Zandi-Mehran, Nazanin; Nazarimehr, Fahimeh; Rajagopal, Karthikeyan; Ghosh, Dibakar; Jafari, Sajad; Chen, Guanrong FFT bifurcation: a tool for spectrum analyzing of dynamical systems. (English) ✄ ✂  ✁ Zbl 07488767 Appl. Math. Comput. 422, Article ID 126986, 13 p. (2022) Summary: This paper presents FFT bifurcation as a tool for investigating complex dynamics. Firstly, two well-known chaotic systems (Rössler and Lorenz) are discussed from the frequency viewpoint. Then, both discrete-time and continuous-time systems are studied. Various systems with different properties are discussed. In discrete-time systems, Logistic map and a biological map are investigated. For continuous- time systems, a system with a stable equilibrium, forced van der Pol system, and a system with a line of equilibria are discussed. For each system under investigation, the proposed FFT bifurcation diagrams are compared with the conventional bifurcation diagrams, showing some interesting information uncovered by the FFT bifurcation. For periodic trajectories, the FFT bifurcations show high power at the dominant frequency and harmonics. By doubling the periods, their dominant frequencies are halved, and more harmonics emerge in the studied frequency intervals. For the chaotic dynamics, the FFT bifurcation shows a wideband power spectrum. The FFT bifurcation shows interesting results in comparison to conventional bifurcation diagrams. MSC: 37Nxx Applications of dynamical systems 37Dxx Dynamical systems with hyperbolic behavior 34Cxx Qualitative theory for ordinary differential equations Keywords: FFT bifurcation; bifurcation diagram; dynamical system; frequency spectrum; hidden dynamics Full Text: DOI References: [1] Dudkowski, D.; Jafari, S.; Kapitaniak, T.; Kuznetsov, N. V.; Leonov, G. A.; Prasad, A., Hidden attractors in dynamical systems, Phys. Rep., 637, 1-50 (2016) · Zbl 1359.34054 [2] Panahi, S.; Nazarimehr, F.; Jafari, S.; Sprott, J. C.; Perc, M.; Repnik, R., Optimal synchronization of circulant and non- circulant oscillators, Appl. Math. Comput., 394, Article 125830 pp. (2021) · Zbl 07332981 [3] Schuster, H. G.; Just, W., Deterministic Chaos: an Introduction (2006), John Wiley \& Sons [4] Benner, P.; Feng, L.; Rudnyi, E. B., Using the superposition property for model reduction of linear systems with a large number of inputs, (Proceedings of the 18th International Symposium on Mathematical Theory of Networks \& Systems (2008)) [5] Bishop, R. C., Metaphysical and epistemological issues in complex systems, Philosophy of Complex Systems:, 105-136 (2011), Elsevier [6] Gu, J.; Li, C.; Chen, Y.; Iu, H. H.; Lei, T., A conditional symmetric memristive system with infinitely many chaotic attractors, IEEE Access, 8, 12394-12401 (2020) [7] Perc, M., Visualizing the attraction of strange attractors, Eur. J. Phys., 26, 579-587 (2005) [8] Silva, P. H.O.; Nardo, L. G.; Martins, S. A.M.; Nepomuceno, E. G.; Perc, M., Graphical interface as a teaching aid for nonlinear dynamical systems, Eur. J. Phys., 39, Article 065105 pp. (2018) [9] Zhang, X.; Li, C.; Min, F.; Iu, H. H.; Gao, H., Broken symmetry in a memristive chaotic oscillator, IEEE Access, 8, 69222-69229 (2020), 9044836 [10] Gu, S.; He, S.; Wang, H.; Du, B., Analysis of three types of initial offset-boosting behavior for a new fractional-order dynamical system, Chaos Solitons Fractals, 143, Article 110613 pp. (2021) [11] Parastesh, F.; Jafari, S.; Azarnoush, H.; Shahriari, Z.; Wang, Z.; Boccaletti, S., Chimeras, Phys. Rep., 898, 1-14 (2021) · Zbl 07404961 [12] Hâncean, M. G.; Slavinec, M.; Perc, M., The impact of human mobility networks on the global spread of COVID-19, J. Complex Netw., 8, 1-14 (2021) [13] Hilborn, R. C., Chaos and Nonlinear Dynamics: an Introduction for Scientists and Engineers (2000), Oxford University Press: Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2022 FIZ Karlsruhe GmbH Page 1