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Chaos, Bifurcations and Diffusion

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Complex and Adaptive Dynamical Systems

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Abstract

Complex system theory deals with dynamical systems containing very large numbers of variables. It extends dynamical system theory, which deals with dynamical systems containing a few variables. A good understanding of dynamical systems theory is therefore a prerequisite when studying complex systems.

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Notes

  1. 1.

    The term isocline stands for “equal slope” in ancient Greek.

  2. 2.

    Note: \(\int e^{-x^2/a}\textrm{d} x=\sqrt{a\pi}\) and \(\lim_{a\to0} \exp(-x^2/a)/\sqrt{a\pi}=\delta(x)\).

Further Reading

  • For further studies we refer to introductory texts for dynamical system theory (Katok and Hasselblatt, 1995), classical dynamical systems (Goldstein, 2002), chaos (Schuster and Just, 2005; Devaney, 1989; Gutzwiller, 1990, Strogatz, 1994), stochastic systems (Ross, 1982; Lasota and Mackey, 1994) and differential equations with time delays (Erneux, 2009). Other textbooks on complex and/or adaptive systems are those by Schuster (2001) and Boccara (2003). For an alternative approach to complex system theory via Brownian agents consult Schweitzer (2003).

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  • The interested reader may want to study some selected subjects in more depth, such as the KAM theorem (Ott, 2002), relaxation oscillators (Wang, 1999), stochastic resonance (Benzit et al., 1981; Gammaitoni et al., 1998), coherence resonance (Pikovsky and Kurths, 1997), Lévy flights (Metzler and Klafter, 2000), the connection of Lévy flights to the patterns of wandering albatrosses (Viswanathan et al., 1996), human traveling (Brockmann et al., 2006) and diffusion of information in networks (Eriksen et al., 2003).

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  • The original literature provides more insight, such as the seminal works of Einstein (1905) and Langevin (1908) on Brownian motion or the first formulation and study of the Lorenz (1963) model.

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  • Benzit, R., Sutera, A., Vulpiani, A. 1981 The mechanism of stochastic resonance. Journal of Physics A 14, L453–L457.

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  • Boccara, N. 2003 Modeling Complex Systems. Springer, Berlin.

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  • Brockmann, D., Hufnagel, L., Geisel, T. 2006 The scaling laws of human travel. Nature 439, 462.

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  • Devaney, R.L. 1989 An Introduction to Chaotic Dynamical Systems. Addison-Wesley, Reading, MA.

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  • Einstein, A. 1905 Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik 17, 549.

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  • Eriksen, K.A., Simonsen, I., Maslov, S., Sneppen, K. 2003 Modularity and extreme edges of the internet. Physical Review Letters 90, 148701.

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  • Erneux, T. 2009 Applied Delay Differential Equations. Springer, New York.

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  • Gammaitoni, L., Hänggi, P., Jung, P., Marchesoni, F. 1998 Stochastic resonance. Review of Modern Physics 70, 223–287.

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  • Goldstein, H. 2002 Classical Mechanics. 3rd Edition, Addison-Wesley, Reading, MA.

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  • Gutzwiller, M.C. 1990 Chaos in Classical and Quantum Mechanics. Springer, New York.

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  • Katok, A., Hasselblatt, B. 1995 Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge.

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  • Langevin, P. 1908 Sur la théorie du mouvement brownien. Comptes Rendus 146, 530–532.

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  • Lasota, A., Mackey, M.C. 1994 Chaos, Fractals, and Noise – Stochastic Aspects of Dynamics. Springer, New York.

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  • Lorenz, E.N. 1963 Deterministic nonperiodic flow. Journal of the Atmospheric Sciences 20, 130–141.

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  • Metzler, R., Klafter J. 2000 The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Physics Reports 339, 1.

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  • Schuster, H.G. 2001 Complex Adaptive Systems. Scator, Saarbrücken.

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  • Schuster, H.G., Just, W. 2005 Deterministic Chaos. 4th Edition, Wiley-VCH, New York.

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  • Schweitzer, F. 2003 Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences. Springer, New York.

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  • Strogatz, S.H. 1994 Nonlinear Systems and Chaos. Perseus Publishing, Cambridge, MA.

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  • Viswanathan, G.M., Afanasyev, V., Buldyrev, S.V., Murphy, E.J., Prince, P.A., Stanley, H.E. 1996 Lévy flight search patterns of wandering albatrosses. Nature 381, 413

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  • Wang, D.L. 1999 Relaxation oscillators and networks In Webster, J.G. (ed.), Encyclopedia of Electrical and Electronic Engineers, pp. 396–405. Wiley, New York.

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Author information

Authors and Affiliations

  1. Universität Frankfurt, Institut für Theoretische Physik, Max-von-Laue-Str. 1, 60438, Frankfurt, Germany

    Prof. Dr. Claudius Gros

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  1. Prof. Dr. Claudius Gros
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Correspondence to Claudius Gros .

Exercises

Exercises

2.1.1 The Lorenz Model

Perform the stability analysis of the fixpoint \((0,0,0)\) and of \(C_{+,-}\ = \ (\pm \sqrt{b(r-1)}, \pm \sqrt{b(r-1)}, r-1)\) for the Lorenz model Eq. (2.21) with \(r,\,b>0\). Discuss the difference between the dissipative case and the ergodic case \(\sigma=-1-b\), see Eq. (2.23).

2.1.2 The Poincaré Map

For the Lorenz model Eq. (2.21) with \(\sigma=10\) and \(\beta=8/3\), evaluate numerically the Poincaré map for (a) \(r=22\) (regular regime) and the plane \(z=21\) and (b) \(r=28\) (chaotic regime) and the plane \(z=27\).

2.1.3 The Hausdorff Dimension

Calculate the Hausdorff dimension of a straight line and of the Cantor set, which is generated by removing consecutively the middle-1/3 segment of a line having a given initial length.

2.1.4 The Driven Harmonic Oscillator

Solve the driven, damped harmonic oscillator

$$\ddot x\,+\,\gamma\, \dot x\, +\,\omega_0^2\,x\ =\ \epsilon\cos(\omega t)$$

in the long-time limit. Discuss the behavior close to the resonance \(\omega\to\omega_0\).

2.1.5 Continuous-Time Logistic Equation

Consider the continuous-time logistic equation

$$\dot y(t) \ =\ \alpha y(t)\Big[1-y(t)\Big].$$

(A) Find the general solution and (B) compare to the logistic map Eq. (2.8) for discrete times \(t=0,\ \varDelta t,\ 2\varDelta t,\ ..\).

2.1.6 Information Flow in Networks

Choose a not-too-big social network and examine numerically the flow of information, Eq. (2.41), through the network. Set the weight matrix \(W_{ij}\) identical to the adjacency matrix \(A_{ij}\), with entries being either unity or zero. Evaluate the steady-state distribution of information and plot the result as a function of vertex degrees.

2.1.7 Stochastic Resonance

Solve the driven double-well problem Eq. (2.64) numerically and try to reproduce Figs. 2.15 and 2.16.

2.1.8 Delayed Differential Equations

The delayed Eq. (2.68) allows for harmonically oscillating solutions for certain sets of parameters a and b. Which are the conditions? Speciallize then for the case \(a=0\).

2.1.9 Car-Following Model

A car moving with velocity \(\dot x(t)\) follows another car driving with velocity \(v(t)\) via

$$\ddot x(t+T) \ =\ \alpha (v(t)-\dot x(t)), \qquad \alpha > 0, \vspace*{-4pt}$$
((2.74))

with \(T>0\) being the reaction time of the driver. Prove the stability of the steady-state solution for a constant velocity \(v(t)\equiv v_0\) of the preceding car.

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Gros, C. (2011). Chaos, Bifurcations and Diffusion. In: Complex and Adaptive Dynamical Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04706-0_2

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  • DOI: https://doi.org/10.1007/978-3-642-04706-0_2

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  • Print ISBN: 978-3-642-04705-3

  • Online ISBN: 978-3-642-04706-0

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