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Discontinuity, Nonlinearity, and Complexity 12(1) (2023) 183--195 | DOI:10.5890/DNC.2023.03.013

Marat Akhmet$^1$, Mehmet Onur Fen$^{2}$, Astrit Tola$^1$

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Abstract

This paper reveals a new way to indicate the presence of chaos in continuous-time models, beside other techniques such as the method of Lyapunov exponents and bifurcation diagrams. The sequential test confirms the existence of an unpredictable solution, and therefore, Poincar\'{e} chaos for differential equations. The main part of the study consists of the description of the novel algorithm, demonstrating its convenience for analysis of chaotic dynamics. The procedure facilities are carefully determined, and they are implemented to Lorenz and R\"{o}ssler systems. The peculiarity of the method lies in the fact that in addition to the indication of just chaos, we clarify the divergence character of a single trajectory, which is based on the unpredictability feature. Potentially it can be more effective than the conventional ways for indication of chaos. The presence of chaos is also approved in the case of zero largest Lyapunov exponent.

References

1. [1]	Akhmet, M. and Fen, M.O. (2016), Unpredictable points and chaos, Communications in Nonlinear Science and Numerical Simulation, 40, 1-5.

2. [2]	Devaney, R.L. (1989), An Introduction to Chaotic Dynamical Systems, Addison-Wesley Publishing Company, Inc.

3. [3]	Li, T.Y. and Yorke, J.A. (1975), Period three implies chaos, The American Mathematical Monthly, 82, 985-992.

4. [4]	Feigenbaum, M.J. (1980), Universal behavior in nonlinear systems, Los Alamos Science/Summer, 1, 16-39.

5. [5]	Gonz{a}les-Miranda, J.M. (2004), Synchronization and Control of Chaos, Imperial College Press.

6. [6]	Hale, J.K. and Ko\c{c}ak, H. (1991), Dynamics and Bifurcations, Springer-Verlag.

7. [7]	Nayfeh, A.H. and Balachandran, B. (1995), Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, John Wiley \& Sons, Inc.

8. [8]	Robinson, C. (1995), Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press.

9. [9]	Sander, E. and Yorke, J.A. (2011), Period-doubling cascades galore, Ergodic Theory and Dynamical Systems, 31(4), 1249-1267.

10. [10]	Schuster, H.G. (1999), Handbook of Chaos Control, Wiley-VCH.

11. [11]	Akhmet, M. and Fen, M.O. (2018), Non-autonomous equations with unpredictable solutions, Communications in Nonlinear Science and Numerical Simulation, 59, 657-670.

12. [12]	Alligood, K.T., Sauer, T.D., and Yorke, J.A. (1996), Chaos: An Introduction to Dynamical Systems, Springer.

13. [13]	Akhmet, M., Fen, M.O., and Alejaily, E.M. (2020), A randomly determined unpredictable function, Kazakh Mathematical Journal, 20, 30-36.

14. [14]	Lorenz, E.N. (1963), Deterministic nonperiodic flow, Journal of Atmospheric Sciences, 20(2), 130-141.

15. [15]	R\"{o}ssler, O.E. (1976), An equation for continuous chaos, Physics Letters A, 57(5), 397-398.

16. [16]	Wei, Z. (2011), Dynamical behaviors of a chaotic system with no equilibria, Physics Letters A, 376(2), 102-108.

17. [17]	Sell, G.R. (1971), Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold Company.

18. [18]	Sparrow, C. (1982), The Lorenz Equations: Bifurcations, Chaos and Strange Attractors, Springer-Verlag.

19. [19]	Franceschini, V. (1980), A Feigenbaum sequence of bifurcations in the Lorenz model, Journal of Statistical Physics, 22(3), 397-406.

20. [20]	Sprott, J.C. (2017), Asymmetric bistability in the R\"{o}ssler system, Acta Physica Polonica B, 48, 97-107.

21. [21]	Palmer, T.N. (1993), Extended-range atmospheric prediction and the Lorenz model, Bulletin of the American Meteorological Society, 74, 49-66.

22. [22]	Lawandy, N.M. and Lee, K. (1987), Stability analysis of two coupled Lorenz lasers and the coupling-induced periodic $\to$ chaotic transition, Optics Communications, 61(2), 137-141.

23. [23]	Cuomo, K.M., Oppenheim, A.V., and Strogatz, S.H. (1993), Synchronization of Lorenz-based chaotic circuits with applications to communications, IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 40(10), 626-633.

24. [24]	Liao, T.L. and Huang, N.S. (1999) An observer-based approach for chaotic synchronization with applications to secure communications, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 46(9), 1144-1150.