Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations

Abstract One of the problems arising in the analysis of periodic nonlinear dynamical systems is to determine multiple equilibria and periodic solutions. These may be formulated in terms of finding zeros of a nonlinear vector function. The iterative methods commonly employed for this purpose concentrate on locating a single zero from a given initial guess. The homotopic and continuation methods are also used for locating multiple zeros but they primarily have been used to follow a single solution branch with a parameter variation. The purpose of this paper is to explore a method based on a differential equation for finding multiple zeros. The equilibria of this differential equation correspond with the zeros of a given vector function. Since the equilibria is asymptotically stable, they can be obtained by examining a large number of trajectories. For this purpose, the cell-to-cell mapping is ideally suited for finding the equilibria, and hence the zeros of a nonlinear vector function. Preliminary results for finding multiple fixed points of a highly nonlinear map are provided in this paper.

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Codimension-Two Bifurcations of Fixed Points in a Class of Discrete Prey-Predator Systems

Discrete Dynamics in Nature and Society ◽

10.1155/2011/862494 ◽

2011 ◽

Vol 2011 ◽

pp. 1-27 ◽

Cited By ~ 7

Author(s):

R. Khoshsiar Ghaziani ◽

W. Govaerts ◽

C. Sonck

Keyword(s):

Normal Form ◽

Fixed Points ◽

Dynamic Behaviour ◽

Numerical Continuation ◽

Continuation Methods ◽

Volterra System ◽

Codimension Two ◽

Bifurcation Points ◽

Two Parameters ◽

Form Stability

The dynamic behaviour of a Lotka-Volterra system, described by a planar map, is analytically and numerically investigated. We derive analytical conditions for stability and bifurcation of the fixed points of the system and compute analytically the normal form coefficients for the codimension 1 bifurcation points (flip and Neimark-Sacker), and so establish sub- or supercriticality of these bifurcation points. Furthermore, by using numerical continuation methods, we compute bifurcation curves of fixed points and cycles with periods up to 16 under variation of one and two parameters, and compute all codimension 1 and codimension 2 bifurcations on the corresponding curves. For the bifurcation points, we compute the corresponding normal form coefficients. These quantities enable us to compute curves of codimension 1 bifurcations that branch off from the detected codimension 2 bifurcation points. These curves form stability boundaries of various types of cycles which emerge around codimension 1 and 2 bifurcation points. Numerical simulations confirm our results and reveal further complex dynamical behaviours.

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