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Differential Equations with a Small Parameter and Multipeak Oscillations
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Journal Article
Differential Equations with a Small Parameter and Multipeak Oscillations
2024
Overview
In this paper, we study a nonlinear dynamical system of autonomous ordinary differential equations with a small parameter
such that two variables
and
are fast and another one
is slow. If we take the limit as
, then this becomes a “
degenerate system
” included in the one-parameter family of two-dimensional subsystems of
fast motions
with the parameter
in some interval. It is assumed that in each subsystem there exists a
structurally stable
limit cycle
. In addition, in the
complete
dynamical system there is some structurally stable periodic orbit
that tends to a limit cycle
for some
as
tends to zero. We can define the first return map, or the Poincaré map, on a local cross section in the hyperplane
orthogonal to
at some point. We prove that the Poincaré map has an invariant manifold for the fixed point corresponding to the periodic orbit
on a guaranteed interval over the variable
, and the interval length is separated from zero as
tends to zero. The proved theorem allows one to formulate some sufficient conditions for the existence and/or absence of multipeak oscillations in the complete dynamical system. As an example of application of the obtained results, we consider some kinetic model of the catalytic reaction of hydrogen oxidation on nickel.
Publisher
Pleiades Publishing
Subject
