Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities
This book focuses on bifurcation theory for autonomous and nonautonomous differential equations with discontinuities of different types – those with jumps present either in the right-hand side, or in trajectories or in the arguments of solutions of equations. The results obtained can be applied to various fields, such as neural networks, brain dynamics, mechanical systems, weather phenomena and population dynamics. Developing bifurcation theory for various types of differential equations, the book is pioneering in the field. It presents the latest results and provides a practical guide to applying the theory to differential equations with various types of discontinuity. Moreover, it offers new ways to analyze nonautonomous bifurcation scenarios in these equations. As such, it shows undergraduate and graduate students how bifurcation theory can be developed not only for discrete and continuous systems, but also for those that combine these systems in very different ways. At the same time, it offers specialists several powerful instruments developed for the theory of discontinuous dynamical systems with variable moments of impact, differential equations with piecewise constant arguments of generalized type and Filippov systems.
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2 Hopf Bifurcation in Impulsive Systems
3 Hopf Bifurcation in Filippov Systems
4 Nonautonomous Bifurcation in Impulsive Bernoulli Equations
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27t-periodic analyze Assume asymptotic behavior asymptotically unstable attractor B-equivalence method Bernoulli equation bifurcation of periodic bifurcation theory center manifold continuous continuous functions curves defined Denote discontinuous dynamical systems discontinuous limit cycle discontinuous right-hand side equations with discontinuous equations with piecewise following system forward and pullback Hopf bifurcation Hopf bifurcation theorem impact impulsive differential equations impulsive systems initial condition initial value x0 Lemma lim Sup limit cycle linear neighborhood nonautonomous bifurcation Nonlinear Physical nonperturbed system nontrivial bounded solutions obtain ordinary differential equations origin oscillators periodic solution piecewise constant argument pitchfork bifurcations planar polar coordinates proof pullback attracting pullback stable real numbers repulsive satisfy the following ſº ſº-substitution solution r(q stable manifold sufficiently small Theorem tions trajectory transcritical bifurcation trivial solution variable vector fields z-axis