Global Analysis on a Discontinuous Dynamical System

2017 ◽  
Vol 27 (05) ◽  
pp. 1750078 ◽  
Author(s):  
Hebai Chen ◽  
Zhenbang Cao ◽  
Denghui Li ◽  
Jianhua Xie
Keyword(s):  
Dynamical System ◽  
Global Analysis ◽  
Melnikov Method ◽  
Melnikov Function ◽  
Asymptotically Stable ◽  
Subharmonic Solution ◽  
Abstract Function ◽  
Numerical Examples ◽  
Harmonic Solution ◽  
Discontinuous Dynamical System

We have studied a Filippov system [Formula: see text] with small [Formula: see text], [Formula: see text] and [Formula: see text] being periodic. Since [Formula: see text] is an abstract function, the subharmonic Melnikov function cannot be computed. In other words, for this system the Melnikov method loses effectiveness. First, we proved that the equation has a unique harmonic solution, a unique [Formula: see text]-subharmonic solution for any [Formula: see text] and they are Lyapunov asymptotically stable. Moreover, this equation has no other type of periodic solutions. Further, the attractor of this system is not chaotic. Finally, some numerical examples are given.

Positive stable realizations of discrete-time linear systems

2012 ◽  
Vol 60 (3) ◽  
pp. 605-616
Author(s):  
T. Kaczorek
Keyword(s):  
Transfer Function ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Asymptotically Stable ◽  
Numerical Examples ◽  
Discrete Time Systems ◽  
Necessary And Sufficient ◽  
Time Systems ◽  
Time Linear

Abstract The problem of existence and determination of the set of positive asymptotically stable realizations of a proper transfer function of linear discrete-time systems is formulated and solved. Necessary and sufficient conditions for existence of the set of the realizations are established. A procedure for computation of the set of realizations are proposed and illustrated by numerical examples.

scholarly journals On asymptotically stable sets

1970 ◽  
Vol 17 (2) ◽  
pp. 181-186 ◽  
Author(s):  
D. Desbrow
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Metric Space ◽  
Physical System ◽  
Asymptotically Stable ◽  
Stable Sets ◽  
Compact Closure ◽  
First Order ◽  
Closed Sets ◽  
The Future

In this paper we study closed sets having a neighbourhood with compact closure which are positively asymptotically stable under a flow on a metric space X. For an understanding of this and the rest of the introduction it is sufficient for the reader to have in mind as an example of a flow a system of first order, autonomous ordinary differential equations describing mathematically a time-independent physical system; in short a dynamical system. In a flow a set M is positively stable if the trajectories through all points sufficiently close to M remain in the future in a given neighbourhood of M. The set M is positively asymptotically stable if it is positively stable and, in addition, trajectories through all points of some neighbourhood of M approach M in the future.

Non Linear Difference Equation of Orlicz Type

2017 ◽  
Vol 14 (1) ◽  
pp. 306-313
Author(s):  
Awad. A Bakery ◽  
Afaf. R. Abou Elmatty
Keyword(s):  
Difference Equation ◽  
Equilibrium Point ◽  
Positive Solution ◽  
Orlicz Function ◽  
Sufficient Conditions ◽  
Linear Difference Equation ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Numerical Examples ◽  
The Difference

We give here the sufficient conditions on the positive solutions of the difference equation xn+1 = α+M((xn−1)/xn), n = 0, 1, …, where M is an Orlicz function, α∈ (0, ∞) with arbitrary positive initials x−1, x0 to be bounded, α-convergent and the equilibrium point to be globally asymptotically stable. Finally we present the condition for which every positive solution converges to a prime two periodic solution. Our results coincide with that known for the cases M(x) = x in Ref. [3] and M(x) = xk, where k ∈ (0, ∞) in Ref. [7]. We have given the solution of open problem proposed in Ref. [7] about the existence of the positive solution which eventually alternates above and below equilibrium and converges to the equilibrium point. Some numerical examples with figures will be given to show our results.

scholarly journals A Stochastic Cobweb Dynamical Model

2008 ◽  
Vol 2008 ◽  
pp. 1-18 ◽  
Author(s):  
Serena Brianzoni ◽  
Cristiana Mammana ◽  
Elisabetta Michetti ◽  
Francesco Zirilli
Keyword(s):  
Dynamical System ◽  
Markov Chain ◽  
Random Variable ◽  
Asymptotically Stable ◽  
Stochastic Dynamical System ◽  
Cobweb Model ◽  
Linear Demand ◽  
Discrete Values ◽  
Analytical Tools

We consider the dynamics of a stochastic cobweb model with linear demand and a backward-bending supply curve. In our model, forward-looking expectations and backward-looking ones are assumed, in fact we assume that the representative agent chooses the backward predictor with probability , and the forward predictor with probability , so that the expected price at time is a random variable and consequently the dynamics describing the price evolution in time is governed by a stochastic dynamical system. The dynamical system becomes a Markov process when the memory rate vanishes. In particular, we study the Markov chain in the cases of discrete and continuous time. Using a mixture of analytical tools and numerical methods, we show that, when prices take discrete values, the corresponding Markov chain is asymptotically stable. In the case with continuous prices and nonnecessarily zero memory rate, numerical evidence of bounded price oscillations is shown. The role of the memory rate is studied through numerical experiments, this study confirms the stabilizing effects of the presence of resistant memory.

Melnikov Method for a Class of Planar Hybrid Piecewise-Smooth Systems

2016 ◽  
Vol 26 (02) ◽  
pp. 1650030 ◽  
Author(s):  
Shuangbao Li ◽  
Wensai Ma ◽  
Wei Zhang ◽  
Yuxin Hao
Keyword(s):  
Melnikov Method ◽  
Homoclinic Solution ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Piecewise Smooth Systems ◽  
Stable And Unstable Manifolds ◽  
Straight Line ◽  
Unstable Manifolds ◽  
Smooth System

In this paper, we extend the well-known Melnikov method for smooth systems to a class of periodic perturbed planar hybrid piecewise-smooth systems. In this class, the switching manifold is a straight line which divides the plane into two zones, and the dynamics in each zone is governed by a smooth system. When a trajectory reaches the separation line, then a reset map is applied instantaneously before entering the trajectory in the other zone. We assume that the unperturbed system is a piecewise Hamiltonian system which possesses a piecewise-smooth homoclinic solution transversally crossing the switching manifold. Then, we study the persistence of the homoclinic orbit under a nonautonomous periodic perturbation and the reset map. To achieve this objective, we obtain the Melnikov function to measure the distance of the perturbed stable and unstable manifolds and present the theorem for homoclinic bifurcations for the class of planar hybrid piecewise-smooth systems. Furthermore, we employ the obtained Melnikov function to detect the chaotic boundaries for a concrete planar hybrid piecewise-smooth system.

Dynamics of Morse-Smale urn processes

1995 ◽  
Vol 15 (6) ◽  
pp. 1005-1030 ◽  
Author(s):  
Michel Benaïm ◽  
Morris W. Hirsch
Keyword(s):  
Dynamical System ◽  
Sample Path ◽  
Asymptotically Stable ◽  
Martingale Differences ◽  
Stable Periodic Orbit ◽  
Sample Paths ◽  
Stable Periodic ◽  
Image Position ◽  
Deterministic Dynamical System ◽  
Uniformly Bounded

AbstractWe consider stochastic processes {xn}n≥0 of the formwhere F: ℝm → ℝm is C2, {λi}i≥1 is a sequence of positive numbers decreasing to 0 and {Ui}i≥1 is a sequence of uniformly bounded ℝm-valued random variables forming suitable martingale differences. We show that when the vector field F is Morse-Smale, almost surely every sample path approaches an asymptotically stable periodic orbit of the deterministic dynamical system dy/dt = F(y). In the case of certain generalized urn processes we show that for each such orbit Γ, the probability of sample paths approaching Γ is positive. This gives the generic behavior of three-color urn models.

GLOBAL CONTACT DYNAMICS OF AN ICE-STRUCTURE INTERACTION MODEL

1992 ◽  
Vol 02 (03) ◽  
pp. 607-620 ◽  
Author(s):  
ARMIN W. TROESCH ◽  
DALE G. KARR ◽  
KLAUS-PETER BEIER
Keyword(s):  
Dynamical System ◽  
Global Analysis ◽  
Ice Sheet ◽  
Interaction Model ◽  
Contact Dynamics ◽  
Total System ◽  
Numerical Techniques ◽  
Analogue Model ◽  
Structure Interaction ◽  
Basin Boundaries

The interaction between a moving ice sheet and an elastic structure is studied using the analogue model of Matlock, et al. [1969]. The ice sheet is represented by a series of teeth with bilinear, discontinuous stiffness. A global analysis of the resulting dynamical system is performed. Using a combination of analytical and numerical techniques, periodic solutions are determined and basin boundaries of the Poincaré map identified. While the total system dynamics are quite complex, two types of threshold solutions are found, each necessary but not sufficient in defining local separatrices.

On the Global Analysis of a Piecewise Linear System that is excited by a Gaussian White Noise

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
Chen Kong ◽  
Xue Gao ◽  
Xianbin Liu
Keyword(s):  
Dynamical System ◽  
Averaging Method ◽  
Global Analysis ◽  
Piecewise Linear ◽  
Gaussian White Noise ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Random Dynamical System ◽  
Piecewise Linear System ◽  
Dynamical Behaviors

The global analysis is very important for a nonlinear dynamical system which possesses a chaotic saddle and a nonchaotic attractor, especially for the one that is driven by a noise. For a random dynamical system, within which, chaotic saddles exist, it is found that if the noise intensity exceeds a critical value, the so called “noise-induced chaos” is observed. Meanwhile, the exit behavior is also found to be influenced significantly by the existence of chaotic saddles. In the present paper, based on the generalized cell-mapping digraph (GCMD) method, the global dynamical behaviors of a piecewise linear system, wherein a chaotic saddle exists and consists of subharmonic solutions in a wide frequency range, are investigated numerically. Further, in order to simplify the system that is driven by a Gaussian white noise excitation, the stochastic averaging method is applied and through which, a five-dimensional Itô system is obtained. Some of the global dynamical behaviors of the original system are retained in the averaged one and then are analyzed. The researches in this paper show that GCMD method is a good numerical tool to investigate the global behaviors of a nonlinear random dynamical system, and the stochastic averaging method is effective for solving the global problems.

High-Order Melnikov Method for Time-Periodic Equations

2017 ◽  
Vol 17 (4) ◽  
pp. 793-818 ◽  
Author(s):  
Fengjuan Chen ◽  
Qiudong Wang
Keyword(s):  
Explicit Formula ◽  
Melnikov Method ◽  
Melnikov Function ◽  
High Order ◽  
New Method ◽  
Integral Formulas ◽  
Multiple Integrals ◽  
Time Periodic

AbstractThis paper discusses a high-order Melnikov method for periodically perturbed equations. We introduce a new method to compute {M_{k}(t_{0})} for all {k\geq 0}, among which {M_{0}(t_{0})} is the traditional Melnikov function, and {M_{1}(t_{0}),M_{2}(t_{0}),\ldots\,} are its high-order correspondences. We prove that, for all {k\geq 0}, {M_{k}(t_{0})} is a sum of certain multiple integrals, the integrand of which we can explicitly compute. In particular, we obtain explicit integral formulas for {M_{0}(t_{0})} and {M_{1}(t_{0})}. We also study a concrete equation for which the explicit formula of {M_{1}(t_{0})} is used to prove the existence of a transversal homoclinic intersection in the case of {M_{0}(t_{0})\equiv 0}.

scholarly journals Global Analysis of Almost Periodic Solution of a Discrete Multispecies Mutualism System

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Hui Zhang ◽  
Bin Jing ◽  
Yingqi Li ◽  
Xiaofeng Fang
Keyword(s):  
Periodic Solution ◽  
Global Analysis ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Asymptotically Stable ◽  
Almost Periodic ◽  
Periodic Sequences ◽  
Almost Periodic Sequences ◽  
Globally Attractive ◽  
Mutualism System

This paper discusses a discrete multispecies Lotka-Volterra mutualism system. We first obtain the permanence of the system. Assuming that the coefficients in the system are almost periodic sequences, we obtain the sufficient conditions for the existence of a unique almost periodic solution which is globally attractive. In particular, for the discrete two-species Lotka-Volterra mutualism system, the sufficient conditions for the existence of a unique uniformly asymptotically stable almost periodic solution are obtained. An example together with numerical simulation indicates the feasibility of the main result.

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