Chaos limit point

This Creative Force, therefore, really represents a force which LIMITS the possibilities belonging to matter when it finds itself in a state which we would call Matiere vierge. This Virgin Matter is wrongly taken by certain people to represent Nothingness or Chaos. If, however, we chose to regard our solar system from the point of view of the [as-... 

The frequency of modulation il is now the main parameter, and we are able to switch the system of SHG between different dynamics by changing the value of il. To find the regions of where a chaotic motion occurs, we calculate a Lyapunov spectrum versus the knob parameter il. The first Lyapunov exponent A,j from the spectrum is of the greatest importance its sign determines the chaos occurrence. The maximal Lyapunov exponent Xj as a function of is presented for GCL in Fig. 6a and for BCL in Fig. 6b. We see that for some frequencies il the system behaves chaotically (A-i > 0) but orderly Ck < 0) for others. The system in the second case is much more damped than in the first case and consequently much more stable. By way of example, for = 0.9 the system of SHG becomes chaotic as illustrated in Fig. 7a, showing the evolution of second-harmonic and fundamental mode intensities. The phase point of the fundamental mode draws a chaotic attractor as seen in the phase portrait (Fig. 7b). However, the phase point loses its chaotic features and settles into a symmetric limit cycle if we change the frequency to = 1.1 as shown in Fig. 8b, while Fig. 8a shows a seven-period oscillation in intensities. To avoid transient effects, the evolution is plotted for 450 < < 500. 

It is readily seen that the set of equations (76) consists of three equations of motion in the real variables ReIm c, w. If, (x) = constant, chaos in the system does not appear since the set (76) becomes a two-dimensional autonomous system. The maximal Lyapunov exponents for the systems (75) and (72)-(74) plotted versus the pulse duration T are presented in Fig. 36. We note that within the classical system (75) by fluently varying the length of the pulse T, we turn order into chaos and chaos into order. For 0 < T < 0.84 and 1.08 < 7) < 7.5, the maximal Lyapunov exponents Li are negative or equal to zero and, consequently, lead to limit cycles and quasiperiodic orbits. In the points where L] = 0, the system switches its periodicity. The situation changes dramatically if,... 

So, as a result, if there is any chaos in ip(t), the auto-correlation test does not show it. We would like to point out, however, that in the semi-classical hmit, i.e. the limit where Planck s constant h is small compared... 

These equations can display a whole range of quantitatively different types of dynamics, including fixed points (nodes or stable foci), limit cycles, chaos, and quasi-periodicity. In this section we briefly describe these different types of... 

The beauty of the magnetic field problem is the degree of control over its structure which can be achieved. In particular, increasing a single parameter (the energy or n value) allows the system to be raised to the semiclassical limit. Varying the field strength, on the other hand, allows the point where the order-to-chaos transition occurs to be modified. 

Chaotic behavior in nonlinear dissipative systems is characterized by the existence of a new type of attractor, the strange attractor. The name comes from the unusual dimensionality assigned to it. A steady state attractor is a point in phase space, whereas a limit cycle attractor is a closed curve. The steady state attractor, thus, has a dimension of zero in phase space, whereas the limit cycle has a dimension of one. A torus is an example of a two-dimensional attractor because trajectories attracted to it wind around over its two-dimensional surface. A strange attractor is not easily characterized in terms of an integer dimension but is, perhaps surprisingly, best described in terms of a fractional dimension. The strange attractor is, in fart, a fractal object in phase space. The science of fractal objects is, as we will see, intimately connected to that of nonlinear dynamics and chaos. 

Kekule then entered the discussion, responding to Naquet in the Comptes rendus of the Academie des Sciences, and wishing to strongly affirm the constancy of atomicity. He proclaimed, "The equivalent can vary, but not the atomicity." To maintain his stance he defended the concept of molecular compounds for substances such as ammonium chloride, and he cited the phenomena of dissociation and water of crystallization as evidence that this suggestion was not foolishly ad hoc. He was also intent to show that if one were to accept the variability of valence, one would enter a slippery slope toward theoretical chaos— for then how could one limit the possible variability Erlenmeyer reprinted Kekule s paper, equipping it with his own parenthetic editorial exclamation points, question marks, and sarcastic footnotes, it severely strained the relationship of the two former friends." ... 

Fig. 15.5. The diagram of the fixed points and the limit cycles for the logistic eqiiatirai as a function of the coupling constant From J. Gleick, Chaos, Viking, New York (1988). Reproduced with permission of the author.

The Liapunov number can be used as a quantitative measure for chaos. The connection between chaos and the Liapunov number is through attractors. An attractor is a set of points S such that for nearly any point surrounding S, the dynamics will approach S as the time approaches infinity. The steady state of a fluid flow can be termed an attractor with dimension zero and a stable limit cycle dimension one. There are attractors that do not have integer dimensirms and are often called strange attractors. There is no tmiversally acceptable definition for strange attractors. The Liapunov number is determined by the principle axes of the ellipsoidal in the phase space, which originates from a ball of points in the phase space. The relatitaiship between the Liapunov number and the characterization of chaos is not universal and is an area of intensive research. 

As pointed out in the preceding paragraph, some equations, by their very nature, complicate matters. As an additional example, consider the specific limitations of the Chao-Seader correlation and others like it, which allow only about 20% methane in the liquid. In a typical flash problem it is conventional for a first set of K constants to be furnished either by the engineer or initialized by the flash program. These K s immediately lead to vapor and liquid compositions. With these compositions the component fugacltles and liquid activity coefficients are calculated, which in turn lead to a seemingly better set of K s. If the first... 

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