On chaotic wave patterns in periodically forced steady-state KdVB and extended KdVB equations

Abstract

The periodically forced KdVB and extended KdVB equations are considered. We investigate the structure of the totality of steady profiles. The existence of profiles that are close to any shuffling of two basic profiles is proved, and hence the existence of spatially chaotic and recurrent solutions. The proofs are based on topological degree theory to analyse chaotic behaviour. These proofs combine ideas suggested by P. Zgliczyński ( Zgliczyński 1996 Topol. Methods Nonlinear Anal . 8 , 169–177 ) with the method of topological shadowing. The results are also applicable to the classical problem of a quite general model of a forced nonlinear oscillator with viscous damping.