The Concept of Synchronization from the Observer's Viewpoint

Abstract

In this paper the concept of synchronization for continuous time dynamical systems from the viewpoint of an observer is considered. It is proved that: this concept is a generalization of the notion of synchronization. It is proved that the future of the points of the set in which two dynamical systems are relative probability synchronized is the same up to the homeomorphism determined by a relative probability synchronization. The persistence of relative probability synchronization under a topological conjugate relation is deduced.

Keywords

• Synchronization,relative probability measure,observer,relative conjugacy

References

1. [1] S. Chen and J. Lu, Synchronization of an uncertain unified chaotic system via addaptive control, Chaos, Solitons and Fractals 14 (2002), 643–647.
2. [2] B. Kılıc and E. Bas, Complex solutions for the Fisher equation and the Benjamin-BonaMahony equation, Cankaya University Journal of Science and Engineering 7 (2010), 87–93.
3. [3] J. Lu, G. Chen, S. Zhang and S. Celikovsky, Bridge the gap between the Lorenz system and the Chen system, International Journal of Bifurcation and Chaos 12 (2002), 2917–2926.
4. [4] R. A. Mashiyev, Some applications to Lebesgue points in variable exponent Lebesgue spaces, Cankaya University Journal of Science and Engineering 7 (2010), 105–113.
5. [5] M. R. Molaei and B. Ghazanfari, Relative probability measures, Fuzzy Sets, Rough Sets, Multivalued Operations and Applications 1 (2008), 89–97.
6. [6] D. B. Pourkargar and M. Shahrokhi, Optimal fuzzy synchronization of generalized Lorenz chaotic systems, The Journal of Mathematics and Computer Science 2 (2011), 27–36.
7. [7] M. R. Molaei, Relative semi-dynamical systems, International Journal of Uncertainty, Fuzziness and Knowledge-based Systems 12 (2004), 237–243.
8. [8] M. R. Molaei, Observational modeling of topological spaces, Chaos, Solitons and Fractals 42 (2009), 615–619.

9. [9] L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338–353.
10. [10] R. Mane, Ergodic Theory and Differentiable Dynamics, Springer-Verlag 1987.
11. [11] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag 1982.
12. [12] E. N. Lorenz, Deterministic nonperiodic flow, Journal of Atmospheric Sciences 20 (1963), 130–141.
13. [13] J. Lu and G. Chen, A new chaotic attractor coined, International Journal of Bifurcation and Chaos 12 (2002), 659–661.
14. [14] J. Lu, T. Zhou and S. Zhang, Chaos synchronization between linearly coupled chaotic systems, Chaos, Solitons and Fractals 14 (2002), 529–541.
15. [15] M. R. Molaei and B. Ghazanfari, Relative entropy of relative measure preserving maps with constant observers, Journal of Dynamical Systems and Geometric Theories 5 (2007), 179–191.