This booklet specializes in bifurcation idea for self sustaining and nonautonomous differential equations with discontinuities of alternative kinds – people with jumps current both within the right-hand facet, or in trajectories or within the arguments of suggestions of equations. the consequences acquired may be utilized to numerous fields, resembling neural networks, mind dynamics, mechanical structures, climate phenomena and inhabitants dynamics. Developing bifurcation conception for varied different types of differential equations, the booklet is pioneering within the box. It offers the most recent effects and gives a realistic consultant to utilising the idea to differential equations with quite a few forms of discontinuity. additionally, it bargains new how one can examine nonautonomous bifurcation eventualities in those equations. As such, it exhibits undergraduate and graduate scholars how bifurcation conception could be built not just for discrete and non-stop platforms, but additionally for those who mix those structures in very alternative ways. while, it deals experts numerous strong tools built for the idea of discontinuous dynamical structures with variable moments of impression, differential equations with piecewise consistent arguments of generalized sort and Filippov systems.