Chaotic solution

At finite velocity kinetic friction behaves quite differently in the sense that the commensurability plays a less significant role. Besides, the system shows rich dynamic properties since Eq (16) may lead to periodic, quasi-periodic, or chaotic solutions, depending on damping coefficient y and interaction strength h. Based on numerical results of an incommensurate case [18,19], we outline a force curve of F in Fig. 23 asafunction ofv, in hopes of gaining a better understanding of dynamic behavior in the F-K model. 

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-peri-odic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations. 

Chaotic solutions are those which are neither periodic nor asymptotic to a periodic solution but are characterized by extreme sensitivity to initial conditions. A solution that is asymptotic to a stable periodic solution is not sensitive to starting point, for, if we start from two nearby values, the trajectories will both converge on the same periodic solution and get closer and closer together. With a chaotic solution, the trajectories starting at two nearby points ultimately diverge no matter how close they may have been at the beginning. If /( )( ) denotes the nth iterate,... 

This set of equations is a nonlinear eigenvalue time delay differential equation. Such equations, even for one variable, often have periodic or chaotic solutions and, from the physics of the problem are also certain of having pulse-like solutions in some systems. 

Fig. 12.11 Two-mode oscillatory behavior in the single nephron model. Black colored regions correspond to a chaotic solution. The figure shows different regions in which 1 4, 1 5 and 1 6 synchronization occurs in the interaction between the fast myogenic oscillations...

Carlson et al. 1994, Schmittbhul et al 1993, Vilotte et al 1994). In fact, Schmitbhul et al (1993) noted that the system shows earthquake-like almost chaotic solutions for very small values of = Nv/y/R while the solutions are soliton-like for 0 of the order of unity. This size-dependent transition has been extensively studied by Vilotte et al (1994) and Anan-thakrishna and Ramachandran (1994), who also showed that the same transition occurs as the parameter a in (4.4) decreases from higher values. As the blocks in the model represent the independent junctions where the earth s crust rests on a moving tectonic plate (for any particular epicentre region, in case of earthquakes), the number N of such blocks in the model is usually quite small. Also, the (dimensionless) tectonic plate velocity v is very small in reality and of the order of 10 (Carlson et al 1994). This... 

Second, if the equations modeling the dynamic behavior of a plant display chaotic solutions (inherent in many non-linear systems), then we cannot be certain that the process behaves the same way when the preconditions defining a reactive pathway take on any values within given arithmetic intervals of values. 

There are two main types of dynamical systems differential equations and iterated nuips (also known as difference equations). Differential equations describe the evolution of systems in continuous time, whereas iterated maps arise in problems where time is discrete. Differential equations are used much more widely in science and engineering, and we shall therefore concentrate on them. Later in the book we will see that iterated maps can also be very useful, both for providing simple examples of chaos, and also as tools for analyzing periodic or chaotic solutions of differential equati ons. 

Zhabotinskii s observation of double oscillations was indeed the first indication of chaotic oscillations, see Figure 3 of Zhabotinskii (1964-2). Wegmann and Rossler (1978) observed the oscillations of electrochemical potential, a variable, in this reaction, Fig. III.10. Referring to the Model (1976-2) of Rossler, see page 44, they explained this chaotic behavior of the reaction as an example of the chaotic solution of the abstract model, Fig. III. 10b. 

A particular class of problems whose solutions are sensitive to initial conditions is known as chaotic problems. The phenomenon of chaos has been observed in fluid mechanics, chemical reactions, and biological systems (Cvitanocic, 1987). A special feature of these systems is unpredictability. The chaotic solutions are so sensitive to initial conditions that two systems with minute differences in their initial states can eventually diverge from each other. Thus their long-term dynamics are unpredictable. 

Abstract We review results about the Fourier Analysis of chaotic solutions of quasi-integrable systems based on the Nekhoroshev theorem. We describe also an application to Asteroids stability. 

In this section we illustrate the characterizations which can be given to the Fourier spectra of chaotic solutions of quasi-integrable systems. We first consider the easy integrable case ... 

Moreover, for chaotic solutions it is also proved that in suitable coordinates some of the degrees of freedom behave essentially quasi-periodically while the others behave chaotically. Therefore, the Fourier representation of the observables G(I,intermediate between a discrete representation, and a continuous one without structures. In Guzzo and Benettin (2001) we introduced a Fourier... 

The fact that the spectrum of an observable of a system computed on a chaotic solution has not the peak structure can be considered a numerical evidence that the system does not satisfy the hypotheses of Nekhoroshev theorem in the neighborhood of that solution, i.e. the Nekhoroshev theorem does not prevent fast evolution of the actions . 

The transition of spectra from structured to unstructured ones is not described by a theorem, but has been studied numerically in Guzzo, Lega and Froeschle (2002) by comparing the geometry of resonances of a given system computed with the Fast Lyapunov Indicator with the structure of the spectra of an observable computed on well selected chaotic solutions. More precisely, in Froeschle et al. (2000) we estimated with the FLI method that the transition between Littlewood and Chirikov regime for the Hamiltonian system ... 

In Guzzo (2002) we computed numerically the peak structure of a test function of the chaotic solutions of a model degenerate system, detecting the main and secondary peak structures for small values of the perturbing parameters. 

A. Preliminaries. In this section we sketch the mathematical ideas which are at the basis of the Fourier representation of chaotic solutions. All mathematical details can be found Guzzo and Benettin (2001), Guzzo (2002). 

The application of the Fourier Analysis of chaotic solutions to a physically interesting system, namely the dynamics of some numbered Asteroids, has been done in Guzzo, Knezevic and Milani (2002). In this section we briefly describe some of the results published in that paper. 

One approach is to use equations with chaotic solutions. Such solutions are unpredictable over the long term yet exhibit interesting structures as they move about on a strange attractor, a fractal object with noninteger dimension. Our books show many examples of computer art produced by these and related methods. In this chapter, we propose another simple method for producing fractal art. 

Therefore, the system tends to a stationary solution and has no time-periodic or chaotic solutions. 

It exhibits solutions in the form of spatially-irregular cells" splitting and merging in a chaotic manner in time. An example of spatio-temporal behavior of a chaotic solution of the Kuramoto-Sivashinsky equation (158) as well as a snapshot of this solution at a particular moment of time are shown in Fig.20. 

Traditional reaction kinetics has dealt with the large class of chemical reactions that are characterised by having a unique and stable stationary point (i.e. all reactions tend to the equilibrium ). The complementary class of reactions is characterised either by the existence of more than one stationary point, or by an unstable stationary point (which could possibly bifurcate to periodic solutions). Other extraordinarities such as chaotic solutions are also contained in the second class. The term exotic kinetics refers to different types of qualitative behaviour (in terms of deterministic models) to sustained oscillation, multistationarity and chaotic effects. Other irregular effects, e.g. hyperchaos (Rossler, 1979) can be expected in higher dimensions. 

Does a certain kinetic differential equation of the form (4.6) admit chaotic solutions ... 

In the theory of differential equations the Peano inequality ensures that the / solutions starting near each other cannot diverge from each other at a rate that is above exponential. Chaotic solutions, however, are examples where the divergence is not less than exponential. To put it another way, no matter how exact the measurement of the initial conditions was, one cannot well estimate the value of the solution at a later time. 

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