Bifurcation Analysis In A Generalized Friction Model With Time-Delayed Feedback

In this paper, a generalized model of the two friction models, each with two different types of control forces with time-delayed feedback proposed by A. Saha et al, is taken into considerations. With the time delay as the bifurcation parameter, the occurrence of Hopf bifurcations and the local stability of the equilibrium is discussed. First, several reasonable suppositions are given. The differential equations are linearized around the zero equilibrium under these suppositions. By obtaining the characteristic equation associated with the linearized equations, analysis of the real parts of the eigenvalues is carried out. In this way, analysis of the local stability of the zero equilibrium is given, which contributes to the verification of some bifurcation phenomena (such as a series of Hopf bifurcations). Moreover, the existence of a particular type of fixed-point bifurcation is observed as well. Further, bifurcation phenomena from certain non-isolated points may happen. The analysis of the local stability of the zero equilibrium also indicates that, under some condition, the generalized model harbors a phenomenon that the equilibrium may undergo finite switches from stability to instability to stability and finally become unstable. Besides, with the method introduced by Faria and Magalháes to compute the normal forms, the normal form on the center manifold is computed. By calculating some relevant essential parameters, the direction of the periodic solutions bifurcating from the zero equilibrium and their stability on the center manifold can be determined. In addition, in this paper, corresponding examples to several different cases appearing before are constructed. Numerical simulations of these instances are carried out one by one, using the mathematical software Matlab; corresponding wave profiles and phase portraits are depicted. These simulations are in excellent accordance with the theoretical results derived before. Apart from the above, more than one periodic solutions of the system may exist at some fixed delays, according to the global Hopf bifurcation diagram given by the tool package BIFTOOL. Finally a brief conclusion is presented, pointing out some drawbacks, putting forward some unsolved problems and some future research work.