Nonautonomous differential systems in two dimensions.

*(English)*Zbl 1196.37037
Battelli, Flaviano (ed.) et al., Handbook of differential equations: Ordinary differential equations. Vol. IV. Amsterdam: Elsevier/North Holland (ISBN 978-0-444-53031-8/hbk). Handbook of Differential Equations, 133-268 (2008).

Let \(C\) denote the space of continuous and bounded maps \(\mathbb R\to M_n\), where \(M_n\) is the set of the \(n\times n\) real matrices. For \(\omega\in C\) denote by \(\tau_t(\omega)\) the \(t\)-translation \(s\to \omega(s+t)\). \(\tau_t\) is called the Bebutov flow. For a uniformly continuous \(A\in C\) denote by \(\Omega\) the closure of \(\{\tau_t(A): t\in\mathbb R\}\). \(\Omega\) is compact and invariant with respect to the Bebutov flow. It induces the flow \(\tilde\tau_t\) on \(\Omega\times \mathbb R^n\) given by \((\omega,x)\to (\tau_t(\omega),\Phi_\omega(t)x)\), where \(\Phi_\omega\) denotes the fundamental matrix of the equation \(x'=\omega(t)x\). The flow \(\tilde\tau_t\) further generates in a natural way the so-called projective flow \(\hat\tau_t\) on \(\Omega\times \mathbb P^{n-1}\), where \(\mathbb P^{n-1}\) denotes the \(n-1\) dimensional projective space. In order to examine properties of projective flows, one can assume without loss of generality that \(A\) attains values in \(sl(n,\mathbb R)\).

The article under review surveys results on dynamical properties of these flows and their consequences for solutions of the corresponding nonautonomous equations, mainly in the case \(n=2\). The article is divided into five sections. An outline of the presented material is given in Section 1. Section 2 provides basic definitions and facts related to the Bebutov flow, minimal sets, almost periodic and almost automorphic flows, ergodic properties, Lyapunov exponents, the exponential dichotomy and the rotation number. Some simplifications in the case \(n=2\) are discussed. Section 3 presents results on projective flows for \(n=2\). It begins with a presentation of some consequences of the Floquet theory to the case of periodic \(A\). In the main part of the section a non-periodic \(A: \mathbb R\to sl(2,\mathbb R)\) generating a minimal, almost periodic flow \(\Omega\) is considered. The results related to the number \(k\) of minimal sets of the induced projective flow are provided. In the case \(k\geq 3\), a change of variables reduces the considered equation to a simple autonomous form. A theorem on almost automorphic extensions of each minimal set is given in the case \(k=2\). In the case \(k=1\), three classes of systems are discussed: strongly elliptic, weakly elliptic, and weakly hyperbolic. In the latter case a variety of possible types of dynamics is described in details. Section 4 is devoted to the inverse spectral problem of the Sturm-Liouville operator \(L(\phi):=(p\phi')'-q\phi\), i.e. a reconstruction of uniformly continuous functions \(p\), \(q\), and \(y\) (with some additional properties) satisfying \(L(\phi)=\lambda y \phi\) for a nonzero \(\phi\). It is assumed the spectrum \(\Sigma(L)\) consists of closed intervals and the Lyapunov exponents of the associated two-dimensional nonautonoumous linear equation vanish for \(\lambda\in \Sigma(L)\). Basic facts on the operator \(L\), including the definition and properties of the Weil \(m\)-functions are given in Subsection 4.1. Subsections 4.2 and 4.3 deal with the Bebutov and projective flows for \(\Omega\) corresponding to the considered equations. Results on the spectrum and resolvent set of \(L\), the rotation number of the equation, the Weil \(m\)-functions, and the set of parameters \(\lambda\) for which the equation admits an exponential dichotomy are presented.

Finally, in Subsection 4.4, advanced results from analytic functions theory and algebraic geometry are used in order to determine relations among \(p\), \(q\), and \(y\) in terms of the poles of meromorphic functions induced by the Weyl \(m\)-functions. Moreover, a characterization of systems which satisfy the imposed assumptions on the spectrum and the Lyapunov exponents is provided. In the last Section 5, a genericity theorem is stated an proved: the set of pairs \((\gamma,A)\in \mathbb R^k\times C_0\) such that the equation \(x'=A(\tau_t(\psi))x\) admits an exponential dichotomy is open and dense in \(\mathbb R^k\times C_0\), where \(C_0\) denotes the set of continuous maps \(A: T^k\to sl(2,\mathbb R)\), \(T^k\) denotes the \(k\)-dimensional torus, and the flow \(\tau_t\) on \(T^k\) is given by \(\tau_t(\psi)=\psi+\gamma t\).

For the entire collection see [Zbl 1173.34001].

The article under review surveys results on dynamical properties of these flows and their consequences for solutions of the corresponding nonautonomous equations, mainly in the case \(n=2\). The article is divided into five sections. An outline of the presented material is given in Section 1. Section 2 provides basic definitions and facts related to the Bebutov flow, minimal sets, almost periodic and almost automorphic flows, ergodic properties, Lyapunov exponents, the exponential dichotomy and the rotation number. Some simplifications in the case \(n=2\) are discussed. Section 3 presents results on projective flows for \(n=2\). It begins with a presentation of some consequences of the Floquet theory to the case of periodic \(A\). In the main part of the section a non-periodic \(A: \mathbb R\to sl(2,\mathbb R)\) generating a minimal, almost periodic flow \(\Omega\) is considered. The results related to the number \(k\) of minimal sets of the induced projective flow are provided. In the case \(k\geq 3\), a change of variables reduces the considered equation to a simple autonomous form. A theorem on almost automorphic extensions of each minimal set is given in the case \(k=2\). In the case \(k=1\), three classes of systems are discussed: strongly elliptic, weakly elliptic, and weakly hyperbolic. In the latter case a variety of possible types of dynamics is described in details. Section 4 is devoted to the inverse spectral problem of the Sturm-Liouville operator \(L(\phi):=(p\phi')'-q\phi\), i.e. a reconstruction of uniformly continuous functions \(p\), \(q\), and \(y\) (with some additional properties) satisfying \(L(\phi)=\lambda y \phi\) for a nonzero \(\phi\). It is assumed the spectrum \(\Sigma(L)\) consists of closed intervals and the Lyapunov exponents of the associated two-dimensional nonautonoumous linear equation vanish for \(\lambda\in \Sigma(L)\). Basic facts on the operator \(L\), including the definition and properties of the Weil \(m\)-functions are given in Subsection 4.1. Subsections 4.2 and 4.3 deal with the Bebutov and projective flows for \(\Omega\) corresponding to the considered equations. Results on the spectrum and resolvent set of \(L\), the rotation number of the equation, the Weil \(m\)-functions, and the set of parameters \(\lambda\) for which the equation admits an exponential dichotomy are presented.

Finally, in Subsection 4.4, advanced results from analytic functions theory and algebraic geometry are used in order to determine relations among \(p\), \(q\), and \(y\) in terms of the poles of meromorphic functions induced by the Weyl \(m\)-functions. Moreover, a characterization of systems which satisfy the imposed assumptions on the spectrum and the Lyapunov exponents is provided. In the last Section 5, a genericity theorem is stated an proved: the set of pairs \((\gamma,A)\in \mathbb R^k\times C_0\) such that the equation \(x'=A(\tau_t(\psi))x\) admits an exponential dichotomy is open and dense in \(\mathbb R^k\times C_0\), where \(C_0\) denotes the set of continuous maps \(A: T^k\to sl(2,\mathbb R)\), \(T^k\) denotes the \(k\)-dimensional torus, and the flow \(\tau_t\) on \(T^k\) is given by \(\tau_t(\psi)=\psi+\gamma t\).

For the entire collection see [Zbl 1173.34001].

Reviewer: Roman Srzednicki (Kraków)

##### MSC:

37B55 | Topological dynamics of nonautonomous systems |

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

34C27 | Almost and pseudo-almost periodic solutions to ordinary differential equations |

34D08 | Characteristic and Lyapunov exponents of ordinary differential equations |

34D09 | Dichotomy, trichotomy of solutions to ordinary differential equations |

37A30 | Ergodic theorems, spectral theory, Markov operators |

47E05 | General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX) |