Chaotic system, definition

In order to allow for the largest possible class of chaotic systems, the degree of sensitivity is not specified in Devaney s definition of chaos. It turns out that many chaotic systems of practical importance are exponentially sensitive to initial conditions. In this case the sensitivity can be characterized quantitatively with the help of Lyapunov exponents. 

In order to study the oscillation frequency of the chaotic model it. is important to develop a means for decomposing a chaotic signal into its phase and amplitude components. This is non-trivial for chaotic systems where there is often no unambiguous definition of phase. In our case, the motion always shows phase coherent dynamics, so that a phase can be defined as an angle in x,y)-phase plane or via the Hilbert-transform [31]. Here, we use an alternative method which is based on counting successive maxima, that allows analysis even if the signal is spiky . In this scheme we estimate the instantaneous phase

[Pg.411]

Although there are many definitions of chaos (Gleick, 1987), for our purposes a chaotic system may be defined as one having three properties deterministic dynamics, aperiodicity, and sensitivity to initial conditions. Our first requirement implies that there exists a set of laws, in the case of homogeneous chemical reactions, rate laws, that is, first-order ordinary differential equations, that govern the time evolution of the system. It is not necessary that we be able to write down these laws, but they must be specifiable, at least in principle, and they must be complete, that is, the system cannot be subject to hidden and/or random influences. The requirement of aperiodicity means that the behavior of a chaotic system in time never repeats. A truly chaotic system neither reaches a stationary state nor behaves periodically in its phase space, it traverses an infinite path, never passing more than once through the same point. 

According to Stuart A. Kauffman (1991) there is no generally accepted definition for the term complexity . However, there is consensus on certain properties of complex systems. One of these is deterministic chaos, which we have already mentioned. An ordered, non-linear dynamic system can undergo conversion to a chaotic state when slight, hardly noticeable perturbations act on it. Even very small differences in the initial conditions of complex systems can lead to great differences in the development of the system. Thus, the theory of complex systems no longer uses the well-known cause and effect principle. 

Formally the unperturbed Hamiltonian is equivalent to the Hamiltonian of the hydrogen atom in constant homogenious electric field. Chaotic dynamics of hydrogen atom in constant electric field under the influence of time-periodic field was treated earlier (Berman et. al, 1985 Stevens and Sundaraml987). To treat nonlinear dynamics of this system under the influence of periodic perturbations we need to rewrite (1) in action-angle variables. Action can be found using its standard definition ... 

Thus, we have found that the mechanisms of escape from a nonhyperbolic attractor and a quasihyperbolic (Lorenz) attractor are quite different, and that the prehistory of the escape trajectories reflects the different structure of their chaotic attractors. The escape process for the nonhyperbolic attractor is realized via several steps, which include transitions between low-period saddle-cycles coexisting in the system phase space. The escape from the Lorenz attractor consist of two qualitatively different stages the first is defined by the stable and unstable manifolds of the saddle center point, and lies on the attractor the second is the escape itself, crossing the saddle boundary cycle surrounding the stable point attractor. Finally, we should like to point out that our main results were obtained via an experimental definition of optimal paths, confirming our experimental approach as a powerful instrument for investigating noise-induced escape from complex attractors. 

We are now ready for a definition of chaos. Summarizing a century of research in chaos, Devaney (1992) gives the following definition of chaos A system is chaotic if (Cl) periodic orbits are dense,... 

The importance of viewpoint and the apparent chaotic nature of ecological systems make discussion of such parameters as ecosystem stability difficult to determine accurately. In Figure 11.32 a system that hits a perturbation is depicted. Although the distances that each have traveled are the same in a two-dimensional picture, from the viewpoint of the observer one system moves farther than the other and by some definitions is less stable. Conversely, if the chaotic nature of systems prevents a return to the original state, recovery cannot be considered an inherent property of the system. 

By our definition, the dynamics in Example 9.5.1 are not chaotic, because the long-term behavior is not aperiodic. On the other hand, the dynamics do exhibit sensitive dependence on initial conditions—if we had chosen a slightly different initial condition, the trajectory could easily have ended up at C instead of C Thus the system s behavior is unpredictable, at least for certain initial conditions. 

We have seen that the logistic map can exhibit aperiodic orbits for certain parameter values, but how do we know that this is really chaos To be called chaotic, a system should also show sensitive dependence on initial conditions, in the sense that neighboring orbits separate exponentially fast, on average. In Section 9.3 we quantified sensitive dependence by defining the Liapunov exponent for a chaotic differental equation. Now we extend the definition to one-dimensional maps. 

Chaotic dynamics are by definition aperiodic dynamics in deterministic systems with sensitive dependence on initial conditions. Although many refer to chaotic dynamics in Eq. (3), this does not conform to standard nonlinear dynamics terminology since such systems have a finite state space and must necessarily cycle. However, in Eq. (5), chaotic dynamics are possible and have been demonstrated numerically and analytically in some example networks [42, 46]. We have no way to predict whether any particular logical structure is capable of generating chaotic dynamics for some set of parameters. A necessary condition for chaotic dynamics is that there is a vertex that lies on at least two... 

For each set of initial conditions, Eqs. (4.1)-(4.3) can be solved to And X ", U ", and The initial conditions are randomly selected from known distribution functions, and we can assume that there is an infinite number of possible combinations. Each combination is called a realization of the granular flow, and the set of all possible realizations forms an ensemble. Note that, because the particles have finite size, they cannot be located at the same point thus X " 4 X for n 4 m. Also, the collision operator will generate chaotic trajectories and thus the particle positions will become uncorrelated after a relatively small number of collisions. In contrast, for particles suspended in a fluid the collisions are suppressed and correlations can be long-lived and of long range. We will make these concepts more precise when we introduce fluid-particle systems later. While the exact nature of the particle correlations is not a factor in the definition of the multi-particle joint PDF introduced below, it is important to keep in mind that they will have... 

In this article we discuss the problem of understanding the long-term stability properties of a solution of a quasi-integrable Hamiltonian system by means of a Fourier analysis on a short observation time. Precisely, even for resonant chaotic motions, we will show how the combined use of Fourier analysis and Nekhoroshev theorem allows to understand the stability properties on a time T exp(T), where T is a suitable observation time, of the order of the resonant period. To be definite, we will refer to quasi-integrable Hamiltonian systems with Hamiltonian of the form ... 

We now delve into the topic of what is meant by chaotic motion. We should first point out that there is no generally agreed upon technical definition of what the word chaotic means for dynamical systems, but it is possible for us to get a sense of what the issues are, nevertheless. It is not a foreign notion to consider that some kinds of systems exhibit regular motion (a pendulum, a onedimensional oscillator, a thrown baseball) and others behave erratically (a balloon with air escaping from its nozzle). It is more unsettling to consider the three ideas that follow. 

The quantitative laws of chemical combination provide clear pointers to the molecular theory of matter, which increases progressively in vividness and realism with the application of Newton s laws to the motions of the particles. The interpretation of phenomena such as the pressure and viscosity of gases and the Brownian motion, and the assignment of definite magnitudes to molecular speeds, masses, and diameters render it clear that a continual interchange of energies must occur between the molecules of a material system, a circumstance which lies at the basis of temperature equilibrium and determines what in ordinary experience is called the flow of heat. It is responsible indeed for far more than this, and a large part of physical chemistry follows from the conception of the chaotic motion of the molecules. This matter must now be examined more deeply. 

Typical definitions of a chaotic dynamical system [103] require at least the following conditions to be satisfied on the phase space D ... 

Theoretically, optimum separation is achieved when the sample dimensionality and system dimensionality are equivalent, resulting in an ordered separation (Figure 3). In the above example, only one separation dimension would be required to analyze the alkane sample. If, however, the dimensionality of the sample exceeds that of the system, sample components will not be resolved in an orderly fashion, but rather a disordered or chaotic separation will result. In Figure 3, three descriptors are required to define the sample - shape, pattern, and size. In a chemical sense, these might be molar mass, polarity, and molecular shape. As more dimensions of separation are applied, greater definition of the mixture components is achieved. Unfortunately, for very complex samples, the sample dimensionality will be... 

In the definitions above, flows are considered to be vectorial quantities (energy and mass) which express the transport in time and surface units. In a non-uniform gas system which is in a condition of non-equilibrium, the velocity of molecules could be expressed by the addition of two terms v = u + c. The first, u, is associated with an eventual convective motion of molecules, whereas c considers chaotic molecular motions. The balance of the molecules contained in a volume element di2 of phase space should be considered in a dt time interval. A molecule moving from r to r+v dr in the absence of collisions may be expressed as ... 

---