Sharp estimates for spectrum of Cauchy operator and periods of solutions of Lipschitz differential equations

TL;DR

This paper provides sharp estimates for the spectrum of the Cauchy operator and solution periods in Lipschitz differential equations, with applications to stability, uniqueness, and optimal norm selection.

Contribution

It introduces precise spectral and period estimates for non-autonomous differential equations, extending results to nonlinear, higher-order, and Hamiltonian systems, with norm optimization insights.

Findings

Lower bounds for periods of oscillatory solutions with periodic coefficients

Conditions for uniqueness of oscillatory solutions

Stability criteria for Hamiltonian and nonlinear systems

Abstract

Estimates for the spectrum of the Cauchy operator and logarithms of solutions of non-autonomous differential equations in the space, expressed in an arbitrary matrix norm, are found. For equations with periodic coefficients, the lower bound for the periods of "oscillatory" solutions is obtained. Similar results are derived for a nonlinear periodic equation with zero equilibrium position; in the autonomous case, they are valid for any periodic solution. All the estimates are attained at a scalar equation and, thereby, are accurate for any norm. The estimates are extended to equations with derivatives of any order. Using the obtained results, a condition for the uniqueness of an oscillatory solution is found; stability criteria for a linear Hamiltonian system with periodic coefficients and for periodic solutions of a nonlinear system (expressed through the period and any norm of the Hessian) are obtained. Types of matrices and functions, for which the Euclidean norm has a minimum value (and, therefore, provides the most accurate estimates for a specific equation), are established.