Abstract
In this paper, from the view of stability and chaos control, we investigate the Rossler chaotic system with delayed feedback. At first, we consider the stability of one of the fixed points, verifying that Hopf bifurcation occurs as delay crosses some critical values. Then, for determining the stability and direction of Hopf bifurcation we derive explicit formulae by using the normal-form theory and center manifold theorem. By designing appropriate feedback strength and delay, one of the unstable equilibria of the Rossler chaotic system can be controlled to be stable, or stable bifurcating periodic solutions occur at the neighborhood of the equilibrium. Finally, some numerical simulations are carried out to support the analytic results.
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References
Rossler, O.E.: An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1976)
Chen, G., Lu, J.: Dynamical Analysis, Control and Synchronization of Lorenz Families. Chinese Science Press, Beijing (2003)
Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421–428 (1992)
Yang, T., Yang, C., Yang, L.: Control of Rossler system to periodic motions using impulsive control methods. Phys. Lett. A 232(5), 356–361 (1997)
Rafikov, M., Balthazar, J.M.: On an optimal control design for Rossler systems. Phys. Lett. A 333, 241–245 (2004)
Agiza, H.N., Yassen, M.T.: Synchronization of Rossler and Chen chaotic dynamical systems using active control. Phys. Lett. A 278(4), 191–197 (2001)
Chen, C., Yan, J., Liao, T.: Sliding mode control for synchronization of Rossler systems with time delays and its application to secure communication. Phys. Scr. 76, 436–441 (2007)
Ghosh, D., Chowdhury, A.R., Saha, P.: Multiple delay Rossler system—Bifurcation and chaos control. Chaos Solitons Fractals 35, 472–485 (2008)
Vasegh, N., Sedigh, A.K.: Delayed feedback control of time-delayed chaotic systems: Analytical approach at Hopf bifurcation. Phys. Lett. A 372, 5110–5114 (2008)
Kittel, A., Parisi, J., Pyragas, K.: Delayed feedback control of chaos by self-adapted delay time. Phys. Lett. A 198, 433–436 (1995)
Song, Y., Wei, J.: Bifurcation analysis for Chen’s system with delayed feedback and its application to control of chaos. Chaos Solitons Fractals 22, 75–91 (2004)
Ruan, S., Wei, J.: On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 10(6), 863–874 (2003)
Qu, Y., Wei, J.: Bifurcation analysis in a time-delay model for prey–predator growth with stage-structure. Nonlinear Dyn. 49, 285–294 (2007)
Jiang, W., Yuan, Y.: Bogdanov–Takens singularity in Van der Pol’s oscillator with delayed feedback. Physica D 227(2), 149–161 (2007)
Wei, J., Li, M.Y.: Global existence of periodic solutions in a tri-neuron network model with delays. Physica D 198, 106–119 (2004)
Campbell, S.A., Yuan, Y.: Zero singularities of codimension two and three in delay differential equations. Nonlinearity 21, 2671–2691 (2008)
Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)
Hale, J.: Theory of Functional Differential Equations. Springer, New York (1977)
Lancaster, P., Tismenetsky, M.: The Theory of Matrices with Applications. Academic Press, London (1985)
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Ding, Y., Jiang, W. & Wang, H. Delayed feedback control and bifurcation analysis of Rossler chaotic system. Nonlinear Dyn 61, 707–715 (2010). https://doi.org/10.1007/s11071-010-9681-y
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DOI: https://doi.org/10.1007/s11071-010-9681-y