Classical bifurcation theory is undergoing a revitalization with the infusion of ideas from singularities of mappings and structural stability. The situation now is similar to that a quarter century ago when Krasnosel'skii introduced topological methods, especially degree theory, into the subject (see Krasnosel'skii [1964]). Like degree theory, the theory of singularities of mappings is playing...

Amplifiers based on Josephson junctions allow for a fast and noninvasive readout of superconducting qubits. Motivated by the ongoing progress toward realization fault-tolerant qubits Majorana bound states, we investigate topological counterpart bifurcation amplifier. We predict that dynamics junction driven in appropriate parameter regime may be used as an additional tool to detect emergence st...

This paper considers a Lotka-Volterra model with time delays and delay dependent parameters. The linear stability conditions are obtained with characteristic root method. The Hopf bifurcation is demonstrated. Using normal form theory and center manifold theory, Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived. Finally,...

A bifurcation theory is applied to the multimachine power system to investigate the effect of iron saturation on the complex dynamics of the system. The second system of the IEEE second benchmark model of Subsynchronous Resonance (SSR) is considered. The system studied can be mathematically modeled as a set of first order nonlinear ordinary differential equations with (!=Xc/XL) as a bifurcation...

We first consider the existence of local Hopf bifurcations, and then derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions, using the normal form theory and center manifold reduction. Further, particular attention is focused on the existence of the global Hopf bifurcation. By using the global Hopf bifurcation theory due to W...

The aim of this chapter is to introduce tools from bifurcation theory which will be necessary in the following sections for the study of neural field equations (NFE) set in the primary visual cortex. In a first step, we deal with elementary bifurcations in low dimensions such as saddle-node, transcritical, pitchfork and Hopf bifurcations. NFEs are dynamical systems defined on Banach spaces and ...

The existence of periodic solutions for evolution equations is of certain interest for both pure and applied mathematicians. Even for bidimensional systems of differential equations the detection of limit cycles by theoretical means is difficult. The bifurcation theory offers a strong tool for finding limit cycles, namely the theory concerning the Hopf bifurcation (when there is a varying param...

| Bifurcations of equilibrium points and limit cycles in unidirectionally coupled three oscillators are studied. According to their symmetrical properties, we classify equilibrium points and limit cycles into three and eight di erent types, respectively. Possible oscillations in unidirectional coupled three oscillators are presented by calculating Hopf bifurcation sets of equilibrium points. We...

The principles of strain localization analysis as applied to geomaterials are presented. Emphasis is given to the effects of fluid and temperature in the occurrence and development of instabilities. RÉSUMÉ. Dans cet article, on présente les principes de l’analyse de localisation des déformations dans les géomatériaux. En particulier, on met l’accent sur les effets du fluide et de la température...