Invariant measures for complex-valued dissipative dynamical systems and applications

Xin Li; Wenxian Shen; Chunyou Sun

doi:10.3934/dcdsb.2017124

Article Contents

2017, Volume 22, Issue 6: 2427-2446. Doi: 10.3934/dcdsb.2017124

This issue Previous Article Limit cycle bifurcations for piecewise smooth integrable differential systems Next Article Nonsmooth frameworks for an extended Budyko model

Invariant measures for complex-valued dissipative dynamical systems and applications

1.
School of Mathematical and Statistics, Lanzhou University, Lanzhou, Gansu, China
2.
Department of Mathematics and Statistics, Auburn University, Auburn, AL, USA

Author Bio: E-mail address: lix13@lzu.edu.cn; E-mail address: wenxish@auburn.edu
E-mail address: sunchy@lzu.edu.cn
E-mail address: sunchy@lzu.edu.cn

Received: May 31, 2016

Revised: January 31, 2017

Published: May 2017

This work was partly supported by the NSFC (Grants No. 11471148,11522109).

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Abstract

In this work, we extend the classical real-valued framework to deal with complex-valued dissipative dynamical systems. With our new complex-valued framework and using generalized complex Banach limits, we construct invariant measures for continuous complex semigroups possessing global attractors. In particular, for any given complex Banach limit and initial data $u_{0}$, we construct a unique complex invariant measure $\mu$ on a metric space which is acted by a continuous semigroup $\{S(t)\}_{t\geq 0}$ possessing a global attractor $\mathcal{A}$. Moreover, it is shown that the support of $\mu$ is not only contained in global attractor $\mathcal{A}$ but also in $\omega(u_{0})$. Next, the structure of the measure $\mu$ is studied. It is shown that both the real and imaginary parts of a complex invariant measure are invariant signed measures and that both the positive and negative variations of a signed measure are invariant measures. Finally, we illustrate the main results of this article on the model examples of a complex Ginzburg-Landau equation and a nonlinear Schrödinger equation and construct complex invariant measures for these two complex-valued equations.

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Mathematics Subject Classification: Primary:35B41, 35Q55, 35Q56, 37L40, 76F20.

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