In this letter, a rigorous solution of the chaotic Lozi mapping is given for some regions of the bifurcation parameters. The relevance of this result is that while chaos has been proved for the Lozi mapping, there has not previously been discussion of the possibility of finding a rigorous formula for this solution. This important result is obtained using the Jordan normal form defined for matri...

The forced response curve is a graph showing the amplitude of the response of a periodically forced system as the forcing frequency is varied. Zhang and Golubitsky (SIADS, 10(4) (2011) 1272–1306) classified the existence and multiplicity of periodic responses to small amplitude periodic forcing of a system near a Hopf bifurcation point, where the forcing frequency, ωF , is close to the Hopf fre...

We study in this paper the bifurcation and singularities of the Ginzburg–Landau equation. The existence and structure of the bifurcated solutions are obtained by T. Ma, J. Park and S. Wang [8] and now we prove that the bifurcated invariant set Σλ contains at least C k n = n···(n−k+1) k! singularity manifolds T k+1 provided λ crosses the second eigenvalue α of −α∆. And we achieved the similar re...

A discontinuous change in the size of an attractor is the most easily observed type of global bifurcation. More generally, an explosion is a discontinuous change in the set of recurrent points. An explosion often results from heteroclinic and homoclinic tangency bifurcations. We prove that, for one-dimensional maps, explosions are generically the result of either tangency or saddle-node bifurca...

We give criteria for the existence of smooth bifurcation branches of fixed boundary CMC surfaces in R, and we discuss stability/instability issues for the surfaces in bifurcating branches. To illustrate the theory, we discuss an explicit example obtained from a bifurcating branch of fixed boundary unduloids inR.

In this paper we deal with reversible vector fields on a 2-dimensional manifold having a codimension one submanifold as its symmetry axis. We classify generically the one parameter families of such vector fields. As a matter of fact, aspects of structural stability and codimension one bifurcation are analysed.

A structured model of bioreactor for an activated sludge process was presented. The stability and bifurcation characteristics of the model are investigated, the bifurcation analysis of the model shows static and complex dynamic behavior (periodic and complex) over a wide range of the model parameters. The model exhibits a new interesting behavior (in some range of parameters) including four sta...

The work is devoted to the systematic research of the periodic, quasi-periodic or chaotic oscillations and the coexistence in the nonlinear Duffing-van der Pol type dynamical systems describing processes and phenomena in nature or engineering. The achieved result is the elaboration of the basic theory for searching nonlinear effects based on the concepts of periodic skeletons, bifurcation theor...

In this paper we demonstrate the power of the computer algebra package conley, which enables one to compute connection and transition matrices, two of the main algebraic tools of the Conley index theory. In particular, we study the Cahn-Hilliard equation on the unit square and extend the results obtained in [Maier-Paape et al., 2007] to a bigger range of the bifurcation parameter. Besides provi...

In this paper, a coupled FHN model with two different delays is investigated. The local stability and the existence of Hopf bifurcation for the system are analyzed. The effect of two different delays on dynamical behavior is discussed. Simulation results are presented to support theoretical analysis. Finally, main conclusions are included.