Global existence, uniqueness and pathwise property of solutions to a stochastic Rössler-Lorentz system

Abstract

The authors integrate two well-known systems, the Rössler and Lorentz systems, to introduce a new chaotic system, called the Lorentz-Rössler system. Then, taking into account the effect of environmental noise, the authors incorporate white noise in both Rössler and Lorentz systems to have a corresponding stochastic system. By deriving the uniform a priori estimates for an approximate system and then taking them to the limit, the authors prove the global existence, uniqueness and the pathwise property of solutions to the Lorentz-Rössler system. Moreover, the authors carried out a number of numerical experiments, and the numerical results demonstrate their theoretic analysis and show some new qualitative properties of solutions which reveal that the Lorentz-Rössler system could be used to design more complex and more secure nonlinear hop-frequence time series.

Keywords

Stochastic differential equations / Rössler-Lorentz systems / Existence, Pathwise property

References

[1] Chen G, Dong X. From Chaos to Order Methodologies Perspectives and Applications, 1994, Singapore: World Scientific

[2] Chen S H, Lü J H. Synchronization of an uncertain unified chaotic systems using backstepping design. Chaos, Solitions and Fractals, 2002, 14: 643-647

[3] Cheng S R. Stochastic population systems. Stochastic Analysis and Applications, 2009, 27: 854-874

[4] Cuomo K M. Circuit implementation of synchronized chaos with application to communications. Phys. Rev. Lett., 1993, 71: 65-68

[5] Fang J Q, Hong Y, Chen G. Switching manifold approach to chaos synchronization. Phys. Rev. E, 1999, 59: 2523-2526

[6] Gorman M, Widmann P J, Robbins K A. Nonlinear dynamics of a convection loop: A quantitative comparison of experiment with theory. Phys. D, 1986, 19: 255-267

[7] Haken H. Analogy between higher instabilities in fluids and lasers. Phys. Lett. A, 1975, 53: 77-78

[8] Hemnti N. Strange attractors inbrushless DC motors. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 1994, 41: 40-45

[9] Itoh M, Yang T, Chua L O. Conditions for impulsive synchronization of chaotic and hyperchaotic systems. Int. J. Bifur. Chaos, 2001, 11: 551-560

[10] Knobloch E. Chaos in the segmented disc dynamics. Phys. Lett. A, 1981, 82(9): 439-440

[11] Liu M, Wang K. Persistence, extinction and global asymptotical stability of a non-autonomous predator-prey model with random perturbation. Appl. Math. Modelling, 2012, 36: 5344-5353

[12] Lorentz E N. Determinstic nonperiodic flows. J. Atmospheric Sci., 1963, 12: 139-163

[13] Mao X. Exponential Stability of Stochastic Differential Equations, 1994, New York: Marcel Dekker Inc.

[14] Mao X. Stochastic Differential Equations and Applications, 1997, Chichester: Horwood Publishing

[15] Peifer M, Schelter B, Winterhalder M Mixing properties of the Rössler system and consequences for coherence and synchronization analysis. Phys. Rev. E., 2005, 72: 1-7

[16] Peitgen H O, Jürgens H, Saupe D. Chaos and Fractals, 1992, Berlin, Heidelberg: Springer-Verlag

[17] Poland D. Cooperative catalysis and chemical chaos: a chemical model for Lorentz equation. Phys. D, 1993, 65: 86-99

[18] Rössler O E. An equation for continuous chaos. Phys. Lett. A, 1976, 57: 397-398

[19] Xing Z W, Peng J G. Boundedness, persistence and extinction of a stochastic non-autonomous logistic system with time delays. Appl. Math. Modelling, 2012, 36: 3379-3386

[20] Yang Q S, Jiang D Q. A note on asymptotic behaviors of stochastic population model with Allee effect. Appl. Math. Modelling, 2011, 35: 4611-4619