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.

The Optimal Regulator Problem for a Stationary Linear System With State-Dependent Noise

1970 ◽  
Vol 92 (2) ◽  
pp. 363-368 ◽  
Author(s):  
P. J. McLane
Keyword(s):  
Linear System ◽  
Sufficient Conditions ◽  
Quadratic Functional ◽  
Optimal Controls ◽  
Final Time ◽  
State Dependent ◽  
Continuous Dynamic ◽  
The Stability ◽  
And Control ◽  
Stationary Linear System

The problem of minimizing a quadratic functional of the system outputs and control for a stationary linear system with state-dependent noise is solved in this paper. Both the finite final time and infinite final time versions of the problem are treated. For the latter case existence conditions are obtained using the second method of Lyapunov. The optimal controls for both problems are obtained using Bellman’s continuous dynamic programming. In light of this, the system dynamics are assumed to determine a diffusion process. For the infinite final time version of the problem noted above, sufficient conditions are obtained for the stability of the optimal system and uniqueness of the optimal control law. In addition, for this problem, an example is treated. The computational results for this example illustrate some of the qualitative features of regulators for linear, stationary systems with state-dependent disturbances.

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.

Stability of Continuous Dynamic Systems With Parametric Excitation

1969 ◽  
Vol 36 (2) ◽  
pp. 212-216 ◽  
Author(s):  
J. R. Dickerson ◽  
T. K. Caughey
Keyword(s):  
Partial Differential Equations ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Dynamic Systems ◽  
Parametric Excitation ◽  
Sufficient Conditions ◽  
Partial Differential ◽  
Continuous Dynamic

A Lyapunov-type approach is used to establish sufficient conditions guaranteeing the asymptotic stability of a class of partial differential equations with parametric excitation.

scholarly journals Periodic perturbations of quadratic planar polynomial vector fields

2002 ◽  
Vol 74 (2) ◽  
pp. 193-198 ◽  
Author(s):  
MARCELO MESSIAS
Keyword(s):  
Vector Fields ◽  
Dynamical Behavior ◽  
Saddle Points ◽  
Heteroclinic Cycle ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Poincaré Compactification ◽  
Stable And Unstable Manifolds ◽  
Perturbed System ◽  
Periodic Perturbations

In this work are studied periodic perturbations, depending on two parameters, of quadratic planar polynomial vector fields having an infinite heteroclinic cycle, which is an unbounded solution joining two saddle points at infinity. The global study envolving infinity is performed via the Poincaré compactification. The main result obtained states that for certain types of periodic perturbations, the perturbed system has quadratic heteroclinic tangencies and transverse intersections between the local stable and unstable manifolds of the hyperbolic periodic orbits at infinity. It implies, via the Birkhoff-Smale Theorem, in a complex dynamical behavior of the solutions of the perturbed system, in a finite part of the phase plane.

HOMOCLINIC BIFURCATION IN SEMI-CONTINUOUS DYNAMIC SYSTEMS

2012 ◽  
Vol 05 (06) ◽  
pp. 1250059 ◽  
Author(s):  
CHUANJUN DAI ◽  
MIN ZHAO ◽  
LANSUN CHEN
Keyword(s):  
Periodic Solution ◽  
Dynamic System ◽  
Dynamic Systems ◽  
Vector Fields ◽  
Homoclinic Bifurcation ◽  
Homoclinic Bifurcations ◽  
Continuous Dynamic System ◽  
Continuous Dynamic ◽  
Theoretical Results

In this paper, a class of homoclinic bifurcations in semi-continuous dynamic systems are investigated. On the basis of rotated vector fields theory, existence of order-1 periodic solution and the rotated vector fields of the semi-continuous dynamic system are discussed. Furthermore, homoclinic cycles and homoclinic bifurcations are described. Finally, an example is provided to show the validity of our theoretical results.

Bounds on Motions of Some Lumped and Continuous Dynamic Systems

1972 ◽  
Vol 39 (1) ◽  
pp. 251-256 ◽  
Author(s):  
R. H. Plaut ◽  
E. F. Infante
Keyword(s):  
Structural Dynamics ◽  
Dynamic Systems ◽  
Sufficient Conditions ◽  
Upper Bounds ◽  
Dynamic Loads ◽  
Continuous Systems ◽  
Time Varying ◽  
Continuous Dynamic ◽  
Dynamics Problems ◽  
General Dynamic

Lumped and continuous systems subjected to general dynamic loads or perturbations are considered. The motions of these systems are assumed to be described by ordinary or partial differential equations with time-varying forcing terms. Upper bounds on the motions are derived with a Liapunov type of approach. The results are applied to some structural dynamics problems. Displacement bounds are determined for elastic columns, plates, and arches, and sufficient conditions for stability of arches against dynamic “snap-through” are obtained.

Geometric approach to the stability analysis of the periodic solution in a semi-continuous dynamic system

2014 ◽  
Vol 07 (02) ◽  
pp. 1450018 ◽  
Author(s):  
Yuan Tian ◽  
Kaibiao Sun ◽  
Lansun Chen
Keyword(s):  
Periodic Solution ◽  
Dynamic Systems ◽  
Sufficient Conditions ◽  
Geometric Theory ◽  
Geometric Approach ◽  
Stage Structure ◽  
Positive Order ◽  
Continuous Dynamic ◽  
The Stability ◽  
Order One Periodic Solution

Integrated pest management (IPM) is a long-term management strategy and has been proved to be more effective in pest control. To well-understand the mechanism and effect of the action of IPM, the geometric theory of the involved semi-continuous dynamic systems is becoming more and more important. In this work, a geometric approach is applied to analyze the stability of the positive order-one periodic solution in semi-continuous dynamic systems. A stability criterion to test the stability of the order-one periodic solution is established. As an application, a stage-structure model involved chemical control is presented to show the efficiency of the proposed method. The sufficient conditions to insure the existence of the periodic solution are provided. In addition, the number and the stability of the periodic solutions are discussed accordingly. The simulations are carried out to verify the results.

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.

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