Implementation of a generalized intermittency scenario in the Rössler dynamical system

Authors

DOI:

https://doi.org/10.20535/SRIT.2308-8893.2026.1.07

Keywords:

ideal dynamical system, regular and chaotic attractors, generalized intermittency scenario

Abstract

The realization of novel scenario involving transitions between different types of chaotic attractors is investigated for the Rössler system. Characteristic features indicative of the presence of generalized intermittency scenario in this system are identified. The properties of “chaos–chaos” transitions following the generalized intermittency scenario are analyzed in detail based on phase-parametric characteristics, Lyapunov characteristic exponents, phase portraits, and Poincaré sections.

References

A. Sommerfeld, “Beitrage zum dynamischen Ausbau der Festigkeitslehre,” Physikalische Zeitschrift, 3, pp. 266–271, 1902.

V.O. Kononenko, Vibrating system with a limited power-supply. London: Iliffe,1969, 236 p.

P. Manneville, Y. Pomeau, “Different ways to turbulence in dissipative dynamical systems,” Physica D. Nonlinear Phenom., vol. 1, issue 2, pp. 219–226, 1980. doi: https://doi.org/10.1016/0167-2789(80)90013-5

Y. Pomeau, P. Manneville, “Intermittent transition to turbulence in dissipative dynamical systems,” Comm. Math. Phys., vol. 74, pp. 189–197, 1980. doi: https://doi.org/10.1007/BF01197757

M.J. Feigenbaum, “Quantitative universality for a class of nonlinear transformations,” J. Stat. Phys., vol. 19, pp. 25–52, 1978. doi: https://doi.org/10.1007/BF01020332

M.J. Feigenbaum, “The universal metric properties of nonlinear transformations,” J. Stat. Phys., vol. 2, pp. 669–706, 1979. doi: https://doi.org/10.1007/BF01107909

A. Shvets, “Overview of Scenarios of Transition to Chaos in Nonideal Dynamic Systems,” 13th Chaotic Modeling and Simulation International Conference. CHAOS 2020. Springer Proceedings in Complexity, pp. 853–864, Springer, Cham, 2021. doi: https://doi.org/10.1007/978-3-030-70795-8_59

A. Shvets, S. Donetskyi, “New Types of Limit Sets in the Dynamic System “Spherical Pendulum—Electric Motor,” Nonlinear Mechanics of Complex Structures. Advanced Structured Materials, vol. 157, pp. 443–455, 2021. doi: https://doi.org/10.1007/978-3-030-75890-5_25

S.V. Donetskyi, A.Yu. Svets, “Generalization of the concept of attractor for pendulum systems with limited excitation,” J. Math. Sci., vol. 273, pp. 220–229, 2023. doi: https://doi.org/10.1007/s10958-023-06550-7

O. Horchakov, A. Shvets, “Generalized scenarios of transition to chaos in ideal dynamic systems,” System Research and Information Technologies, no. 3, pp. 64–73, 2024. doi: https://doi.org/10.20535/srit.2308-8893.2024.3.04

A. Shvets, “Generalised Intermittency in Non-ideal and “Classical” Dynamical Systems,” Analytical and Approximate Methods for Complex Dynamical Systems. Understanding Complex Systems, pp. 75–87, Springer, Cham, 2025. doi: https://doi.org/10.1007/978-3-031-77378-5_5

E. Hairer, S.P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems. Berlin, Springer-Verlag, 1987.

G. Benettin, L. Galgani, J.-M. Strelcyn, “Kolmogorov entropy and numerical experiments,” Phys. Rev. A, 14, 2338, 1976. doi: https://doi.org/10.1103/PHYSREVA.14.2338

G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, “Lyapunov Characterictic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application,” Meccanica, vol. 15, pp. 21–30, 1980. doi: https://doi.org/10.1007/BF02128237

Handbook of Applications of Chaos Theory; Edited by C.H. Skiadas, Char. Skiadas. Chapman and Hall/CRC, 2016, 952 p. doi: https://doi.org/10.1201/b20232

M. Hénon, “A two-dimensional mapping with a strange attractor,” Communications in Mathematical Physics, vol. 50, pp. 69–77, 1976. doi: https://doi.org/10.1007/BF01608556

S.P. Kuznetsov, Dynamic Chaos. M.: Fizmatlit, 2006, 292 p.

T.S. Krasnopolskaya, A.Yu. Shvets, “Properties of chaotic oscillations of the liquid in cylindrical tanks,” Prikl. Mekh., 28(6), pp. 52–61, 1992.

T.S. Krasnopol'skaya, A.Y. Shvets, “Parametric resonance in the system: Liquid in tanks + electric motor,” Int. Appl. Mech., vol. 29, pp. 722–730, 1993. doi: https://doi.org/10.1007/BF00847371

O.E. Rössler, “An Equation for Continuous Chaos,” Physics Letters A, vol. 57, issue 5, pp. 397–398, 1976. doi: https://doi.org/10.1016/0375-9601(76)90101-8

O.E. Rössler, “Continuous Chaos – Four Prototype Equations,” Annals of the New York Academy of Sciences, 316, pp. 376–392, 1979. doi: https://doi.org/10.1111/j.1749-6632.1979.tb29482.x

O.E. Rössler, C. Letellier, “The Phenomenon of Chaos,” Chaos. Understanding Complex Systems. Springer, Cham, 2020. doi: https://doi.org/10.1007/978-3-030-44305-4_1

E.N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci., vol. 20, issue 2, pp. 130–141, 1963. doi: https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2

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2026-03-31

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Mathematical methods, models, problems and technologies for complex systems research