed to classify the basins for attractors of dynamical systems in arbitrary dimensions. There are four basic classes depending on their size and extent, and each class can be further quantified to facilitate comparisons. The calculation uses a Monte Carlo method and is applied to numerous common dissipative chaotic maps and flows in various dimensions.

Perpetual points and hidden attractors in dynamical systems

We discuss the use of perpetual points for tracing the hidden and the rare attractors of dynamical systems. The analysis of perpetual points and their co-existence due to the parameters values is presented and the impact of these points on the behavior of the systems is shown. The results are obtained for single as well as coupled externally excited van der Pol–Duffing oscillators. The presented results can be generalized to other systems having different dynamics.

The number of attractors in dynamical systems associated with cycles

The number of attractors in dynamical systems of binary vectors associated with cycles is calculated.

Hidden attractors in dynamical systems: systems with no equilibria, multistability and coexisting attractors

From a computational point of view it is natural to suggest the classification of attractors, based on the simplicity of finding basin of attraction in the phase space: an attractor is called a hidden attractor if its basin of attraction does not intersect with small neighborhoods of equilibria, otherwise it is called a self-excited attractor. Self-excited attractors can be localized numerically by the standard computational procedure, in which after a transient process a trajectory, started from a point of unstable manifold in a neighborhood of unstable equilibrium, is attracted to the state of oscillation and traces it. Thus one can determine unstable equilibria and check the existence of self-excited attractors. In contrast, for the numerical localization of hidden attractors it is necessary to develop special analytical-numerical procedures in which an initial point can be chosen from the basin of attraction analytically. For example, hidden attractors are attractors in the systems with no-equilibria or with the only stable equilibrium (a special case of multistability and coexistence of attractors); hidden attractors arise in the study of well-known fundamental problems such as 16th Hilbert problem, Aizerman and Kalman conjectures, and in applied research of Chua circuits, phase-locked loop based circuits, aircraft control systems, and others.

Regular attractors of autonomous and nonautonomous dynamical systems

This paper studies regular global attractors of the dynamical systems corresponding to dissipative evolu� tion equations and their nonautonomous perturba� tions. We prove that the nonautonomous dynamical system (process) resulting from a small nonautono� mous perturbation of an autonomous dynamical sys� tem (semigroup) having a regular attractor has a regu� lar nonautonomous attractor as well. Moreover, the symmetric Hausdorff deviation of the perturbed attractors from the unperturbed ones is bounded above by O(e κ ), where e is a perturbation parameter and 0 < κ < 1. We apply the obtained results to weakly dissipa� tive wave equations in a bounded domain in 3 per�

The Exit Problem from a Neighborhood of the Global Attractor for Dynamical Systems Perturbed by Heavy-Tailed Lévy Processes

We consider a finite-dimensional deterministic dynamical system with the global attractor 𝒜 which supports a unique ergodic probability measure P. The measure P can be considered as the uniform long-term mean of the trajectories staying in a bounded domain D containing 𝒜. We perturb the dynamical system by a multiplicative heavy tailed Lévy noise of small intensity ϵ > 0 and solve the asymptotic first exit time and location problem from D in the limit of ϵ↘0. In contrast to the case of Gaussian perturbations, the exit time has an algebraic exit rate as a function of ϵ, just as in the case when 𝒜 is a stable fixed point studied earlier in [9, 14, 19, 26]. As an example, we study the first exit problem from a neighborhood of the stable limit cycle for the Van der Pol oscillator perturbed by multiplicative α-stable Lévy noise.

INTERTWINED BASINS OF ATTRACTION

This paper deals with intertwined basins of attraction for dynamical systems in a metric space. After giving a general definition of intertwining property, which is preserved by a topological equivalence between dynamical systems, we present a sufficient condition to guarantee the existence of intertwined basins for dynamical systems in ℝn.

Chaotic Behavior and Strange Attractors in Dynamical Systems

In this paper, we study the invariant set of dynamical systems in which attractor and non-attractor sets exist. We aim to carve out a small section of the theory of chaotic dynamical systems - that of attractors - and outline its fundamental concepts from a computational mathematics perspective. The motivation for this paper is primarily to define what an attractor is and to clarify what distinguishes its various types (non-strange, strange non-chaotic, and strange chaotic). We discuss the Henon and Lorenz attractors as important examples of this type of chaotic system.

Аттракторы динамических систем, ассоциированных с циклами

Аттракторы динамических систем, ассоциированных с циклами

Human like trajectories for humanoid robots

Human like trajectories for humanoid robots

CRITERIA OF TURBULENT TRANSITION IN PARALLEL FLOWS

Based on the energy gradient method, criteria for turbulent transition are proposed for pressure driven flow and shear driven flow, respectively. For pressure driven flow, the necessary and sufficient condition for turbulent transition is the presence of the velocity inflection point in the averaged flow. For shear driven flow, the necessary and sufficient condition for turbulent transition is the existence of zero velocity gradient in the averaged flow profile. It is shown that turbulent transition can be effected via a singularity of the energy gradient function which may be associated with the chaotic attractor in dynamic system. The role of disturbance in the transition is also clarified in causing the energy gradient function to approach the singularity. Finally, it is interesting that turbulence can be controlled by modulating the distribution of the energy gradient function.

Nash Equilibria in Multi-Agent Motor Interactions

Social interactions in classic cognitive games like the ultimatum game or the prisoner's dilemma typically lead to Nash equilibria when multiple competitive decision makers with perfect knowledge select optimal strategies. However, in evolutionary game theory it has been shown that Nash equilibria can also arise as attractors in dynamical systems that can describe, for example, the population dynamics of microorganisms. Similar to such evolutionary dynamics, we find that Nash equilibria arise naturally in motor interactions in which players vie for control and try to minimize effort. When confronted with sensorimotor interaction tasks that correspond to the classical prisoner's dilemma and the rope-pulling game, two-player motor interactions led predominantly to Nash solutions. In contrast, when a single player took both roles, playing the sensorimotor game bimanually, cooperative solutions were found. Our methodology opens up a new avenue for the study of human motor interactions within a game theoretic framework, suggesting that the coupling of motor systems can lead to game theoretic solutions.

Intertwined basins of attraction of dynamical systems

In this paper, we investigate a kind of intertwining phenomenon and give a new definition of intertwined attractors of the dynamical systems and get some related results.

Anomaly of fractal dimensions observed in stochastically switched systems

We studied an anomaly in fractal dimensions measured from the attractors of dynamical systems driven by stochastically switched inputs. We calculated the dimensions for different switching time lengths in two-dimensional linear dynamical systems, and found that changes in the dimensions due to the switching time length had a singular point when the system matrix had two different real eigenvalues. Using partial dimensions along each eigenvector, we explicitly derived a generalized dimension D(q) and a multifractal spectrum f(alpha) to explain this anomalous property. The results from numerical calculations agreed well with those from analytical equations. We found that this anomaly is caused by linear independence, inhomogeneity of eigenvalues, and overlapping conditions. The mechanism for the anomaly could be identified for various inhomogeneous systems including nonlinear ones, and this reminded us of anomalies in some physical values observed in critical phenomena.

Approximation of solutions to nonregular nonlinear equations by attractors of dynamic systems in a Banach space

1. PROBLEM DEFINITION Consider the equation F (x) = 0, x ∈ X, (1.1) where F : X → X is a nonlinear operator, acting in a Banach space X. Assume that the space X is reflexive and has a Gateaux differentiable norm. The simplest examples of such spaces are lp, Lp with 1 < p < ∞, as well as the Hilbert spaces ([1], pp. 35–37). Denote by x∗ a solution to equation (1.1) and put ΩR(x) = {z ∈ X : ‖z − x‖ < R}. Assume that the operator F is Frechet differentiable and ‖F (x1)− F (x2)‖ ≤ L‖x1 − x2‖ ∀x1, x2 ∈ ΩR(x). (1.2) Hereinafter the symbol ‖ · ‖ stands for the norms of various spaces. The derivative F ′(x) is not necessarily boundedly invertible, so the initial problem (1.1) is, generally, ill-posed. Heeding the fact that F (x) is defined inaccurately, a stable approximation of solutions to the problem is based on regularization methods (see, e. g., [2], pp. 126–128; [3], pp. 9–11; [4], pp. 15–18). Assume that instead of the exact operator F in (1.1) we know only its approximation F : X → X; let the latter be Frechet differentiable and satisfy inequality (1.2) (with the same constant L) and the conditions ‖F (x∗)‖ ≤ δ; ‖F ′(x)− F ′(x)‖ ≤ δ ∀x ∈ ΩR(x). (1.3)