Using an example of a radiophysical generator model, scenarios for the formation of various chaotic attractors are described, including chaos and hyperchaos. It is shown that as a result of a secondary Neimark–Sacker bifurcation, a hyperchaos with two positive Lyapunov exponents can occur in the system. A comparative analysis of chaotic attractors born as a result of loss of smoothness of an invariant curve, as a result of period-doubling bifurcations, and as a result of secondary Neimark–Sacker bifurcation was carried out.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
System (1) is four-dimensional and in fact it has four Lyapunov exponents, but the fourth exponent is always negative, and we did not add it to the plots.
In accordance with [40, 41], if for a flow dynamical system at variation of the parameter, the largest Lyapunov exponent is equal zero and two of the following are negative and equal to each other before bifurcation, and if two of the largest Lyapunov exponents are equal zero and the third is negative after bifurcation, it indicates the Neimark–Sacker bifurcation.
Also sometimes it is called adiabatic initial conditions, i.e., for each new value of the parameter, the initial conditions were chosen as the final state attained for the previous value of the parameter.
Schuster, H.G.: Deterministic chaos: an introduction, p. 220. Physik-Verlag, Weinheim (1984)
Mosekilde, E., Maistrenko, Y., Postnov, D.: Chaotic Synchronization: Applications to Living Systems, vol. 42. World Scientific, Singapore (2002)
Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Westview Press, Boulder (2014)
Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.-M.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: a method for computing all of them. Meccanica 15, 9–30 (1980)
Pikovsky, A., Politi, A.: Lyapunov Exponents: A Tool to Explore Complex Dynamics. Cambridge University Press, Cambridge (2016)
Rossler, O.E.: An equation for hyperchaos. Phys. Lett. A 71, 155–157 (1979)
Tamasevicius, A., Namajunas, A., Cenys, A.: Simple 4D chaotic oscillator. Electron. Lett. 32, 957–958 (1996)
Blokhina, E.V., Kuznetsov, S.P., Rozhnev, A.G.: High-dimensional chaos in a gyrotron. IEEE Trans. Electron Dev. 54, 188–193 (2007)
Rozental’, R.M., Isaeva, O.B., Ginzburg, N.S., Zotova, I.V., Sergeev, A.S., Rozhnev, A.G.: Characteristics of chaotic regimes in a space-distributed gyroklystron model with delayed feedback. Russ. J. Nonlinear Dyn. 14, 155–168 (2018)
Chen, Z., Yang, Y., Qi, G., Yuan, Z.: A novel hyperchaos system only with one equilibrium. Phys. Lett. A 360, 696–701 (2007)
Wu, W., Chen, Z., Yuan, Z.: The evolution of a novel fourdimensional autonomous system: among 3-torus, limit cycle, 2-torus, chaos and hyperchaos. Chaos Solitons Fractals 39, 2340–2356 (2009)
Li, Q., Tang, S., Zeng, H., Zhou, T.: On hyperchaos in a small memristive neural network. Nonlinear Dyn. 78, 1087–1099 (2014)
Li, Q., Zeng, H., Li, J.: Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria. Nonlinear Dyn. 79, 2295–2308 (2015)
Biswas, D., Banerjee, T.: A simple chaotic and hyperchaotic time-delay system: design and electronic circuit implementation. Nonlinear Dyn. 83, 2331–2347 (2016)
Fonzin, T.F., Kengne, J., Pelap, F.B.: Dynamical analysis and multistability in autonomous hyperchaotic oscillator with experimental verification. Nonlinear Dyn. 93, 653–669 (2018)
Kapitaniak, T., Thylwe, K.E., Cohen, I., Wojewoda, J.: Chaos-hyperchaos transition. Chaos Solitons Fractals 5, 2003–2011 (1995)
Harrison, M.A., Lai, Y.C.: Route to high-dimensional chaos. Phys. Rev. E 59, R3799 (1999)
Kapitaniak, T., Maistrenko, Y., Popovych, S.: Chaos–hyperchaos transition. Phys. Rev. E 62, 1972 (2000)
Yanchuk, S., Kapitaniak, T.: Symmetry-increasing bifurcation as a predictor of a chaos–hyperchaos transition in coupled systems. Phys. Rev. E 64, 056235 (2001)
Nikolov, S., Clodong, S.: Hyperchaos–chaos–hyperchaos transition in modified Rössler systems. Chaos Solitons Fractals 28, 252–263 (2006)
Harikrishnan, K.P., Misra, R., Ambika, G.: On the transition to hyperchaos and the structure of hyperchaotic attractors. Eur. Phys. J. B 86, 1–12 (2013)
Li, Q., Tang, S., Yang, X.S.: Hyperchaotic set in continuous chaos–hyperchaos transition. Commun. Nonlinear Sci. Numer. Simul. 19, 3718–3734 (2014)
Munteanu, L., Brian, C., Chiroiu, V., Dumitriu, D., Ioan, R.: Chaos–hyperchaos transition in a class of models governed by Sommerfeld effect. Nonlinear Dyn. 78, 1877–1889 (2014)
Gonchenko, S.V., Ovsyannikov, I.I., Simò, C., Turaev, D.: Three-dimensional Hénon-like maps and wild Lorenz-like attractors. Int. J. Bifurc. Chaos 15, 3493–3508 (2005)
Gonchenko, A.S., Gonchenko, S.V., Shilnikov, L.P.: Towards scenarios of chaos appearance in three-dimensional maps. Russ. J. Nonlinear Dyn. 8, 3–28 (2012). (Russian)
Gonchenko, A., Gonchenko, S., Kazakov, A., Turaev, D.: Simple scenarios of onset of chaos in three-dimensional maps. Int. J. Bifurc. Chaos 24, 1440005 (2014)
Stankevich, N.V., Dvorak, A., Astakhov, V., Jaros, P., Kapitaniak, M., Perlikowski, P., Kapitaniak, T.: Chaos and hyperchaos in coupled antiphase driven toda oscillators. Regul. Chaotic Dyn. 23, 120–126 (2018)
Kuznetsov, A.P., Sedova, Y.V.: Coupled systems with hyperchaos and quasiperiodicity. J. Appl. Nonlinear Dyn. 5, 161–167 (2016)
Rech, P.C.: Hyperchaos and quasiperiodicity from a four-dimensional system based on the Lorenz system. Eur. Phys. J. B 90, 251 (2017)
Amabili, M., Karagiozis, K., Païdoussis, M.P.: Hyperchaotic behaviour of shells subjected to flow and external force. In: ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting collocated with 8th International Conference on Nanochannels, Microchannels, and Minichannels, American Society of Mechanical Engineers, pp. 1209–1217 (2010)
Kuznetsov, A.P., Migunova, N.A., Sataev, I.R., Sedova, YuV, Turukina, L.V.: From chaos to quasi-periodicity. Regul. Chaotic Dyn. 20, 189–204 (2015)
Stankevich, N.V., Astakhov, O.V., Kuznetsov, A.P., Seleznev, E.P.: Exciting chaotic and quasi-periodic oscillations in a multicircuit oscillator with a common control scheme. Tech. Phys. Lett. 44, 428–431 (2018)
Anishchenko, V.S., Nikolaev, S.M.: Generator of quasi-periodic oscillations featuring two-dimensional torus doubling bifurcations. Tech. Phys. Lett. 31, 853–855 (2005)
Anishchenko, V.S., Nikolaev, S.M., Kurths, J.: Peculiarities of synchronization of a resonant limit cycle on a two-dimensional torus. Phys. Rev. E 76, 046216 (2007)
Kuznetsov, A.P., Stankevich, N.V.: Autonomous systems with quasiperiodic dynamics. Examples and their properties: review. Izv. VUZ Appl. Nonlinear Dyn. 23, 71–93 (2015). (in Russia)
Kuznetsov, N.V.: The Lyapunov dimension and its estimation via the Leonov method. Phys. Lett. A 380, 2142–2149 (2016)
Kuznetsov, N.V., Leonov, G.A., Mokaev, T.N., Prasad, A., Shrimali, M.D.: Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system. Nonlinear Dyn. 92, 267–285 (2018)
Zhusubaliyev, Z.T., Mosekilde, E.: Formation and destruction of multilayered tori in coupled map systems. Chaos 18, 037124 (2008)
Zhusubaliyev, Z.T., Laugesen, J.L., Mosekilde, E.: From multi-layered resonance tori to period-doubled ergodic tori. Phys. Lett. A 374, 2534–2538 (2010)
Vitolo, R., Broer, H., Simó, C.: Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms. Nonlinearity 23, 1919–1947 (2010)
Broer, H., Simó, C., Vitolo, R.: Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems. Regul. Chaotic Dyn. 16, 154–184 (2011)
Komuro, M., Kamiyama, K., Endo, T., Aihara, K.: Quasi-periodic bifurcations of higher-dimensional tori. Int. J. Bifurc. Chaos 26, 1630016 (2016)
Wieczorek, S., Krauskopf, B., Lenstra, D.: Mechanisms for multistability in a semiconductor laser with optical injection. Opt. Commun. 183, 215–226 (2000)
Stankevich, N.V., Volkov, E.I.: Multistability in a three-dimensional oscillator: tori, resonant cycles and chaos. Nonlinear Dyn. 94, 2455–2467 (2018)
Authors thank Igor Sataev, Alexey Kazakov and Serhiy Yanchuk for fruitful discussion of this problem.
The work was carried out with the financial support of the Russian Foundation of Basic Research, Grant No. 18-32-00285 (Introduction, Sects. 2, 3, 4.1) and Russian Science Foundation, Grant No. 17-12-01008 (Sect. 4.2). Analysis of multistability was carried out in the frame of project of the Russian Foundation of Basic Research, Grant No. 19-02-00610.
Conflict of interest
The authors declare that they have no conflict of interest.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Stankevich, N., Kuznetsov, A., Popova, E. et al. Chaos and hyperchaos via secondary Neimark–Sacker bifurcation in a model of radiophysical generator. Nonlinear Dyn 97, 2355–2370 (2019). https://doi.org/10.1007/s11071-019-05132-0
- Secondary Neimark–Sacker bifurcation
- Quasiperiodic oscillations
- Lyapunov exponents
Mathematics Subject Classification