A method to calculate basin bifurcation sets for a two-dimensional noninvertible map

Abstract-In this paper we propose a numerical method to calculate basin bifurcation sets in a parameter space. It is known that the basin bifurcations always result from the contact of a basin boundary with the critical curve segment. A numerical example for a two-dimensional quadratic noninvertible map is illustrated and new results of basin bifurcations are shown

On the embedding of a (p-1)-dimensional non invertible map into a p-dimensional invertible map

This paper concerns the description of some properties of p-dimensional invertible real maps Tb, turning into a (p - 1)-dimensional non invertible ones T0, p = 2, 3, when a parameter b of the first map is equal to a critical value, say b=0. Then it is said that the noninvertible map is embedded into the invertible one. More particularly properties of the stable, and the unstable manifolds of a saddle fixed point are considered in relation with this embedding. This is made by introducing the notion of folding as resulting from the crossing through a commutation curve when p = 2, or a commutation surface when p = 3

Complex Dynamical Behavior of a Bounded Rational Duopoly Game with Consumer Surplus

In this chapter, we assume that two bounded rational firms not only pursue profit maximization but also take consumer surplus into account, so the objections of all the firms are combination of their profits and the consumer surplus. And then a dynamical duopoly Cournot model with bounded rationality is established. The existence and stability of the boundary equilibrium points and the Nash equilibrium of the model are discussed, respectively. And then the stability condition of the Nash equilibrium is given. The complex dynamical behavior of the system varies with parameters in the parameter space is studied by using the so-called 2D bifurcation diagram. The coexistence of multiple attractors is discussed through analyzing basins of attraction. It is found that not only two attractors can coexist, but also three or even four attractors may coexist in the established model. Then, the topological structure of basins of attraction and the global dynamics of the system are discussed through invertible map, critical curve and transverse Lyapunov exponent. At last, the synchronization phenomenon of the built model is studied

Itinerant memory dynamics and global bifurcations in chaotic neural networks

We have considered itinerant memory dynamics in a chaotic neural network composed of four chaotic neurons with synaptic connections determined by two orthogonal stored patterns as a simple example of a chaotic itinerant phenomenon in dynamical associative memory. We have analyzed a mechanism of generating the itinerant memory dynamics with respect to intersection of a pair of ␣ branches of periodic points and collapse of a periodic in-phase attracting set. The intersection of invariant sets is numerically verified by a novel method proposed in this paper. 3,11,12 Among such studies we focus on the associative memory dynamics in this paper. Adachi and Aihara analyzed the dynamics of associative memory networks composed of chaotic neurons in detail and examined characteristics of the retrieval process

Research Advances in Chaos Theory

The subject of chaos has invaded practically every area of the natural sciences. Weather patterns are referred to as chaotic. There are chemical reactions and chaotic evolution of insect populations. Atomic and molecular physics have also seen the emergence of the study of chaos in these microscopic domains. This book examines the issue of chaos in nonlinear and dynamical systems, quantum mechanics, biology, and economics

Homoclinic bifurcation analysis for logistic map

In this study, we have developed the method to obtain the homoclinic bifurcation parameter of an arbitrary targeted fixed point in the logistic map Tr. We have considered the geometrical structure of Tr around x =0.5 and derived the core condition of the bifurcation occurrence. As the result of numerical experiment, we have calculated the exact bifurcation parameter of the fixed point of Trℓ with ℓ≤256. We have also discussed the Feigenbaum constants found in the bifurcation parameter and the fixed point coordinate sequences. This fact implies the local stability of the fixed point and global structure around it are in association via the constants