scholarly journals On a conjecture for three-dimensional competitive Lotka-Volterra systems with a heteroclinic cycle

2009 ◽  
pp. 473-490 ◽  
Author(s):  
Mats Gyllenberg ◽  
Ping Yan
Keyword(s):  
Three Dimensional ◽  
Heteroclinic Cycle ◽  
Volterra Systems

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.

Bifurcating bright and dark solitary waves for the perturbed cubic-quintic nonlinear Schrödinger equation

1998 ◽  
Vol 128 (3) ◽  
pp. 585-629 ◽  
Author(s):  
Todd Kapitula
Keyword(s):  
Schrödinger Equation ◽  
Solitary Waves ◽  
Schrodinger Equation ◽  
Three Dimensional ◽  
Small Error ◽  
Heteroclinic Cycle ◽  
Perturbation Techniques ◽  
Ginzburg Landau ◽  
The Waves ◽  
Quintic Nonlinear Schrödinger Equation

The existence of bright and dark multi-bump solitary waves for Ginzburg–Landau type perturbations of the cubic-quintic Schrodinger equation is considered. The waves in question are not perturbations of known analytic solitary waves, but instead arise as a bifurcation from a heteroclinic cycle in a three-dimensional ODE phase space. Using geometric singular perturbation techniques, regions in parameter space for which 1-bump bright and dark solitary waves will bifurcate are identified. The existence of N-bump dark solitary waves (N ≧ 1) is shown via an application of the Exchange Lemma with Exponentially Small Error. N-bump bright solitary waves are shown to exist as a consequence of the work of Kapitula and Maier-Paape.

EXACT HETEROCLINIC CYCLE FAMILY AND QUASI-PERIODIC SOLUTIONS FOR THE THREE-DIMENSIONAL SYSTEMS DETERMINED BY CHAZY CLASS IX

2011 ◽  
Vol 21 (05) ◽  
pp. 1357-1367 ◽  
Author(s):  
JIBIN LI ◽  
XIAOHUA ZHAO
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Periodic Solutions ◽  
Level Set ◽  
Three Dimensional ◽  
Heteroclinic Cycle ◽  
Dimensional System ◽  
Three Dimensional System ◽  
Infinitely Many Periodic Solutions

For differential equation in the Chazy class IX, their corresponding three-dimensional system is studied in this paper by using dynamical system methods and Cosgrove's results. In a level set, the exact explicit parametric representations of a heteroclinic cycle family and uncountably infinitely many periodic solutions as well as quasi-periodic solutions, are all obtained.

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