Dynamics of Dissipative Systems with Hamiltonian Structures

Xiaoming Zhang, Zhenbang Cao, Jianhua Xie, Denghui Li, Celso Grebogi

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In this work, we study a class of dissipative, nonsmooth n degree-of-freedom dynamical systems. As the dissipation is assumed to be proportional to the momentum, the dynamics in such systems is conformally symplectic, allowing us to use some of the Hamiltonian structure. We initially show that there exists an integral invariant of the Poincar ́e–Cartan type in such systems. Then, we prove the existence of a generalized Liouville Formula for conformally symplectic systems with rigid constraints using the integral invariant. A two degree-of-freedom system is analyzed to support the relevance of our results.
Original languageEnglish
Article number2150217
Number of pages9
JournalInternational Journal of Bifurcation and Chaos
Issue number14
Early online date28 Aug 2021
Publication statusPublished - 30 Nov 2021

Bibliographical note

This work is supported by the National Natural Science Foundation of China (11732014).
Data Availability
The data that support the findings of this study are available within the article.


  • Integral invariant
  • impact system
  • generalized Liouvile Formula


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