Symbolic dynamics

Symbolic dynamics is one of the most powerful tools in the theory of chaotic systems. It is a qualitative method for characterizing the dynamics of a given nonlinear system. The power of symbolic dynamics shows whenever a new system N, whose properties are not yet known, can be mapped via symbolic dynamics onto an old dynamical system O that has already been thoroughly studied, and whose properties are understood. If such a mapping exists, the two systems N and O axe dynamically equivalent. Obviously, symbolic dynamics has the potential to save a lot of work. 

Since we have an alphabet, and we are allowed to form words, there should also be a grammar which tells us the rules of how to form words. The grammar of the three-disk system is remarkably simple. It has only one rule Any combination of letters is allowed except for two identical consecutive letters. The restriction in the grammatical rule is immediately obvious since two identical letters in succession corresponds to two successive bounces off the same disk. This is obviously impossible in a force-free situation. 

Since our grammatical rule does not impose any restrictions on the length of a symbol sequence, we are obviously allowed to write down infinitely long sequences. There are two kinds periodic and aperiodic sequences. With the help of the mapping A — 0, B — 1 and C — 2 the symbol sequences can be interpreted as real numbers in base-3 notation. On the basis of this analogy, we call finite (or infinite periodic) sequences rational and infinite aperiodic sequences irrational . Thus, the itinerary of the trajectory shown in Fig. 2.11(a) is rational. An ex- 

Symbolic dynamics is a powerful, but qualitative, tool. We can easily imagine a system trajectory that is close to the trajectory shown in Fig. 2.11(a), but not quite identical. Both trajectories axe then characterized by the same word, ABC AC. Thus, symbol sequences do not specify system trajectories uniquely. In other words, there is no one-to-one correspondence between the impact parameters b and the words constructed from the alphabet A, B, C.  

The application of symbolic dynamics to the three-disk scattering system helped us to focus right away on the essential questions (i) Is it possible to have an infinite number of scatterings (ii) Are there trapped trajectories, and, if yes, how many  

Recall that the KAM theorem states that as long as a nonlinear perturbation to an integrable system - such as that represented by the nonzero q term added to the integrable system defined by equations 4.45 - is sufficiently small, most trajectories will continue to lie on smooth KAM-curves. 

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Molecular aspect Symbol Dynamic relation U < Tr) Static relation H U = Tr) r,s... 

That the use of symbolic dynamics to study the behavior of complex or chaotic systems in fact heralds a new epoch in physics wris boldly suggested by Joseph Ford in the foreword to this Physics Reports review. Ford writes, Just as in that earlier period [referring to 1922, when The Physical Review had published a review of Hilbert Space Operator Algebra] physicists will shortly be faced with the arduous task of learning some new mathematics... For make no mistake about it, the following review heralds a new epoch. Despite its modest avoidance of sweeping claims, its theorems point like arrows toward the physics of the second half of the twentieth century. ... 

The dynamical system F/, 7, consisting of the set of admissible sequences under / and the shift map, 7, is the n -order symbolic dynamical system induced by f The power of the method resides in the following - provided that the... 

Universality in Unimodal Maps A seminal work on the 2-symbol dynamics of one-dimensional unimodal mappings due to Metropolis, Stein Stein [metro73]. Specifically, they studied the iterates of various mappings within periodic windows, labeling the attractor sequences by strings of the form RLLRL , where R and L indicate whether f xo) falls to the right or left of xq, respectively. Each periodic sequence therefore corresponds to a unique finite length word made up of R s and L s. 

As should have been made plausible by the above discussion, symbolic dynamics provides an intuitive conceptual bridge between continuous and discrete dynamical systems. On the one hand, except for the fact that the coarse-grained discrete dynamics of symbolic systems is typically nonlocal in character (see the following section), symbolic dynamical systems are essentially CA-in-disguise. On the other hand, by exploiting the fundamental CA property of continuity in the Cantor-set topology under the metric defined by equation 4.49, real-valued representations of CA dynamics may be readily obtained. We discuss these two alternative points of... 

Vol. 1539 M. Coomaert, A. Papadopoulos, Symbolic Dynamics and Hyperbolic Groups. VIII, 138 pages. 1993. Vol. 1540 H. Komatsu (Ed.), Functional Analysis and Related Topics, 1991. Proceedings. XXI, 413 pages. 1993. 

Intramolecular dynamics and chemical reactions have been studied for a long time in terms of classical models. However, many of the early studies were restricted by the complexities resulting from classical chaos, Tlie application of the new dynamical systems theory to classical models of reactions has very recently revealed the existence of general bifurcation scenarios at the origin of chaos. Moreover, it can be shown that the infinite number of classical periodic orbits characteristic of chaos are topological combinations of a finite number of fundamental periodic orbits as determined by a symbolic dynamics. These properties appear to be very general and characteristic of typical classical reaction dynamics. 

We will start by describing the relevant aspects of the classical dynamics, introducing the concepts of the invariant set and the repeller, the Smale horseshoe and its symbolic dynamics, and the bifurcations at their origin [19]. We then turn to the semiclassical quantization based on the Gutzwiller trace formula and the zeta functions. We proceed to show how this new theoretical framework allows us to explain the distribution of resonances in several molecules like Hgl2, CO2, and H3, to calculate their lifetimes, and to provide a synthesis with respect to previous work. 

Other classically chaotic scattering systems have been shown to have repellers described by a symbolic dynamics similar to (4.10). One of them is the three-disk scatterer in which a point particle undergoes elastic collisions on three hard disks located at the vertices of an equilateral triangle. In this case, the symbolic dynamics is dyadic (M = 2) after reduction according to C)V symmetry. Another example is the four-disk scatterer in which the four disks form a square. The C4 symmetry can be used to reduce the symbolic dynamics to a triadic one based on the symbols 0,1,2), which correspond to the three fundamental periodic orbits described above [14]. 

In the presence of reflection symmetry with respect to the diagonal of the potential-energy surface, as in symmetric molecules or in the four-disk scatterer, Burghardt and Gaspard have shown that a further symmetry reduction can be performed in which the symbolic dynamics still contains three symbols A = 0, +, - [10]. The orbit 0 is the symmetric-stretch periodic orbit as before. The orbit + is one of the off-diagonal orbits 1 or 2 while - represents a half-period of the asymmetric-stretch orbit 12. Note that the latter has also been denoted the hyperspherical periodic orbit in the literature. 

The central feature of the symbolic dynamics is that it provides a complete classification of the periodic orbits Once the symbolic dynamics has been... 

On the other hand, we should mention that, at the level of classical mechanics, periodic-orbit analysis provides a topological characterization of the system in terms of a symbolic dynamics, which appears as a common feature for a given class of systems. 

R. Bowen, Symbolic Dynamics (Collection of Works), Mir, Moscow, 1979 (in Russian). 

As done in Refs. [29,31,32], by using the symbolic dynamics, we can specify the location of the triple colhsion orbits. We skip the detail of this method. The triple collision orbits that experience a triple colhsion in the future will form curves on the Poincare section. Then we call them the triple collision curves. 

A useful technique is the method of symbolic dynamics. We will encounter this topic in Section 2.4. Symbolic dynamics enables us to establish connections between dynamical systems. Often it is possible to map a given dynamical system onto an older one which has been studied before. If such a mapping is possible on the symbolic level, the two systems are dynamically equivalent. 

The request for medications (explicit or implicit) by dependent patients, often conveys a need to be fed. Aside from direct medication effects, the gratification experienced by the patient when given drugs may account for symptomatic improvement. It is also important to note that once the therapist prescribes medications (however appropriate this may be) in some way he or she has strayed from a position of neutrality, and this may have consequences for the therapeutic relationship. Conversely, the choice not to prescribe or recommend medications may be seen as "withholding." The symbolic-dynamic issues discussed here are also important concerns in the treatment of clients with borderline personality disorders. 

B. Symbolic Dynamics of Bifurcations and Chaotic Dynamics The Analysis of Real Biochemical Networks... 

We first reformulate unimolecular decay in terms of symbolic dynamics so as to permit utilization of modern concepts in ergodic theory. In doing so we, at least initially, replace the continuous time dynamics by a discrete time mapping. Specifically, we consider dynamics at multiples of a fixed time increment S, defining T"x as the propagation of a phase space point x for n time increments [i.e., x(t = nS) = T"x]. In what follows, time parameters associated with the discrete dynamics are measured in units of S. These include t, t, t, and t<, which are also used in connection with the flow. In the later context the conversion to -independent units is implicit. Note that within the assumed discrete dynamics, S and S+ are broadened from surfaces to volumes S-s and S+s comprising all points that enter or have left R during a time interval b. 

FIGURE 6.19 (Upper panel) Steady-state shear viscosity versus shear rate (soUd symbols), dynamic viscosity versus frequency (open symbols), and transient viscosity calculated from Eq. (6.65) versus the inverse of the time of shearing (solid line). (Lower panel) Dynamic storage and loss modulus master curve for the same entangled polybutadiene solution (Roland and Robertson, 2006). 

If the number of dof of the motor task is greater than, say, 4, a computer is needed to obtain Eq. (6.1) explicitly. A number of commercial software packages are available for this puipose including AUTOLEV by On-Line Dynamics Inc., SD/FAST by Symbolic Dynamics Inc., ADAMS by Mechanical Dynamics Inc., and DADS by CADSI. 

Fig. 13 Global structure factor versus wave vector for different times for a quench from the mixed state at /N = 0.314 to X-N = 5. Lines represent Monte Carlo results, symbols dynamic SCF theory results, (a) Compares dynamic SCF theory using a local Onsager coefficient with Monte Carlo simulations. Local dynamics obviously overestimates the growth rate and shifts the wavevector that corresponds to maximal growth rate to larger values, (b) Compares dynamic SCF theory using anon-local Onsager coefficient that mimics Rouses dynamics with Monte Carlo results showing better agreement. From [29]...

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