#### For citation:

Govorukhin V. N., Tsybulin V. G., Tyaglov M. Y. Multistability and memory effects in dynamical system with cosymmetric potential. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2020, vol. 28, iss. 3, pp. 259-273. DOI: 10.18500/0869-6632-2020-28-3-259-273

# Multistability and memory effects in dynamical system with cosymmetric potential

The purpose of present study is the analysis of strong multistability in a dynamical system with cosymmetry. We study the dynamics and realization of steady-states in a mechanical system with two degrees of freedom. The minimum potential energy of the system is achieved on a curve in the form of an ellipse, which gives rise to a continuum family of equilibria and strong multistability. This problem belongs to the class of dynamical systems with cosymmetry. Methods. To analyze the system, we used methods of computational qualitative analysis of dynamical systems and cosymmetry theory. Results. The behavior of the system is studied for different values of the initial potential energy, semi-axes of the ellipse and the friction coefficient. In the conservative cosymmetric case, the existence of chaotic regions in the phase space is established. In the presence of friction, a complex dependence of the realization of equilibria on the family on the initial data was numerically found, which is due to the memory effect of conservative chaos. The results of the analysis of the system in case of violation of cosymmetry are presented and the memory effects of the destroyed family of equilibria are demonstrated. Conclusion. With strong multistability, the effects of memory about the properties of the system under a small violation of them have a significant effect on the dynamics. Despite the determinism in the presence of friction (all trajectories tend to equilibria), there is a strong dependence of the implementation of equilibria on the initial data, which is characteristic of chaos. With a small violation of cosymmetry, the system also demonstrates the memory of the disappeared continual family of equilibria: from all the initial data, the trajectories first tend to the neighborhood of the family, and then slowly drift along it to one of the remaining equilibria.

1. Feudel U. Complex dynamics in multistable systems // Int. J. of Bifurcation and Chaos. 2008. Vol. 18, no. 6. P. 1607–1626.

2. Felk E.V., Kuznetsov A.P., Savin A.V. Multistability and transition to chaos in the degenerate Hamiltonian system with weak nonlinear dissipative perturbation // Physica A: Statistical Mechanics and its Applications. 2014. Vol. 410. P. 561–557.

3. Shabunin A.V. Multistability of traveling waves in an ensemble of harmonic oscillators with long-range couplings. Izvestiya VUZ. Applied Nonlinear Dynamics, 2018, vol. 26, no. 1 pp. 48–63 (in Russian).

4. Govorukhin V.N., Yudovich V.I. Bifurcations and selection of equilibria in a simple cosymmetric model of filtrational convection // Chaos. 1999. Vol. 9. P. 403–412.

5. Gotthans T., Petrzela J. New class of chaotic systems with circular equilibrium // Nonlinear Dyn. 2015. Vol. 81. P. 1143–1149.

6. Li C., Sprott J.C., Hu W., Xu Y. Infinite multistability in a self-reproducing chaotic system // Intern. J. of Bifurcation and Chaos. 2017. Vol. 27, no. 10. 1750160.

7. Budyansky A.V., Frischmuth K., Tsybulin V.G. Cosymmetry approach and mathematical modeling of species coexistence in a heterogeneous habitat // Discrete & Continuous Dynamical Systems. 2019. Vol. 24. P. 547–561.

8. Riaza R. Transcritical bifurcation without parameters in memristive circuit // SIAM J. Appl. Math. 2018. Vol. 78, no. 1. P. 395–417.

9. Golubitsky M., Swift J., Knobloch E. Symmetries and pattern selection in Rayleigh–Benard convection // Physica D. 1984. Vol. 10. P. 249–276.

10. Yudovich V.I. Cosymmetry, degeneracy of the solutions of operator equations and the onset of filtrational convection. Math. Notes, 1991, vol. 49, pp. 540–545.

11. Yudovich V.I. Secondary cycle of equilibria in a system with cosymmetry, its creation by bifurcation and impossibility of symmetric treatment of it // Chaos. 1995. Vol. 5, no. 2. P. 402–411.

12. Yudovich V.I. Bifurcations under perturbations violating cosymmetry. Physics-Doklady, 2004, vol. 49, pp. 522–526.

13. Bratsun D.A., Lyubimov D.V., Roux B. Co-symmetry breakdown in problems of thermal convection in porous medium // Physica D. 1995. Vol. 82. P. 398–417.

14. Govorukhin V.N., Shevchenko I.V. Multiple equilibria, bifurcations and selection scenarios in cosymmetric problem of thermal convection in porous medium // Physica D. 2017. Vol. 361. P. 2–58.

15. Liebscher S., Harterich J., Webster K., Georgi M. Ancient dynamics in Bianchi models: Approach to periodic cycles // Commun. Math. Phys. 2011. Vol. 305. P. 59–83.

16. Frischmuth K., Kovaleva E.S., Tsybulin V.G. Family of equilibria in a population kinetics model and its collapse // Nonlinear Analysis: Real World Applic. 2011. Vol. 12. P. 146–155.

17. Korneev I.A., Vadivasova T.E., Semenov V.V. Hard and soft excitation of oscillations in memristorbased oscillators with a line of equilibria // Nonlinear Dynamics. 2017. Vol. 89. P. 2829–2843.

18. Nonlinear Dynamical Systems with Self-Excited and Hidden Attractors. Eds V.H. Pham, S. Vaidyanathan, C. Volos, T. Kapitaniak. Springer, New York, 2018.

19. Govorukhin V.N., Tsybulin V.G., Karasozen B. Dynamics of numerical methods for cosymmetric ordinary differential equations // Int. J. Bifurcation and Chaos. 2001. Vol. 11. P. 2339–2357.

20. Kurakin L.G., Yudovich V.I. Bifurcations accompanying monotonic instability of an equilibrium of a cosymmetric dynamical system // Chaos. 2000 Vol. 10. P. 311–330.

21. Yudovich V.I. Cycle-creating bifurcation from a family of equilibria of a dynamical system and its delay. J. Appl. Math. and Mech., 1998, vol. 62, no. 1, pp. 19–29.

22. Karasozen B., Tsybulin V.G. Destruction of the family of steady states in the planar problem of Darcy convection // Physics Letters A. 2008. Vol. 372. P. 5639–5643.

23. Govorukhin V.N. On the action of internal heat sources on convective motion in a porous medium heated from below. J. Applied Mechanics and Technical Physics, 2014, vol. 55, pp. 225–233.

24. Tsybulin V.G., Karasozen B., Ergenc T. Selection of steady states in planar Darcy convection // Physics Letters A. 2006. Vol. 356. P. 189–194.

25. Govorukhin V.N., Shevchenko I.V. Selection of steady regimes of a one-parameter family in the problem of plane convective flow through a porous medium. Fluid Dynamics, 2013, vol. 48, pp. 523–532.

26. Hairer E., Lubich C., Wanner G. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, 2006.

- 1742 reads