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Connected branches of periodic orbits originating at a Hopf bifurcation point of a differential system are considered. A computable estimate for the range of amplitudes of periodic orbits contained in the branch is provided under the assumption that the nonlinear terms satisfy a linear estimate in a ball. If the estimate is global, then the branch is unbounded. The results are formulated in an equivariant setting where the system can have multiple branches of periodic orbits characterized by different groups of symmetries. The nonlocal analysis is based on the equivariant degree method, which allows us to handle both generic and degenerate Hopf bifurcations. This is illustrated by examples. [ABSTRACT FROM AUTHOR]

We consider the dynamics of a scalar piecewise linear "saw map" with infinitely many linear segments. In particular, such maps are generated as a Poincaré map of simple two-dimensional discrete time piecewise linear systems involving a saturation function. Alternatively, these systems can be viewed as a feedback loop with the so-called stop hysteresis operator. We analyze chaotic sets and attractors of the "saw map" depending on its parameters. [ABSTRACT FROM AUTHOR]

Pyragas time-delayed feedback control has proven itself as an effective tool to noninvasively stabilize periodic solutions. In a number of publications, this method was adapted to equivariant settings. In this paper, we consider -symmetric systems of van der Pol and optical oscillators coupled in a cube-like configuration. These systems undergo equivariant Hopf bifurcations giving rise to multiple branches of unstable periodic solutions. We introduce a delayed control term, which ensures stabilization of a selected branch. Group theoretic restrictions which help to shape our choice of control are discussed. Furthermore, we explicitly describe the domains in a two-dimensional parameter space for which the periodic solutions of the delayed system are stable. [ABSTRACT FROM AUTHOR]