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DCDS-B

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.

DCDS-B

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.

For more information please click the “Full Text” above

For more information please click the “Full Text” above

keywords:

DCDS-B

Periodic
canard trajectories with multiple segments following the unstable
part of critical manifold

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.

DCDS-S

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.

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