Linear stability analysis of systems with Preisach memory
Alexander Pimenov Dmitrii I. Rachinskii
Discrete & Continuous Dynamical Systems - B 2009, 11(4): 997-1018 doi: 10.3934/dcdsb.2009.11.997
We consider differential equations coupled with the input-output memory relation defined by the Preisach operator. The differential equation relates an instant value of the rate of change of the output of the Preisach operator with an instant value of its input. We propose an algorithm for the linearisation of the evolution operator of the system and apply it to define the characteristic multiplier of periodic solutions, which determines their stability. Examples of the system considered include models of terrestrial hydrology and electronic oscillators with hysteresis.
keywords: local stability analysis operator-differential equation Preisach model. Periodic solution
Peter E. Kloeden Alexander M. Krasnosel'skii Pavel Krejčí Dmitrii I. Rachinskii
Discrete & Continuous Dynamical Systems - B 2013, 18(2): i-iii doi: 10.3934/dcdsb.2013.18.2i
Alexei Vadimovich Pokrovskii was an outstanding mathematician, a scientist with very broad mathematical interests, and a pioneer in the mathematical theory of systems with hysteresis. He died unexpectedly on September 1, 2010 at the age 62. For the previous nine years he had been Professor and Head of Applied Mathematics at University College Cork in Ireland.

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Periodic canard trajectories with multiple segments following the unstable part of critical manifold
Alexander M. Krasnosel'skii Edward O'Grady Alexei Pokrovskii Dmitrii I. Rachinskii
Discrete & Continuous Dynamical Systems - B 2013, 18(2): 467-482 doi: 10.3934/dcdsb.2013.18.467
We consider a scalar fast differential equation which is periodically driven by a slowly varying input. Assuming that the equation depends on $n$ scalar parameters, we present simple sufficient conditions for the existence of a periodic canard solution, which, within a period, makes $n$ fast transitions between the stable branch and the unstable branch of the folded critical curve. The closed trace of the canard solution on the plane of the slow input variable and the fast phase variable has $n$ portions elongated along the unstable branch of the critical curve. We show that the length of these portions and the length of the time intervals of the slow motion separated by the short time intervals of fast transitions between the branches are controlled by the parameters.
keywords: stability. canard trajectory degree theory Periodic solution
Arnold tongues for bifurcation from infinity
Victor S. Kozyakin Alexander M. Krasnosel’skii Dmitrii I. Rachinskii
Discrete & Continuous Dynamical Systems - S 2008, 1(1): 107-116 doi: 10.3934/dcdss.2008.1.107
We consider discrete time systems $x_{k+1}=U(x_{k};\lambda)$, $x\in\R^{N}$, with a complex parameter $\lambda$. The map $U(\cdot;\lambda)$ at infinity contains a principal linear term, a bounded positively homogeneous nonlinearity, and a smaller part. We describe the sets of parameter values for which the large-amplitude $n$-periodic trajectories exist for a fixed $n$. In the related problems on small periodic orbits near zero, similarly defined parameter sets, known as Arnold tongues, are more narrow.
keywords: Arnold tongue positively homogeneous nonlinearity discrete time system bifurcation at infinity Poincare map. Periodic trajectory saturation

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