Journal
NONLINEAR DYNAMICS
Volume 47, Issue 1-3, Pages 263-273Publisher
SPRINGER
DOI: 10.1007/s11071-006-9072-6
Keywords
nonlinear systems; periodic; quasiperiodic; Lyapunov-Floquet transformation; normal forms
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In this paper a general technique for the analysis of nonlinear dynamical systems with periodic-quasiperiodic coefficients is developed. For such systems the coefficients of the linear terms are periodic with frequency omega while the coefficients of the nonlinear terms contain frequencies that are incommensurate with omega. No restrictions are placed on the size of the periodic terms appearing in the linear part of system equation. Application of Lyapunov-Floquet transformation produces a dynamically equivalent system in which the linear part is time-invariant and the time varying coefficients of the nonlinear terms are quasiperiodic. Then a series of quasiperiodic near-identity transformations are applied to reduce the system equation to a normal form. In the process a quasiperiodic homological equation and the corresponding 'solvability condition' are obtained. Various resonance conditions are discussed and examples are included to show practical significance of the method. Results obtained from the quasiperiodic time-dependent normal form theory are compared with the numerical solutions. A close agreement is found.
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