Abstract
Chaos is ubiquitous. Chaotic motions are unpredictable. Quantifying chaos is a central issue for understanding chaotic phenomena. Experimental evidence and theoretical studies predict some qualitative and quantitative measures for quantifying chaos. In this chapter we discuss some measures such as universal sequence (U-sequence), Lyapunov exponent, renormalization group theory, invariant measure, Poincaré section, for quantifying chaotic motions. On the other hand, there are some universal numbers applicable for particular class of systems, for example, the Feigenbaum number, Golden mean, etc. The Lorenz system is a paradigm of deterministic dissipative chaotic systems. The universality is an important feature in chaotic dynamics.
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References
Simmons, G.F.: Introduction to Topology and Modern Analysis. Tata McGraw-Hill (2004)
Munkres, J.R.: Topology. Prentice Hall (2000)
Copson, E.T.: Metric Spaces. Cambridge University Press (1968)
Reisel, R.B.: Elementary Theory of Metric Spaces. Springer-Verlag (1982)
Bahi, J.M., Guyeux, C.: Discrete Dynamical Systems and Chaotic Mechanics: Theory and Applications. CRC Press (2013)
Ott, E.: Chaos in Dynamical Systems, 2nd edn. Cambridge University Press (2002)
Yorke, J.A., Grebogi, C., Ott, E., Tedeschini-Lalli, L.: Scaling behavior of windows in dissipative dynamical systems. Phys. Rev. Lett. 54, 1095–1098 (1985)
Grebogi, C., Ott, E., Yorke, J.A.: Chaos, strange attractors, and fractal basin boundaries in nonlinear dynamics. Science 238, 632–638 (1987)
Metropolis, N., Stein, M.L., Stein, P.R.: On finite limit sets for transformations on the unit interval. J. Comb. Theory (A) 15, 25–44 (1973)
Devaney, R.L.: An Introduction to Chaotic Dynamical Systems. Westview Press, Cambridge (2003)
Feigenbaum, M.J.: The universal metric properties of nonlinear transformations. J. Stat. Phys. 21, 669–706 (1979)
Feigenbaum, M.J.: Universal behavior in nonlinear systems. Los Alamos Sci. 1, 4–27 (1980)
Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Phys. D 16, 285–317 (1985)
Royden, H.L.: Real Analysis. Pearson Education (1988)
Hasselblatt, B., Katok, A.: A First Course in Dynamics with a Panorama of Recent Development. Cambridge University Press (2003)
Coudéne, Y.: Ergodic Theory and Dynamical Systems. Springer (2016)
Barreira, L., Valls, C.: Dynamical Systems: An Introduction. Springer (2012)
Li, T., Yorke, J.A.: Period three implies chaos. Am. Math. Mon. 82, 985–992 (1975)
Glendinning, P.: Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations. Cambridge University Press (1994)
McCauley, J.L.: Chaos, Dynamics and Fractals. Cambridge University Press (1993)
Hilborn, R.C.: Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers. Oxford University Press, Oxford (2000)
Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)
Oppo, G.L., Politi, A.: Collision of Feigenbaum cascades. Phys. Rev. A 30, 435–441 (1984)
Layek, G.C., Pati, N.C.: Period-bubbling transition to chaos in thermo-viscoelastic fluid systems. Int. J. Bifurc. Chaos 30(6), 2030013 (2020)
Berge, P., Pomeau, Y., Vidal, C.: Order Within Chaos. Hermann, Paris (1984)
Pomeau, Y., Manneville, P.: Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189 (1980)
Shil’nikov, L.P.: A case of the existence of a countable number of periodic motions. Sov. Math. Dokl. 6, 163–166 (1965)
Layek, G.C., Pati, N.C.: Bifurcations and chaos in convection taking non-Fourier heat flux. Phys. Lett. A 381, 3568–3575 (2017)
Pati, N.C., Rech, P.C., Layek, G.C.: Multistability for nonlinear acoustic-gravity waves in a rotating atmosphere. Chaos Interdicpl. J. Nonlinear Sci. 31(2), 023108 (2021)
Gallas, J.A.C.: Structure of the parameter space of the Henon map. Phys. Rev. Lett. 70, 2714 (1993)
Rech, P.C.: Organization of the periodicity in the parameter space of a glycolysis discrete-time mathematical model. J. Math. Chem. 632–637 (2019)
Layek, G.C., Pati, N.C.: Organized structures of two bidirectionally coupled logistic maps. Chaos 29, 093104 (2019)
Jain, P., Banerjee, S.: Border-collision bifurcation in one-dimensional discontinuous maps. Int. J. Bifurc. Chaos 13(11), 3341–3351 (2011)
Avrutin, V., Schanz, M., Banerjee, S.: Co-dimension-three bifurcations: explanation of the complex one-, two-, and three-dimensional bifurcation structures in non-smooth maps. Phys. Rev. E. 75(6), 066205 (2007)
Bak, P., Chen, K.: Self-organized criticality. Sci. Am. 264(1), 46–53 (1991)
Cvitanovic, P.: Universality in Chaos: A Reprint Selection. Adam Hilgar, Bristol (1984)
Benettin, G., Cercignani, C., Galgani, L., Giorgilli, A.: Universal properties in conservative dynamical systems. Lett. Nuovo Cimento 28, 1–4 (1980)
Rössler, O.E.: An equation for hyperchaos. Phys. Lett. A 71, 155–157 (1979)
Pati, N.C., Layek, G.C., Pal, N.: Bifurcation and organized structures in a predator-prey model with hunting cooperation. Chaos Soliton Fract. 0960-0779 (2020)
Pati, N.C., Garai, S., Hossain, M., Layek, G.C., Pal, N.: Fear induced multistability in a predator-prey model. IJBC 31(10), 2150150 (2021)
Yudovich, V.I.: Co-symmetry, degeneration of solutions of operator equations, and onset of filtration convection, Mathematical notes of the Academy of Sciences of the USSR 49, 540–545 (1991)
Govorukhin, V.N., Shevchenko, I.V.: Multiple equilibria, bifurcations and selection scenarios in co-symmetric problem of thermal convection in porous medium, Physica D: Nonlinear Phenomena 361, 4258 (2017)
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Layek, G.C. (2024). Chaos. In: An Introduction to Dynamical Systems and Chaos. University Texts in the Mathematical Sciences. Springer, Singapore. https://doi.org/10.1007/978-981-99-7695-9_12
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