Globally chaotic dynamics

Recently, however, experimental studies of reaction processes have cast doubt on the local equilibrium assumption. When that assumption is not valid, understanding of reaction processes requires the smdy of global aspects of the phase space in multidimensional chaotic dynamics [1]. 

We are beginning to understand the real dynamics of global diffusion in the phase space of many-dimensional Hamiltonian systems. From here we are going to travel around the vast world created by the chaotic dynamics of nonlinear systems. 

The mechanism of these transitions is nontrivial and has been discussed in detail elsewhere Q, 12) it involves the development of a homoclinic tangencv and subsequently of a homoclinic tangle between the stable and unstable manifolds of the saddle-type periodic solution S. This tangle is accompanied by nontrivial dynamics (chaotic transients, large multiplicity of solutions etc.). It is impossible to locate and analyze these phenomena without computing the unstable, saddle-tvpe periodic frequency locked solution as well as its stable and unstable manifolds. It is precisely the interactions of such manifolds that are termed global bifurcations and cause in this case the loss of the quasiperiodic solution. 

Let us now consider the behavior of the system when the Kerr coupling constant is switched on (e12 / 0). For brevity and clarity, we restrict our discussion to the question of how the attractors in Fig. 20 change when both oscillators interact with each other. To answer this question, let us have a look at the joint auto-nomized spectrum of Lyapunov exponents for the two oscillators A,j, A,2, L3, A-4, L5 versus the interaction parameter 0 < ( 2 < 0.7. The spectrum is seen in Fig. 32 and describes the dynamical properties of our oscillators in a global sense. The dynamics of individual oscillators can be glimpsed at the appropriate phase portraits. Let us now fix our attention on a detailed analysis of Fig. 32. For the limit value ei2 = 0, the dynamics of the uncoupled oscillators has already been presented in Fig. 20. In the case of very weak interaction 0 < C 2 < 0.0005, the system of coupled oscillators manifests chaotic behavior. For C 2 = 0.0005 we obtain the spectrum 0.06,0.00, —0.21, 0.54, 0.89. It is interesting to... 

M. Komuro, A Mechanism of Chaotic Itinerancy in Globally Coupled Maps, International Conference on New Directions in Dynamical Systems (NDDS 2002), A satellite conference of ICM 2002, August 5-15, 2002, Ryukoku University and Kyoto University, Kyoto, Japan. 

We are beginning to understand chaotic structure in a way not seen before. Numerical methods of measuring chaotic and regular behaviour such as Fast Liapunov Indicators, sup-maps, twist-angles, Frequency Map Analysis, fourier spectal analysis are providing lucid maps of the global dynamical behaviour of multidimensional systems. Fourier spectral analysis of orbits looks to be a powerful tool for the study of Nekhoroshev type stability. Identification of the main resonances and measures of the diffusion of trajectories can be found easily and quickly. Applied to the full N-body problem without simplification, use of these tools is beginning to explain the observed behaviour of real physical systems. 

The role of mixing has been studied in systems with more complex reaction schemes or considering more complex fluid-dynamical properties, and in the context of chemical engineering or microfluidic applications (for reviews on microfluidics see e.g. Squires (2005) or Ottino and Wiggins (2004)). Muzzio and Liu (1996) studied bi-molecular and so-called competitive-consecutive reactions with multiple timescales in chaotic flows. Reduced models that predict the global behavior of the competitive-consecutive reaction scheme were introduced by Cox (2004) and by Vikhansky and Cox (2006), and a method for statistical description of reactive flows based on a con-... 

Fig. 5.4. Control of chaotic front dynamics by extended time-delay autosynchronization, a) Space-time plot of the uncontrolled charge density, and current density J vs. time, b) Same with global voltage control with exponentially weighted current density (denoted by the black curve). Parameters as in Fig. 1, U = 1.15 V, t = 2.29 ns, K = 3 x 10- Vmm2/A, R = 0.2, a = 10 s-b [73]...

M. Komuro, A Mechanism of Chaotic Itinerary on Globally Coupled Maps (in Japanese), RIMS Kokyuroku Vol. 1118 (1999), Singular Phenomena of Dynamical Systems, pp. 96-114. 

Due to the chaotic nature of molecular dynamics, which implies a sensitivity to perturbations of the initial condition or the differential equations themselves, it is to be expected that the global error due to using a numerical method will always grow rapidly (exponentially) in time. As we shall see in later chapters, this does not necessarily mean that a long trajectory is entirely without value. In molecular dynamics it turns out that the real importance of the trajectory is that it provides a mechanism for calculating averages that maintain physical parameters. The simplest example of such a parameter is the energy. 

In typical molecular dynamics applications with multiple bodies and complicated force laws, the motion is assumed to be chaotic. Ergodic coverage of would then have to arise from the global properties of a chaotic system (sensitivity to initial conditions and transitivity). Liouville s equation,... 

Chemical systems with complex kinetics exhibit a fascinating range of dynamical phenomena. These include periodic and aperiodic (chaotic) temporal oscillation as well as spatial patterns and waves. Many of these phenomena mimic similar behavior in living systems. With the addition of global feedback in an unstirred medium, the prototype chemical oscillator, the Belousov-Zhabotinsky reaction, gives rise to clusters, i.e., spatial domains that oscillate in phase, but out of phase with other domains in the system. Clusters are also thought to arise in systems of coupled neurons. 

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