Bifurcations in four-dimensional switched systems

Global Dynamics and Bifurcations in a Four-dimensional Replicator System

In this paper, the four-dimensional cyclic replicator system u̇i = ui[−(Bu)i + ∑4 j=1 uj(Bu)j ], 1 ≤ i ≤ 4, with b1 = b3 is considered, in which the first row of the matrix B is (0 b1 b2 b3) and the other rows of B are cyclic permutations of the first row. Our aim is to study the global dynamics and bifurcations in the system, and to show how and when all but one species go to extinction. By red...

Floquet Exponents and Bifurcations in Switched Converters

Analytics of homoclinic bifurcations in Three-Dimensional Systems

An analytical approach to determine critical parameter values of homoclinic bifurcations in three-dimensional systems is reported. The homoclinic orbit is supposed to be a limit of a unique periodic orbit. Hence, the multiple scales perturbation method is performed to construct an approximation of the periodic solution and its frequency. Then, two simple criteria are used. The first criterion i...

Visualizing bifurcations in High Dimensional Systems: the Spectral bifurcation Diagram

Bifurcation diagrams which allow one to visualise changes in the behaviour of low dimensional nonlinear maps as a parameter is altered are common. Visualisation in higher dimensional systems is more difficult. A straightforward method to visualise bifurcations in flows of high dimensional nonlinear dynamical systems is presented, using the Lorenz ’96 systems with dimension 8 and dimension 40 as...

Two-dimensional invariant manifolds in four-dimensional dynamical systems

This paper explores the visualization of two-dimensional stable and unstable manifolds of the origin (a saddle point) in a four-dimensional Hamiltonian system arising from control theory. The manifolds are computed using an algorithm that finds sets of points that lie at the same geodesic distance from the origin. By coloring the manifolds according to this geodesic distance, one can gain insig...