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Abstract
Chialvo map modelling a neuron has been investigated numerically as it exhbit generically some of the intriguing/ complex features of several excitable biological systems. Bifurcation diagram obtained for this map shows pattern of repeated period-doubling route to chaos. Appearance of complicated periodic windows within bifurcation suggests presence of complexity within the map. Regular and chaotic attractors are drawn here in different parameter space. Lyapunov exponents describing the stability of the systems have been computed for different cases, clearly revealing complexity involved dynamics. Numerical simulation have been extended to calculate topological entropy as a measure of complexity and presented through graphics. Topological entropy graphs showing significant increase within certain parameter range. Correlation dimension of certain chaotic attractors also obtained.
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References
W. Weaver, Science and complexity, American Scientist, 36(4): 536, (1948).
D. M. Hefferman, Multistability, intermittency and remerging Feigenbaum trees in an externally pumped ring cavity laser system, Phys. Lett. A 108: 413 – 422, (1985).
H. A. Simon, The architecture of complexity, Proceedings of the American Philosophical Society, 106(6): 467–482, (1962).
R. Adler, A. Konheim and J. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 :309–319 (1965).
Walby, S. Complexity theory, systems theory, and multiple intersecting social inequalities, Philosophy of the Social Sciences, 37(4): 449-470, (2007).
A. A. Elsadany, Dynamical complexities in a discrete-time food chain. Computational Ecology and Software, 2(2), 124–139, (2012)
J. Gribbin, Deep Simplicity: Chaos, Complexity and the Emergence of Life, Penguin Press Science, (2004).
M. Andrecut and S. A. Kauffman, Chaos in a Discrete Model of a Two-Gene System, Physics Letters A 367: 281-287 (2007).
G. Benettin, L. Galgani, A. Giorgilli and J. M. Strelcyn, Lyapunov Characteristic Exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1 & II: Theory, Meccanica 15: 9–20 & 21–30 (1980).
P. Bryant, R. Brown and H. Abarbanel, Lyapunov exponents from observed time series, Physical Review Letters 65 (13): 1523 – 1526 (1990).
] P. Grassberger and Itamar Procaccia, Measuring the Strangeness of Strange Attractors, Physica D: Nonlinear Phenomena 9 (1‒2): 189‒208 (1983).
L. M. Saha, S. Prasad and G. H. Erjaee, Interesting dynamic behavior in some discrete maps. IJST, A3 (Special Issue-Mathematics): 383-389 (2012).
M. Sandri, Numerical calculation of Lyapunov Exponent Mathematica. J. 6: 78-84 (1996).
S. Gribble, Topological Entropy as a Practical Tool for Identification and Characterization of Chaotic System. Physics 449 Thesis, 1995.
R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc.: 184, 125−136: (1973).
N. J. Balmforth, E. A. Spiegel, C. Tresser, Topological entropy of one dimensional maps: approximations and bounds, Phys. Rev. Lett.72: 80 – 83, (1994).
K. Iwai, Continuity of topological entropy of one dimensional map with degenerate critical points, J. Math. Sci. Univ. Tokyo 5: 19 – 40 (1998).
L. Stewart, E. S. Edward, Calculating topological entropy, J. Stat. Phys. 89, 1017 – 1033, (1997).
S.Das, Recurrence quantification and bifurcation analysis of electrical activity in resistive/memristive synapse coupled Fitzhugh–Nagumo type neurons, Chaos Sol. and Frac. 165, 11277(23) (2022).
Dante R. Chialvo and A. Vania Apkarian, Modulated noisy biological dynamics: Three examples, Journal of Statistical Physics. 70 (1): 375–391 (1993).
Dante R. Chialvo, Generic excitable dynamics on a two-dimensional map, Chaos, Solitons & Fractals, 5 (3): 461–479 (1995).
F. Wang and H. Cao, Mode locking and quasiperiodicity in a discrete-time Chialvo neuron model, Communications in Nonlinear Science and Numerical Simulation. 56: 481–489 (2018).
Nikolai F. Rulkov, Modeling of spiking-bursting neural behavior using two-dimensional map, Physical Review E. 65 (4): 041922(1-9), (2022).
P. Pilarczyk, J. Signerska-Rynkowska and G. Graff, Topological-numerical analysis of a two-dimensional discrete neuron model, Chaos, 33(4): 043110 (2023).
M. Martelli, Introduction to Discrete Dynamical Systems and Chaos, John Wiley & Sons, Inc, New York, 1999.
H. Nagashima and Y. Baba, Introduction to Chaos: Physics and Mathematics of Chaotic Phenomena. Overseas Press India Private Limited, 2005.