Search results for "heteroclinic orbit"
showing 9 items of 9 documents
Heteroclinic contours and self-replicated solitary waves in a reaction–diffusion lattice with complex threshold excitation
2008
Abstract The space–time dynamics of the network system modeling collective behavior of electrically coupled nonlinear cells is investigated. The dynamics of a local cell is described by the FitzHugh–Nagumo system with complex threshold excitation. Heteroclinic orbits defining traveling wave front solutions are investigated in a moving frame system. A heteroclinic contour formed by separatrix manifolds of two saddle-foci is found in the phase space. The existence of such structure indicates the appearance of complex wave patterns in the network. Such solutions have been confirmed and analyzed numerically. Complex homoclinic orbits found in the neighborhood of the heteroclinic contour define …
Regular and singular pulse and front solutions and possible isochronous behavior in the Extended-Reduced Ostrovsky Equation: Phase-plane, multi-infin…
2016
In this paper we employ three recent analytical approaches to investigate several classes of traveling wave solutions of the so-called extended-reduced Ostrovsky Equation (exROE). A recent extension of phase-plane analysis is first employed to show the existence of breaking kink wave solutions and smooth periodic wave (compacton) solutions. Next, smooth traveling waves are derived using a recent technique to derive convergent multi-infinite series solutions for the homoclinic orbits of the traveling-wave equations for the exROE equation. These correspond to pulse solutions respectively of the original PDEs. We perform many numerical tests in different parameter regime to pinpoint real saddl…
Looking More Closely at the Rabinovich-Fabrikant System
2016
Recently, we looked more closely into the Rabinovich–Fabrikant system, after a decade of study [Danca & Chen, 2004], discovering some new characteristics such as cycling chaos, transient chaos, chaotic hidden attractors and a new kind of saddle-like attractor. In addition to extensive and accurate numerical analysis, on the assumptive existence of heteroclinic orbits, we provide a few of their approximations.
Smooth and non-smooth traveling wave solutions of some generalized Camassa–Holm equations
2014
In this paper we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of a recently-derived integrable family of generalized Camassa-Holm (GCH) equations. A recent, novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of three of the GCH NLPDEs, i.e. the possible non-smooth peakon, cuspon and compacton solutions. Two of the GCH equations do not support singular traveling waves. The third equation supports four-segmented, non-smooth $M$-wave solutions, while the fourth supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes. Moreover, sm…
Regular and singular pulse and front solutions and possible isochronous behavior in the short-pulse equation: Phase-plane, multi-infinite series and …
2014
In this paper we employ three recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of a family of so-called short-pulse equations (SPE). A recent, novel application of phase-plane analysis is first employed to show the existence of breaking kink wave solutions in certain parameter regimes. Secondly, smooth traveling waves are derived using a recent technique to derive convergent multi-infinite series solutions for the homoclinic (heteroclinic) orbits of the traveling-wave equations for the SPE equation, as well as for its generalized version with arbitrary coefficients. These correspond to pulse (kink or shock) solutions respectively o…
Generalized Camassa-Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions
2021
In this paper, we consider a member of an integrable family of generalized Camassa–Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new symmetries via the nonclassical method, which cannot be obtained by Lie symmetry method. We employ the multiplier method to construct conservation laws for this family of GCH equations. Using the conservation laws of the underlying equation, double reduction is also constructed. Finally, we investigate traveling waves of the GCH equations. We derive convergent series solutions both for the homoclinic and heteroclinic orbits of the…
Some topological invariants for three-dimensional flows
2001
We deal here with vector fields on three manifolds. For a system with a homoclinic orbit to a saddle-focus point, we show that the imaginary part of the complex eigenvalues is a conjugacy invariant. We show also that the ratio of the real part of the complex eigenvalue over the real one is invariant under topological equivalence. For a system with two saddle-focus points and an orbit connecting the one-dimensional invariant manifold of those points, we compute a conjugacy invariant related to the eigenvalues of the vector field at the singularities. (c) 2001 American Institute of Physics.
Corrigendum to “Smooth and non-smooth traveling wave solutions of some generalized Camassa–Holm equations” [19 (6) (2014) 1746–1769]
2015
Corrigendum Corrigendum to ‘‘Smooth and non-smooth traveling wave solutions of some generalized Camassa–Holm equations’’ [19 (6) (2014) 1746–1769] M. Russo , S. Roy Choudhury , T. Rehman , G. Gambino b University of Central Florida, Department of Mathematics, 4000 Central Florida Blvd., Orlando, USA University of Palermo, Department of Mathematics and Computer Science, Via Archirafi 34, 90123 Palermo, Italy
Corrigendum to “Regular and singular pulse and front solutions and possible isochronous behavior in the short-pulse equation: Phase-plane, multi-infi…
2022
Section 7 of the original paper contained several errors which are corrected here. Equations (54) and (55) are incorrect. In the following, the corrected versions of these equations are given and the subsequent results of Section 7 are also revised.