Periodic fixed point

For certain parameter values tliis chemical system can exlribit fixed point, periodic or chaotic attractors in tire tliree-dimensional concentration phase space. We consider tire parameter set... 

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 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],... 

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.

We see that the phase-plane is broken up into a sequence of fixed points and a series of both open and closed constant-energy curves. The origin (0= =0) and its periodic equivalents (0 27rn, = 0), are stable fixed points (or elliptic... 

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 ... 

Theorem 5 [goles87a] If the synaptic-weight matrix A is symmetric, and the number of sites in the lattice is finite, then the orbits of the generalized threshold rule (equation 5.121) are either fixed points or cycles of period two. 

Since theorem 5 applies to this system, we immediately conclude that majority can yield only either fixed points or cycles of period two. It turns out that one can actually prove a stronger result for this particular rule. Let a finite state be any state consisting of a finite number of nonzero sites. Then we have the following theorem. 

The second part is to show that if we assume a finite state a is not a fixed point - which, according to theorem 5, necessarily makes it period two, or that... 

Equation 5.121 is then reproduced in full by setting bi =5-1/2 for all i. Since aij is symmetric, the condition of theorem 5 is met. We conclude that (f)2d majority can only yield either fixed points or cycles of period two. 

Although it is certainly not immediately obvious from the rule itself, it turns out that, just as is the case for generalized threshold rules, the only possible asymptotic states of finite symmetric muib -threshold rules are either fixed points or cycles of period two [golesQO]. Unlike their binary brethren, however, multi-threshold rules possess some intriguing additional properties. 

It turns out that, in the CML, the local temporal period-doubling yields spatial domain structures consisting of phase coherent sites. By domains, we mean physical regions of the lattice in which the sites are correlated both spatially and temporally. This correlation may consist either of an exact translation symmetry in which the values of all sites are equal or possibly some combined period-2 space and time symmetry. These coherent domains are separated by domain walls, or kinks, that are produced at sites whose initial amplitudes are close to unstable fixed points of = a, for some period-rr. Generally speaking, as the period of the local map... 

Frustrated Loops. The period of a loop depends on whether or not it is frustrated . Consider the parity of the number of value inversions contained in a loop. If it is odd, the loop is said to be frustrated. In this case, a signal propagating around the loop has to go around twice before the loop returns to its initial state. Letting I be the number of site in a loop, the period of the loop is therefore equal to 2 X i for configurations with no symmetries, or an odd factor of 21 for certain special cases. In particular, a frustrated loop cannot have fixed points. 

Landau proposed in 1944 that turbulence arises essentially through the emergence of an ever increasing number of quasi-periodic motions resulting from successive bifurcations of the fluid system [landau44]. For small TZ, the fluid motion is, as we have seen, laminar, corresponding to a stable fixed point in phase space. As Ti is... 

Langton wais able to provide a tentative answer to his question by examining the behavior of the entire rule space of elementary one-dimensional cellular automata rules (see discussion in section 3.2) as parameterized by a single parameter A. He found that as A is increased from its minimal to maximal values, a path is effectively traced in the rule space that progresses from fixed point behavior to simple period-... 

One can interpret this physically as follows suppose that the trajectory of the harmonic oscillator be represented by a point on a rotating wheel. The eye observes a circle (the path of the point) if the wheel rotates rapidly this corresponds to continuous illumination. On the other hand, if one illuminates the rotating wheel with stroboscopic flashes separated by a period 2n, a given mark on the wheel appears as a fixed point. Thus, under continuous illumination one sees ... 

As time increases, the plane wave exp[i( x — mt)] moves with velocity m/. If we consider a fixed point x and watch the plane wave as it passes that point, we observe not only the periodic rise and fall of the amplitude of the unmodified plane wave exp[i(A x — m/)], but also the overlapping rise and fall of the amplitude due to the modulating function 2 cos[(AA jc — Ao)i)/l. Without the modulating function, the plane wave would reach the same maximum... 

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

Periodicity in space means that it repeats at regular intervals, known as the wavelength, A. Periodicity in time means that it moves past a fixed point at a steady rate characterised by the period r, which counts the crests passing per unit time. By definition, the velocity v = A/r. It is custom to use the reciprocals of wavelength 1/X — (k/2-ir) or 9, known as the wavenumber (k = wave vector) and 1/t — v, the frequency, or angular frequency u = 2itv. Since a sine or cosine (harmonic) wave repeats at intervals of 2n, it can be described in terms of the function... 

The more usual case is for intermittent samples to be obtained, representing worker exposures at fixed points in time. If we assume that the concentration C, is fixed (or averaged) over the period of time 7), the TWA concentration is computed by... 

Figure 3. Classical phase portraits (upper panel), residual quantum wavefunctions (middle panel), and ionization probability versus time (in units of the period T) (bottom panel). The parameters are (A) F = 5.0, iv = 0.52 (B) F = 20, iv = 1.04 and (C) F = 10 and u> = 2.0. Note that the peak structure of the final wavefunction reflects both stable and unstable classical fixed points. For case C, the peaks are beginning to coalesce reflecting the approach of the single-well effective potentiai (see text). 

When viewed in three-dimensional space R2xS, the hyperbolic fixed point x0yh turns to hyperbolic periodic orbit of the system... 

Remark 1. The concept for local stable and unstable manifolds becomes clear when one represents the stable and unstable manifolds of the hyperbolic fixed point (periodic orbit) locally. For details see (Wiggins, 1989) or (Wiggins, 1988). 

Right) Three 1 1 periodic orbits (PO) relevant to our work. The corresponding fixed points appear, respectively, as a circle (PO 1), a square (PO 2), and a triangle (PO 3) in the SOS at the left part of the figure. 

---