Cycle limit

The result is independent of the starting point chosen. The number 0.0174577 represents an attractor or a fixed point for the operation. In the SCF method the fixed fixed point point is identical with the single Slater-determinant function (a point in the Hilbert space, cf. Appendix B) - a result of the SCF iterative procedure. 

In thermodynamics, the equilibrium state of an isolated sgfstem (at some fixed external parameters) may be regarded as an attractor, that any non-equilibrium state attains after a sufficiently long time. 

Sometimes an attractor represents something other than just a point at which the evolution of the qfstem definitely ends up. 

Consider a set of two differential equations with time t as variable. Usually their solution [jc(t) and y(t) depends on the initial conditions assumed. Fig. 15.3.a. 

Now let us take a particular set of two non-linear differential equations. As seen from Fig. 15.3.b, this time the behaviour of the solution as a function of time is completely different for high values of t the solution does not depend on the ini- 

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Attractors can be simple time-independent states (points in F), limit cycles (simple closed loops in F) corresponding to oscillatory variations of tire chemical concentrations with a single amplitude, or chaotic states (complicated trajectories in F) corresponding to aperiodic variations of tire chemical concentrations. To illustrate... 

Figure C3.6.5 The first two periodic orbits in the main subhannonic sequence are shown projected onto the (c, C2) plane. This sequence arises from a Hopf bifurcation of the stable fixed point for the parameters given in the text. The arrows indicate the direction of motion, (a) The limit cycle or period-1 orbit at k 2 = 0.11. (b) The first subhannonic or period-2 orbit at k 2 = 0.095.

The quasiperiodic route to chaos is historically important. It arises from a succession of Hopf birfurcations. As already noted, a single Hopf bifurcation results in a limit cycle. The next Hopf bifurcation produces a phase flow tliat can be represented on tire surface of a toms (douglmut). This flow is associated witli two frequencies if tire ratio of tliese frequencies is irrational tlien tire toms surface is densely covered by tire phase trajectory, whereas if... 

Figure C3.6.7 Cubic (jir = 0) and linear (r = 0) nullclines for tire FitzHugh-Nagumo equation, (a) The excitable domain showing trajectories resulting from sub- and super-tlireshold excitations, (b) The oscillatory domain showing limit cycle orbits small inner limit cycle close to Hopf point large outer limit cycle far from Hopf point.

Stability, Bifurcations, Limit Cycles Some aspects of this subject involve the solution of nonlinear equations other aspects involve the integration of ordinaiy differential equations apphcations include chaos and fractals as well as unusual operation of some chemical engineering eqmpment. Ref. 176 gives an excellent introduction to the subject and the details needed to apply the methods. Ref. 66 gives more details of the algorithms. A concise survey with some chemical engineering examples is given in Ref. 91. Bifurcation results are closely connected with stabihty of the steady states, which is essentially a transient phenomenon. 

Chapter 5 provides some examples of purely analyti( al tools useful for describing CA. It discusses methods of inferring cycle-state structure from global eigenvalue spectra, the enumeration of limit cycles, the use of shift transformations, local structure theory, and Lyapunov functions. Some preliminary research on linking CA behavior with the topological characteristics of the underlying lattice is also described. 

The simplest possible attraetor is a fixed point, for which all trajectories starting from the appropriate basin-of-attraction eventually converge onto a single point. For linear dissipative dynamical systems, fixed-point attractors are in fact the only possible type of attractor. Non-linear systems, on the other hand, harbor a much richer spectrum of attractor-types. For example, in addition to fixed-points, there may exist periodic attractors such as limit cycles for two-dimensional flows or doubly periodic orbits for three-dimensional flows. There is also an intriguing class of attractors that have a very complicated geometric structure called strange attractors [ruelleSO],... 

In contrast to dissipative dynamical systems, conservative systems preserve phase-space volumes and hence cannot display any attracting regions in phase space there can be no fixed points, no limit cycles and no strange attractors. There can nonetheless be chaotic motion in the sense that points along particular trajectories may show sensitivity to initial conditions. A familiar example of a conservative system from classical mechanics is that of a Hamiltonian system. 

Period-2 Limit Cycle At the second critical value, a = Q2 = 3, itself becomes unstable and is replaced by a stable attracting period-2 limit cycle, x 2) - This new bifurcation - called the pitchfork bifurcation - is shown schematically in figure 4.4 below. 

Fig. 4.4 A schematic representation of the pitchfork bifurcation a stable fixed point bifurcates into a period-2 limit cycle plus an unstable fixed point.

Period-n Limit Cycles As a increases, the system undergoes an infinite sequence of successive period-doublings via pitchfork bifurcations. In general,... 

While the period of the limit-cycles approaches infinity as n —> oo, the distance between successive critical q s rapidly decreases ... 

Since the phase space of a dissipative dynamical system contracts with time, we know that, in the long time limit, t oo, the motion will be confined to some fixed attractor, A. Moreover, becaust of the contraction, the dimension, D, of A, must be lower than that of the actual phase space. While D adds little information in the case of a noiichaotic attractor (we know immediately, and trivially, for example, that all fixed-points have D = 0, limit cycles have D = 1, 2-tori have D = 2, etc.), it is of significant interest for strange attractors, whose dimension is typically non-integer valued. Three of the most common measures of D are the fractal dimension, information dimension and correlation dimension. 

The spatial and temporal dimensions provide a convenient quantitative characterization of the various classes of large time behavior. The homogeneous final states of class cl CA, for example, are characterized by d l = dll = dmeas = dmeas = 0 such states are obviously analogous to limit point attractors in continuous systems. Similarly, the periodic final states of class c2 CA are analogous to limit cycles, although there does not typically exist a unique invariant probability measure on... 

The technique for enumerating limit cycles for general rules consists essentially of two parts ... 

Generalizing the second observation to cycles of arbitrary length, we note that since each primitive string in a limit cycle with least period equal to q must be a fixed point of the g order rule, (and thus also a fixed point of [4>Y, for any period p with q p), each primitive limit cycle may be expressed by a term equal to its period ... 

This same technique can be used to find the number of limit cycles of period p on lattice size n for any two-neighbor CA rule. 

Let P a a ) be the probability of transition from state a to state a. In general, the set of transition probabilities will define a system that is not describ-able by an equilibrium statistical mechanics. Instead, it might give rise to limit cycles or even chaotic behavior. Fortunately, there exists a simple condition called detailed balance such that, if satisfied, guarantees that the evolution will lead to the desired thermal equilibrium. Detailed balance requires that the average number of transitions from a to a equal the number of transitions from a to a ... 

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