Persistence and imperfection of nonautonomous bifurcation patterns. (English) Zbl 1218.37035
The article deals with some bifurcation phenomena with bounded entire solutions to nonautonomous dynamical systems with basic (bifurcation) and additional (perturbation) parameters \(\lambda\) and \(\varepsilon\). In particular, the following equations are considered: the nonautonomous parabolic equation
\[ \dot u + Au = f(t,u,\lambda,\varepsilon), \]
where \(A\) is a sectorial operator in a Hilbert space \(X\), \(f(t,u,\lambda,\varepsilon)\) a nonlinearity; further, the Carathéodory differential equation
\[ \dot x = f(t,x,\lambda,\varepsilon); \]
and the difference equations of type
\[ x_{k+1} = f_k(x_k,\lambda,\varepsilon). \]
For all these equations the problem about bounded solutions in a neighborhood of a bifurcation point is reduced (of course under special hypotheses) to the analysis of an auxiliary equation, and this analysis allows the author to give the picture for the behavior of bounded solutions under perturbations of both parameters. References in the article are sufficiently complete. However, the author ignores or does not know numerous investigations by Krasnosel’skiĭ and his colleagues about bifurcation and perturbated bifurcation of almost periodic solutions for ordinary and parabolic differential equations. Abstract bifurcation and perturbated bifurcation problems are also considered in the monograph by M. A. Krasnosel’skiĭ, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskij and V. Ya. Stetsenko [Approximate solution of operator equations. Groningen: Wolters-Noordhoff Publishing (1972; Zbl 0231.41024)].
\[ \dot u + Au = f(t,u,\lambda,\varepsilon), \]
where \(A\) is a sectorial operator in a Hilbert space \(X\), \(f(t,u,\lambda,\varepsilon)\) a nonlinearity; further, the Carathéodory differential equation
\[ \dot x = f(t,x,\lambda,\varepsilon); \]
and the difference equations of type
\[ x_{k+1} = f_k(x_k,\lambda,\varepsilon). \]
For all these equations the problem about bounded solutions in a neighborhood of a bifurcation point is reduced (of course under special hypotheses) to the analysis of an auxiliary equation, and this analysis allows the author to give the picture for the behavior of bounded solutions under perturbations of both parameters. References in the article are sufficiently complete. However, the author ignores or does not know numerous investigations by Krasnosel’skiĭ and his colleagues about bifurcation and perturbated bifurcation of almost periodic solutions for ordinary and parabolic differential equations. Abstract bifurcation and perturbated bifurcation problems are also considered in the monograph by M. A. Krasnosel’skiĭ, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskij and V. Ya. Stetsenko [Approximate solution of operator equations. Groningen: Wolters-Noordhoff Publishing (1972; Zbl 0231.41024)].
Reviewer: Peter Zabreiko (Minsk)
MSC:
| 37C60 | Nonautonomous smooth dynamical systems |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34D05 | Asymptotic properties of solutions to ordinary differential equations |
| 47J15 | Abstract bifurcation theory involving nonlinear operators |
Keywords:
nonautonomous bifurcation; imperfect bifurcation; unfolding; semilinear parabolic equation; Carathéodory differential equation; nonautonomous difference equation; exponential dichotomyCitations:
Zbl 0231.41024References:
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