Search results for " DYNAMICAL SYSTEM"

Internal perturbations of homoclinic classes:non-domination, cycles, and self-replication

2010

Conditions are provided under which lack of domination of a homoclinic class yields robust heterodimensional cycles. Moreover, so-called viral homoclinic classes are studied. Viral classes have the property of generating copies of themselves producing wild dynamics (systems with infinitely many homoclinic classes with some persistence). Such wild dynamics also exhibits uncountably many aperiodic chain recurrence classes. A scenario (related with non-dominated dynamics) is presented where viral homoclinic classes occur. A key ingredient are adapted perturbations of a diffeomorphism along a periodic orbit. Such perturbations preserve certain homoclinic relations and prescribed dynamical prope…

The Lyapunov dimension, convergency and entropy for a dynamical model of Chua memristor circuit

2018

For the study of chaotic dynamics and dimension of attractors the concepts of the Lyapunov exponents was found useful and became widely spread. Such characteristics of chaotic behavior, as the Lyapunov dimension and the entropy rate, can be estimated via the Lyapunov exponents. In this work an analytical approach to the study of the Lyapunov dimension, convergency and entropy for a dynamical model of Chua memristor circuit is demonstrated.

Estimation of Lyapunov dimension for the Chen and Lu systems

2015

Nowadays various estimates of Lyapunov dimension of Lorenz-like systems attractors are actively developed. Within the frame of this study the question arises whether it is possible to obtain the corresponding estimates of dimension for the Chen and Lu systems using the reduction of them to the generalized Lorenz system. In the work (Chen and Yang, 2013) Leonov's method was applied for the estimation of Lyapunov dimension, and as a consequence the Lyapunov dimension of attractors of the Chen and Lu systems with the classical parameters was estimated. In the present work an inaccuracy in (Chen and Yang, 2013) is corrected and it is shown that the revised domain of parameters, where the estima…

Up, down, two-sided Lorenz attractor, collisions, merging and switching

2021

We present a slightly modified version of the well known "geometric Lorenz attractor". It consists in a C1 open set O of vector fields in R3 having an attracting region U containing: (1) a unique singular saddle point sigma; (2) a unique attractor Lambda containing the singular point; (3) the maximal invariant in U contains at most 2 chain recurrence classes, which are Lambda and (at most) one hyperbolic horseshoe. The horseshoe and the singular attractor have a collision along the union of 2 co-dimension 1 sub-manifolds which divide O in 3 regions. By crossing this collision locus, the attractor and the horseshoe may merge in a two-sided Lorenz attractor, or they may exchange their nature:…

Non-linear systems under parametric a-stable Levý white noises

2005

Vehicular Motion and Traffic Breakdown: Evaluation of Energy Balance

2009

Microscopic traffic models based on follow–the–leader behaviour are strongly asymmetrically interacting many–particle systems. The well–known Bando’s optimal velocity model includes the fact that (firstly) the driver is always looking forward interacting with the lead vehicle and (secondly) the car travels on the road always with friction. Due to these realistic assumptions the moving car needs petrol for the engine to compensate dissipation by rolling friction. We investigate the flux of mechanical energy to evaluate the energy balance out of the given nonlinear dynamical system of vehicular particles. In order to understand the traffic breakdown as transition from free flow to congested t…

Robust regulation with an H<inf>∞</inf> constrain for linear two-time scale systems

2010

In this paper, the problem of robust multi-objective control design with an H ∞ constrain is studied for a class of linear two-time scale systems. The design is based on a new modelling approach under the assumption of norm-boundedness of the fast dynamics. In this method, a portion of the fast dynamics is treated as a norm-bounded perturbation in the design by its maximum possible gain. In this view, the problem of robust multi-objective control design is performed only for the certain dynamics of the two-time scale system, whose order is less than that of the original system. One illustrative example is used to demonstrate the validity of the proposed approach.

Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity

2011

In this article, we study the asymptotic behaviour of solutions of a first-order stochastic lattice dynamical system with an additive noise. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions so that uniqueness of the Cauchy problem fails to be true. Using the theory of multi-valued random dynamical systems, we prove the existence of a random compact global attractor.

Euler integral as a source of chaos in the three–body problem

2022

In this paper we address, from a purely numerical point of view, the question, raised in [20, 21], and partly considered in [22, 9, 3], whether a certain function, referred to as "Euler Integral", is a quasi-integral along the trajectories of the three-body problem. Differently from our previous investigations, here we focus on the region of the "unperturbed separatrix", which turns to be complicated by a collision singularity. Concretely, we reduce the Hamiltonian to two degrees of freedom and, after fixing some energy level, we discuss in detail the resulting three-dimensional phase space around an elliptic and an hyperbolic periodic orbit. After measuring the strength of variation of the…

Pattern formation driven by cross–diffusion in a 2D domain

2012

Abstract In this work we investigate the process of pattern formation in a two dimensional domain for a reaction–diffusion system with nonlinear diffusion terms and the competitive Lotka–Volterra kinetics. The linear stability analysis shows that cross-diffusion, through Turing bifurcation, is the key mechanism for the formation of spatial patterns. We show that the bifurcation can be regular, degenerate non-resonant and resonant. We use multiple scales expansions to derive the amplitude equations appropriate for each case and show that the system supports patterns like rolls, squares, mixed-mode patterns, supersquares, and hexagonal patterns.