Abstract
Inertial manifolds and limit cycles of dynamical systems in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math>
Highlights
IntroductionWhere A is a symmetric n × n matrix with eigenvalues 0 ≤ λ1 ≤ λ2 ≤ · · · ≤ λn and the function F belongs to C1+α(Rn, Rn) for some α ∈ (0, 1)
We consider ordinary differential equations x = −Ax + F(x), x ∈ Rn, n ≥ 3, (1.1)where A is a symmetric n × n matrix with eigenvalues 0 ≤ λ1 ≤ λ2 ≤ · · · ≤ λn and the function F belongs to C1+α(Rn, Rn) for some α ∈ (0, 1)
The spectral gap method is based on the presence of a natural self-adjoint linear component −A of the vector field of ordinary differential equation (ODE) with dominating third eigenvalue, λ3(A) > λ2(A), which somewhat restricts the range of applications
Summary
Where A is a symmetric n × n matrix with eigenvalues 0 ≤ λ1 ≤ λ2 ≤ · · · ≤ λn and the function F belongs to C1+α(Rn, Rn) for some α ∈ (0, 1). The theory of inertial (that is, invariant and globally exponentially attracting) manifolds was developed in the 1980s as a tool for studying the final (at large times) dynamics of semilinear parabolic equations with a vector field structure of the form (1.1) in an infinite-dimensional Hilbert space X (see [6, Ch. 8], [13] and the references therein). In this case, as usual, it is assumed that A is an unbounded self-adjoint positive linear operator in X with a compact resolvent. In contrast to the bifurcation theory, our approach proves the existence of stable self-sustained oscillations of a “large amplitude”
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