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Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars. (English) Zbl 1149.39021

The article deals with the analysis of solutions to Volterra functional equations \[ x(t) = F(x_t), \quad t > 0,\tag{1} \]
in particular, to delay differential equations
\[ \dot x(t) = F(x_t), \quad t > 0, \]
\(x_t(\theta) = x(t + \theta)\), \(t > 0\), \(-h \leq \theta \leq 0\). The authors’ approach is based on the passage to the equivalent problem for differential equations of the type \[ \frac{d ju(t)}{dt} = A_0^{\odot *} ju(t) + G(u(t)), \quad t > 0 \tag{2} \]
in a suitable space \(X\) of functions defined on \([-h,0]\). Here \(A_0\) is the infinitesimal generator of the strongly continuous semigroup of operators \(T_0(t)\) in \(X\) (this semigroup is defined by (1) due to the standard scheme of N. N. Krasovskiĭ), \(A_0^\odot\) the restriction of \(A_0^*\) to the sun-subspace (subspace of strong continuity) of the conjugate semigroup \(T_0^*(t)\), \(A^{\odot *}\) the conjugate to \(A_0^\odot\), \(j\) the canonical embedding of \(X\) into \(X^{\odot *}\), and \(G\) is a small nonlinearity.
The authors consider the case when \(jX = X^{\odot\odot}\). They describe the stability properties of a steady state \(\bar{\varphi}\) for (2) in terms of the spectral properties of \(A\), stable, unstable, and center manifolds for (2), sufficient conditions for Hopf bifurcation. All these results about solutions to (2) are carried over to delay differential equations and Volterra functional equations of type (1). As examples the authors consider equations of a model for cannibalistic interaction and a structured metapopulation model.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
47D06 One-parameter semigroups and linear evolution equations
92D25 Population dynamics (general)
34K20 Stability theory of functional-differential equations
34K18 Bifurcation theory of functional-differential equations
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