scholarly journals Chaos in perturbed Lotka-Volterra systems

2001 ◽  
Vol 42 (3) ◽  
pp. 399-412
Author(s):  
J. R. Christie ◽  
K. Gopalsamy ◽  
Jibin Li
Keyword(s):  
Sufficient Conditions ◽  
Chaotic Behaviour ◽  
Volterra Equations ◽  
Unperturbed System ◽  
Heteroclinic Cycle ◽  
Volterra System ◽  
Perturbed System ◽  
Volterra Systems ◽  
Perturbed Systems ◽  
Special Case

AbstractLotka-Volterra systems have been used extensively in modelling population dynamics. In this paper, it is shown that chaotic behaviour in the sense of Smale can exist in timeperiodically perturbed systems of Lotka-Volterra equations. First, a slowly varying threedimensional perturbed Lotka-Volterra system is considered and the corresponding unperturbed system is shown to possess a heteroclinic cycle. By using Melnikov's method, sufficient conditions are obtained for the perturbed system to have a transverse heteroclinic cycle and hence to possess chaotic behaviour in the sense of Smale. Then a special case involving a reduction to a two-dimensional Lotka-Volterra system is examined, leading finally to an application with a model for the self-organisation of macromolecules.

ON THE NUMBER OF LIMIT CYCLES IN NEAR-HAMILTONIAN POLYNOMIAL SYSTEMS

2007 ◽  
Vol 17 (06) ◽  
pp. 2033-2047 ◽  
Author(s):  
MAOAN HAN ◽  
GUANRONG CHEN ◽  
CHENGJUN SUN
Keyword(s):  
Limit Cycles ◽  
Polynomial System ◽  
Sufficient Conditions ◽  
Polynomial Systems ◽  
Unperturbed System ◽  
Perturbed System ◽  
Liénard Systems ◽  
Center Point ◽  
Perturbed Systems ◽  
Almost All

In this paper we study a general near-Hamiltonian polynomial system on the plane. We suppose the unperturbed system has a family of periodic orbits surrounding a center point and obtain some sufficient conditions to find the cyclicity of the perturbed system at the center or a periodic orbit. In particular, we prove that for almost all polynomial Hamiltonian systems the perturbed systems with polynomial perturbations of degree n have at most n(n + 1)/2 - 1 limit cycles near a center point. We also obtain some new results for Lienard systems by applying our main theorems.

HETEROCLINIC BIFURCATIONS OF A PREY–PREDATOR FISHERY MODEL WITH IMPULSIVE HARVESTING

2013 ◽  
Vol 06 (05) ◽  
pp. 1350031 ◽  
Author(s):  
CHUNJIN WEI ◽  
LANSUN CHEN
Keyword(s):  
Dynamic Systems ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Heteroclinic Cycle ◽  
Heteroclinic Bifurcation ◽  
Perturbed System ◽  
State Dependent ◽  
Continuous Dynamic ◽  
Fishery Model ◽  
Impulsive Harvesting

In this paper, we consider a prey–predator fishery model with Allee effect and state-dependent impulsive harvesting. First, we investigate the existence of order-1 heteroclinic cycle. Second, choosing p as a control parameter, we obtain the sufficient conditions for the existence and uniqueness of order-1 periodic solution of system (2.3) by using the geometry theory of semi-continuous dynamic systems. Finally, on the basis of the theory of rotated vector fields, heteroclinic bifurcation to perturbed system of system (2.3) is also studied. The methods used in this paper are novel to prove the existence of order-1 heteroclinic cycle and heteroclinic bifurcations.

scholarly journals Qualitative Analyses of Differential Systems with Time-Varying Delays via Lyapunov–Krasovskiĭ Approach

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1196
Author(s):  
Cemil Tunç ◽  
Osman Tunç ◽  
Yuanheng Wang ◽  
Jen-Chih Yao
Keyword(s):  
Sufficient Conditions ◽  
Zero Solution ◽  
Unperturbed System ◽  
Time Varying ◽  
Time Varying Delay ◽  
Gronwall’S Inequality ◽  
Perturbed System ◽  
Linear Delay ◽  
Delay Differential ◽  
Varying Delay

In this paper, a class of systems of linear and non-linear delay differential equations (DDEs) of first order with time-varying delay is considered. We obtain new sufficient conditions for uniform asymptotic stability of zero solution, integrability of solutions of an unperturbed system and boundedness of solutions of a perturbed system. We construct two appropriate Lyapunov–Krasovskiĭ functionals (LKFs) as the main tools in proofs. The technique of the proofs depends upon the Lyapunov–Krasovskiĭ method. For illustration, two examples are provided in particular cases. An advantage of the new LKFs used here is that they allow to eliminate using Gronwall’s inequality. When we compare our results with recent results in the literature, the established conditions are more general, less restrictive and optimal for applications.

scholarly journals Synchronization control of stochastic delayed Lotka–Volterra systems with hardware simulation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Lan Wang ◽  
Yiping Dong ◽  
Da Xie ◽  
Hao Zhang
Keyword(s):  
Sufficient Conditions ◽  
Synchronization Control ◽  
Simulation Tool ◽  
Volterra System ◽  
Stochastic Effects ◽  
Volterra Systems ◽  
Field Programmable ◽  
Global Positive Solution ◽  
System With Time Delay ◽  
Theoretical Results

AbstractIn this paper, the synchronization control of a non-autonomous Lotka–Volterra system with time delay and stochastic effects is studied. The purpose is to firstly establish sufficient conditions for the existence of global positive solution by constructing a suitable Lyapunov function. Some synchronization criteria are then derived by designing an appropriate full controller and a pinning controller, respectively. Finally, an example is presented to illustrate the feasibility and validity of the main theoretical results based on the Field-Programmable Gate Array hardware simulation tool.

Singular perturbation of symbolic dynamics via thermodynamic formalism

2008 ◽  
Vol 28 (4) ◽  
pp. 1261-1289 ◽  
Author(s):  
TAKEHIKO MORITA ◽  
HARUYOSHI TANAKA
Keyword(s):  
Singular Perturbation ◽  
Finite Type ◽  
Symbolic Dynamics ◽  
Thermodynamic Formalism ◽  
Unperturbed System ◽  
Large Parameter ◽  
Subshift Of Finite Type ◽  
Perturbed System ◽  
Perturbed Systems

AbstractWe consider singular perturbation of a mixing subshift of finite type by means of thermodynamic formalism. In our formulation, the perturbed systems are described by a family of potentials {Φ(α,⋅)} with large parameter α on a fixed subshift of finite type, and the original (unperturbed) system is characterized as the system at infinity obtained by collapsing the perturbed system upon taking $\alpha \to \infty $. We apply our formulation to the collapse of cookie-cutter systems and dispersing open billiards.

scholarly journals Relative asymptotic equivalence of dynamic equations on time scales

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Cosme Duque ◽  
Hugo Leiva ◽  
Abdessamad Tridane
Keyword(s):  
Lyapunov Exponents ◽  
Time Scales ◽  
Sufficient Conditions ◽  
Dynamic Equations ◽  
Unperturbed System ◽  
Asymptotic Equivalence ◽  
Perturbed System ◽  
Exponential Trichotomy

AbstractThis paper aims to study the relative equivalence of the solutions of the following dynamic equations $y^{\Delta }(t)=A(t)y(t)$ y Δ ( t ) = A ( t ) y ( t ) and $x^{\Delta }(t)=A(t)x(t)+f(t,x(t))$ x Δ ( t ) = A ( t ) x ( t ) + f ( t , x ( t ) ) in the sense that if $y(t)$ y ( t ) is a given solution of the unperturbed system, we provide sufficient conditions to prove that there exists a family of solutions $x(t)$ x ( t ) for the perturbed system such that $\Vert y(t)-x(t) \Vert =o( \Vert y(t) \Vert )$ ∥ y ( t ) − x ( t ) ∥ = o ( ∥ y ( t ) ∥ ) , as $t\rightarrow \infty $ t → ∞ , and conversely, given a solution $x(t)$ x ( t ) of the perturbed system, we give sufficient conditions for the existence of a family of solutions $y(t)$ y ( t ) for the unperturbed system, and such that $\Vert y(t)-x(t) \Vert =o( \Vert x(t) \Vert )$ ∥ y ( t ) − x ( t ) ∥ = o ( ∥ x ( t ) ∥ ) , as $t\rightarrow \infty $ t → ∞ ; and in doing so, we have to extend Rodrigues inequality, the Lyapunov exponents, and the polynomial exponential trichotomy on time scales.

scholarly journals The existence of consensus of a leader-following problem with Caputo fractional derivative

2019 ◽  
Vol 39 (1) ◽  
pp. 77-89 ◽  
Author(s):  
Ewa Schmeidel
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Sufficient Conditions ◽  
Stable Solution ◽  
Volterra Equations ◽  
System Of Nonlinear Equations ◽  
Volterra System ◽  
Resolvent Kernel ◽  
Leader Following ◽  
Asymptotically Stable Solution

In this paper, consensus of a leader-following problem is investigated. The leader-following problem describes a dynamics of the leader and a number of agents. The trajectory of the leader is given. The dynamics of each agent depends on the leader trajectory and others agents' inputs. Here, the leader-following problem is modeled by a system of nonlinear equations with Caputo fractional derivative, which can be rewritten as a system of Volterra equations. The main tools in the investigation are the properties of the resolvent kernel corresponding to the Volterra equations, and Schauder fixed point theorem. By study of the existence of an asymptotically stable solution of a suitable Volterra system, the sufficient conditions for consensus of the leader-following problem are obtained.

scholarly journals On Stieltjes-Volterra integral equations

1978 ◽  
Vol 18 (3) ◽  
pp. 321-334 ◽  
Author(s):  
S.G. Pandit
Keyword(s):  
Integral Equation ◽  
Difference Equation ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Equation System ◽  
Pointwise Estimates ◽  
Volterra System ◽  
Perturbed System ◽  
The Difference ◽  
Image Position

A Stieltjes-Volterra integral equation systemis firstly considered. Pointwise estimates and boundedness of its solutions are obtained under various conditions on the function K. To do this, the well-known Gronwall-Bellman integral inequality is generalized. For a particular choice of u, it is shown that the integral equation reduces to a difference equation. The problem of existence (and non-existence), uniqueness (and non-uniqueness) of the difference equation is discussed. Gronwall-Bellman inequality is further generalized to n linear terms and is subsequently applied to obtain sufficient conditions in order that a certain stability of the unperturbed Volterra systemimplies the corresponding local stability of the (discontinuously) perturbed system

scholarly journals Complete Controllability Conditions for Linear Singularly Perturbed Time-Invariant Systems with Multiple Delays via Chang-Type Transformation

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 71 ◽  
Author(s):  
Olga Tsekhan
Keyword(s):  
Sufficient Conditions ◽  
Singularly Perturbed ◽  
Degenerate System ◽  
Multiple Delays ◽  
Complete Controllability ◽  
Singularly Perturbed System ◽  
System With Delay ◽  
Perturbed System ◽  
Time Invariant ◽  
Controllability Conditions

The problem of complete controllability of a linear time-invariant singularly-perturbed system with multiple commensurate non-small delays in the slow state variables is considered. An approach to the time-scale separation of the original singularly-perturbed system by means of Chang-type non-degenerate transformation, generalized for the system with delay, is used. Sufficient conditions for complete controllability of the singularly-perturbed system with delay are obtained. The conditions do not depend on a singularity parameter and are valid for all its sufficiently small values. The conditions have a parametric rank form and are expressed in terms of the controllability conditions of two systems of a lower dimension than the original one: the degenerate system and the boundary layer system.

PBW deformations of quantum symmetric algebras and their group extensions

2016 ◽  
Vol 15 (03) ◽  
pp. 1650049 ◽  
Author(s):  
Piyush Shroff ◽  
Sarah Witherspoon
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Extensions ◽  
Enveloping Algebra ◽  
Symmetric Algebras ◽  
Necessary And Sufficient ◽  
Special Case

We examine PBW deformations of finite group extensions of quantum symmetric algebras, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.

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