Continuous and discrete dynamics near manifolds of equilibria.

*(English)*Zbl 0535.34002
Lecture Notes in Mathematics. 1058. Berlin etc.: Springer-Verlag. IX, 142 p. DM 24.00; $ 9.40 (1984).

The monograph is devoted to the study of solutions of nonautonomous ordinary differential equations and difference equations near the manifolds of stationary solutions. In Chapter I, ordinary differential equations are studied, while in Chapter II, difference equations are analysed. The study is applied to a selection model of population genetics. To read the book, a basic knowledge of ordinary differential equations (qualitative theory of dynamical systems such as \(\omega\)-limit sets, invariance, Lyapunov functions, stability, asymptotic behavior and so on) are needed.

In three appendices, some auxiliary notions and results are given. The typical result which is generalized, modified and applied to the biological problem, is the following: Let x(t) be any solution of an autonomous differential equation \(x=f(x)\), \(f\in C^ 3({\mathbb{R}}^ n,{\mathbb{R}}^ n)\), which has an m-dimensional differentiable manifold M of stationary solutions. Suppose that (i) the \(\omega\)-limit set of x(t) is not empty, (ii) the distance between x(t) and M decays to zero as \(t\to \infty\), (iii) for each \(\bar x\in M\) n-m eigenvalues of the Jacobian of f at \(\bar x\) have real parts different from zero. Then the solution x(t) converges as \(t\to \infty\) to some point on M.

In three appendices, some auxiliary notions and results are given. The typical result which is generalized, modified and applied to the biological problem, is the following: Let x(t) be any solution of an autonomous differential equation \(x=f(x)\), \(f\in C^ 3({\mathbb{R}}^ n,{\mathbb{R}}^ n)\), which has an m-dimensional differentiable manifold M of stationary solutions. Suppose that (i) the \(\omega\)-limit set of x(t) is not empty, (ii) the distance between x(t) and M decays to zero as \(t\to \infty\), (iii) for each \(\bar x\in M\) n-m eigenvalues of the Jacobian of f at \(\bar x\) have real parts different from zero. Then the solution x(t) converges as \(t\to \infty\) to some point on M.

Reviewer: N.Luca

##### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34C30 | Manifolds of solutions of ODE (MSC2000) |

34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |

92D25 | Population dynamics (general) |

34D20 | Stability of solutions to ordinary differential equations |