Abstract
In this paper, the general characteristics and the topological consideration of the global behaviors of higher order nonlinear dynamical systems and the characteristics of the application of cell-to-cell mapping method in this analysis are expounded. Specifically, the global analysis of a system of two weakly coupled van der Pol oscillators using cell-to-cell mapping method is presented.
The analysis shows that for this system, there exist two stable limit cycles in 4-dimensional state space, and the whole 4-dimensional state space is divided into two almost equal parts which are, respectively, the two asymptotically stable domains of attraction of the two periodic motions of the two stable limit cycles. The validities of these conclusions about the global behaviors are also verified by direct long term numerical integration. Thus, it can be seen that the cell-to-cell mapping method for global analysis of fourth order nonlinear dynamical systems is quite effective.
Similar content being viewed by others
References
Hsu, C. S., A theory of cell-to-cell mapping for dynamical systems, J. Appl. Mech., 47 (1980), 931–939.
Hsu, C. S. and R. S. Guttalu, An unravelling algorithm for global analysis of dynamical systems: An application of cell-to-cell mappings, J. Appl. Mech., 47 (1980), 940–948.
Hsu, C. S., A generalized theory of cell-to-cell mapping for nonlinear dynamical systems, J. Appl. Mech., 48 (1981), 634–642.
Hsu, C. S. R. S. Guttalu and W. H. Zhu, A method of analyzing generalized cell mapping., J. Appl. Mech., 49 (1982), 885–895.
Hsu, C. S., A probabilistic theory of nonlinear dynamical systems based on the cell state space concept, J. Appl. Mech., 49 (1982), 895–902.
Kauderer, H., Nichtlineare Mechanik, Springer (1958).
Yang, T. K. and R. M. Rosenberg, On forced vibrations of a particle in the plane, Int. J. Non-Linear Mech., 3 (1968), 47–63.
Pavlidis, T., Biological Oscillators: Their Mathematical Analysis, Academic Press (1973).
Rand, R. H. and P. J. Holmes, Bifurcation of periodic motions in two weakly coupled van der Pol oscillators, Int. J. Non-Linear Mech., 15 (1980), 387–399.
Hirsch, M. W., C. C. Pugh and M. Shub, Invariant Manifolds, springer Lecture Notes in Math. No. 583, Springer Verlag (1977).
Chillingworth, D. R. J., Differential Topology with a View to Applications, Pitman, London (1976).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hsu, C.S., Jian-xue, X. The global analysis of higher order nonlinear dynamical systems and the application of cell-to-cell mapping method. Applied Mathematics and Mechanics 6, 1035–1044 (1985). https://doi.org/10.1007/BF03250502
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF03250502