Abstract
In the present paper, a custom algorithm based on the method of orthogonal collocation on finite elements is presented and used for the location of global homoclinic point-to-point asymptotic connecting orbits. This kind of global bifurcation occurs in a large variety of problems in Applied Sciences, being associated to specific, significant physical aspects of the problem under consideration. In order to confront the difficulties faced when the location of such orbits is attempted, high order boundary conditions are constructed through scale order approximations, and used instead of the more common first order ones. The effectiveness of the implemented algorithm is justified by means of the specific applications and the figures presented.
Highlights
In recent years, the improvement of hardware capabilities, such as operational CPU frequency, the increasing amount of RAM equipped, the use of parallelization and the improvement of symbolic mathematical software has enabled researchers to numerically compute global asymptotic orbits more since their computation constitutes a computationally demanding task, even in the case of low dimensional systems
An efficient custom algorithm of orthogonal collocation on finite elements implemented in MathworksMatlab has been presented together with two applications in different fields of Applied Mathematics, be them the well-known Lorenz system and a model of a three-species food chain with group defence ecosystem
Global homoclinic asymptotic point-to-point connecting orbits have been obtained numerically, regarding the specific applications. In both cases an initial approximation of the homoclinic connecting orbits of interest has been acquired by continuing limit cycles, which have emerged from a Hopf bifurcation, numerically up to a high value of the fundamental period of them
Summary
The improvement of hardware capabilities, such as operational CPU frequency, the increasing amount of RAM equipped, the use of parallelization and the improvement of symbolic mathematical software has enabled researchers to numerically compute global asymptotic orbits more since their computation constitutes a computationally demanding task, even in the case of low dimensional systems. Homoclinic point-to-point connecting orbits arise in various occasions where hysteresis and saturation phenomena are encountered. These orbits act as separatrices for the nonlinear state space in 2D conservative ODEs, since they divide the phase space into regions of periodic. We present an algorithm for the numerical computation of global homoclinic asymptotic point-to-point connecting orbits (homoclinic P2P orbits for short on), where the evaluation of high order boundary conditions (BC on) is involved. The analysis is carried out with the aid of MathworksMatlab and the symbolic engine of Maplesoft Maple, by means of which we obtain the respective graphs
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
