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Robust homoclinic orbits in planar systems with Preisach hysteresis operator
We construct examples of robust homoclinic orbits for systems of ordinary differential equations coupled with the Preisach hysteresis operator. Existence of such orbits is demonstrated for the first time. We discuss a generic mechanism that creates robust homoclinic orbits and a method for finding them. An example of a homoclinic orbit in a population dynamics model with hysteretic response of the prey to variations of the predator is studied numerically
Hopf bifurcations and simple structures of periodic solution sets in systems with the Preisach nonlinearity
We survey a number of recent results and suggest some new ones on periodic solutions of systems with hysteresis. The main focus of this work is the situation when simple one-parameter structures of periodic regimes appear. We consider forced oscillations, cycles of autonomous systems and Hopf bifurcations from the equilibrium and from infinity
Well-posedness of parabolic equations containing hysteresis with diffusive thresholds
We study complex systems arising, in particular, in population dynamics,
developmental biology, and bacterial metabolic processes, in which each
individual element obeys a relatively simple hysteresis law (a non-ideal
relay). Assuming that hysteresis thresholds fluctuate, we consider the arising
reaction-diffusion system. In this case, the spatial variable corresponds to
the hysteresis threshold. We describe the collective behavior of such a system
in terms of the Preisach operator with time-dependent measure which is a part
of the solution for the whole system. We prove the well-posedness of the system
and discuss the long-term behavior of solutions.Comment: 30 pages, 1 figur
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