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K. J. Somaiya College of Science and Commerce, Mumbai 400077, Maharashtra, India

and

Kiran M. Kolwankar

https://orcid.org/0000-0001-5050-7713

Department of Physics, Ramniranjan Jhunjhunwala College, Ghatkopar West, Mumbai 400086, Maharashtra, India

E-mail Address: Kiran.Kolwankar@rjcollege.edu.in

Corresponding author.

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https://doi.org/10.1142/S0218127425500269Cited by:0 (Source: Crossref)

Abstract

Coupled logistic map systems have been used in a wide variety of fields including population dynamics, COVID detection, memristors, chemical physics, detection of natural resources such as oil and digital currency forecasting, to name a few. One phenomenon which is of considerable interest in coupled systems is synchronization and then the question of the critical coupling constant beyond which the system synchronizes. The usual method to estimate this coupling constant is via a linear stability analysis of the synchronization state which is local. It is necessary to develop global methods to study the synchronization transition and one possible way can be through the study of invariant measures. With this general aim, we study dissipatively coupled chaotic logistic map systems. The invariant measure of such a system of two coupled logistic maps is generated numerically in a regime where the transition to chaotic asynchronous state has taken place and is shown to have a multifractal nature. We have considered both types of systems in which the parameters of the logistic maps are identical as well as nonidentical. The invariant measure restricted to the synchronization manifold is then analyzed using the wavelet leaders algorithm preceded by fractional integration. The obtained multifractal spectrum is studied while varying the coupling strength. The results obtained are discussed.

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References

Aguirre, J., Viana, R. L. & Sanjuán, M. A. F. [2009] “Fractal structures in nonlinear dynamics,” Rev. Mod. Phys. 81, 333–386. Crossref, Web of Science, Google Scholar
Altan, A., Karasu, S. & Bekiros, S. [2019] “Digital currency forecasting with chaotic meta-heuristic bio-inspired signal processing techniques,” Chaos Solit. Fract. 126, 325–336. Crossref, Web of Science, Google Scholar
Altan, A. & Karasu, S. [2020] “Recognition of COVID-19 disease from X-ray images by hybrid model consisting of 2D curvelet transform, chaotic salp swarm algorithm and deep learning technique,” Chaos Solit. Fract. 140, 110071. Crossref, Web of Science, Google Scholar
Bao, B., Rong, K., Li, H., Li, K., Hua, Z. & Zhang, X. [2021] “Memristor-coupled logistic hyperchaotic map,” IEEE Trans. Circuits Syst.-II: Exp. Briefs 68, 2992–2996. Crossref, Web of Science, Google Scholar
Bashkirtseva, I. & Ryashko, L. [2021] “Chaotic transients, riddled basins, and stochastic transitions in coupled periodic logistic maps,” Chaos 31, 053101. Crossref, Web of Science, Google Scholar
Bezanson, J., Edelman, A., Karpinski, S. & Shah, V. B. [2017] “Julia: A fresh approach to numerical computing,” SIAM Rev. 59, 65–98. Crossref, Web of Science, Google Scholar
Cánovas, J. [2023] “The dynamics of coupled logistic maps,” Netw. Heterog. Media 18, 275–290. Crossref, Web of Science, Google Scholar
Drożdż, S. & Oświecimka, P. [2015] “Detecting and interpreting distortions in hierarchical organization of complex time series,” Phys. Rev. E 91, 030902. Crossref, Web of Science, Google Scholar
Frisch, U. & Parisi, G. [1985] “Fully developed turbulence and intermittency,” in Turbulence and Predictability in Geophysical Flows and Climate Dynamics: Proceedings of the International Summer School of Physics “Enrico Fermi” Course: LXXXVII (North-Holland, Amsterdam), pp. 71–88. Google Scholar
Gröger, M. & Hunt, B. R. [2013] “Coupled skinny baker’s maps and the Kaplan–Yorke conjecture,” Nonlinearity 26, 2641. Crossref, Web of Science, Google Scholar
Gu, Y., Tung, M., Yuan, J.-M., Deng, F. H. & Narducci, L. M. [1984] “Crises and hysteresis in coupled logistic maps,” Phys. Rev. Lett. 52, 701–704. Crossref, Web of Science, Google Scholar
Hasan, M. N., Greenwood, T. E., Parker, R. G., Kong, Y. L. & Wang, P. [2023] “Fractal patterns in the parameter space of a bistable Duffing oscillator,” Phys. Rev. E 108, L022201. Crossref, Web of Science, Google Scholar
Hua, Z., Jin, F., Xu, B. & Huang, H. [2018] “2D logistic-sine-coupling map for image encryption,” Signal Process. 149, 148–161. Crossref, Web of Science, Google Scholar
Jaffard, S., Lashermes, B. & Abry, P. [2007] “Wavelet leaders in multifractal analysis,” in Wavelet Analysis and Applications, Applied and Numerical Harmonic Analysis (Birkhäuser, Basel), pp. 201–246. Crossref, Google Scholar
Jaffard, S., Melot, C., Leonarduzzi, R., Wendt, H., Abry, P., Roux, S. G. & Torres, M. E. [2016] “p-exponent and p-leaders, Part I: Negative pointwise regularity,” Physica A 448, 300–318. Crossref, Web of Science, Google Scholar
Jalla, D. & Kolwankar, K. M. [2022] “Universal multifractal stationary densities in expanding piecewise linear coupled maps,” Fractals 30, 2250040. Link, Web of Science, Google Scholar
Jost, J. & Kolwankar, K. M. [2005] “Fractal stationary density in coupled maps,” in Fractals in Engineering: New Trends in Theory and Applications (Springer, London), pp. 57–64. Crossref, Google Scholar
Jost, J. & Kolwankar, K. M. [2006] “Global analysis of synchronization in coupled maps,” Int. J. Bifurcation and Chaos 16, 3695–3703. Link, Web of Science, Google Scholar
Kaneko, K. [1989] “Spatiotemporal chaos in one- and two-dimensional coupled map lattices,” Physica D 37, 60–82. Crossref, Web of Science, Google Scholar
Kaneko, K. [1992] “Overview of coupled map lattices,” Chaos 2, 279–282. Crossref, Web of Science, Google Scholar
Kolwankar, K. M. & Lévy Véhel, J. [2001] “Measuring functions smoothness with local fractional derivatives,” Fract. Calc. Appl. Anal. 4, 285–301. Google Scholar
Kopel, M. [1996] “Simple and complex adjustment dynamics in Cournot duopoly models,” Chaos Solit. Fract. 7, 2031–2048. Crossref, Web of Science, Google Scholar
Lashermes, B., Jaffard, S. & Abry, P. [2005] “Wavelet leader based multifractal analysis,” Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing (IEEE), pp. iv–161. Crossref, Google Scholar
Layek, G. & Pati, N. [2019] “Organized structures of two bidirectionally coupled logistic maps,” Chaos 29, 093104. Crossref, Web of Science, Google Scholar
Leonarduzzi, R., Wendt, H., Abry, P., Jaffard, S., Melot, C., Roux, S. G. & Torres, M. E. [2016] “p-exponent and p-leaders, Part II: Multifractal analysis: Relations to detrended fluctuation analysis,” Physica A 448, 319–339. Crossref, Web of Science, Google Scholar
Linero-Bas, A. & Muñoz-Guillermo, M. [2021] “A full description of $ω$ $ω$ -limit sets of Cournot maps having non-empty interior and some economic applications,” Mathematics 9, 452. Crossref, Google Scholar
Lloyd, A. L. [1995] “The coupled logistic map: A simple model for the effects of spatial heterogeneity on population dynamics,” J. Theor. Biol. 173, 217–230. Crossref, Web of Science, Google Scholar
Maistrenko, Y. L., Maistrenko, V., Popovich, A. & Mosekilde, E. [1998] “Transverse instability and riddled basins in a system of two coupled logistic maps,” Phys. Rev. E 57, 2713–2724. Crossref, Web of Science, Google Scholar
Morgül, Ö. [1998] “On the synchronization of logistic maps,” Phys. Lett. A 247, 391–396. Crossref, Web of Science, Google Scholar
Muzy, J.-F., Bacry, E. & Arneodo, A. [1991] “Wavelets and multifractal formalism for singular signals: Application to turbulence data,” Phys. Rev. Lett. 67, 3515–3518. Crossref, Web of Science, Google Scholar
Oświecimka, P., Drożdż, S., Frasca, M., Gebarowski, R., Yoshimura, N., Zunino, L. & Minati, L. [2020] “Wavelet-based discrimination of isolated singularities masquerading as multifractals in detrended fluctuation analyses,” Nonlin. Dyn. 100, 1689–1704. Crossref, Web of Science, Google Scholar
Pikovsky, A., Rosenblum, M. & Kurths, J. [2001] Synchronization: A Universal Concept in Nonlinear Sciences, Vol. 12 ( Cambridge Nonlinear Science Series, Cambridge University Press, Cambridge, UK). Crossref, Google Scholar
Samko, S., Kilbas, A. & Marichev, O. [1993] Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, Yverdon). Google Scholar
Sathiyadevi, K., Karthiga, S., Chandrasekar, V., Senthilkumar, D. & Lakshmanan, M. [2019] “Frustration induced transient chaos, fractal and riddled basins in coupled limit cycle oscillators,” Commun. Nonlin. Sci. Numer. Simul. 72, 586–599. Crossref, Web of Science, Google Scholar
Wang, Y., Liu, Z., Zhang, L. Y., Pareschi, F., Setti, G. & Chen, G. [2021] “From chaos to pseudorandomness: A case study on the 2-D coupled map lattice,” IEEE Trans. Cybern. 53, 1324–1334. Crossref, Web of Science, Google Scholar
Wendt, H., Abry, P. & Jaffard, S. [2007] “Bootstrap for empirical multifractal analysis,” IEEE Signal Process. Mag. 24, 38–48. Google Scholar