Deterministic dynamic system

The authors then ask the following question Do there exist deterministic dynamical systems that are, in a precise sense, equivalent to a monotonous Markov process The question can be reformulated in a more operational way as follows Does there exist a similarity transformation A which, when applied to a distribution function p, solution of the Liouville equation, transforms the latter into a function p that can also be interpreted as a distribution function (probability density) and whose evolution is governed by a monotonous Markov process An affirmative answer to this question requires the following conditions on A (MFC) ... 

Deterministic dynamic systems generated by autonomous ordinary differential equations. 

For this purpose let us consider the following SDOF deterministic dynamical system ... 

Despite bearing no direct relation to any physical dynamical system, the onedimensional discrete-time piecewise linear Bernoulli Shift map nonetheless displays many of the key mechanisms leading to deterministic chaos. The map is defined by (see figure 4.2) ... 

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. 

For deterministic dynamics the state zt+At at time t + At is of course completely determined by the state of the system zt a time step At earlier. Therefore, the single-time-step transition probability p(zt -> zt+At) can be written in terms of a delta function... 

For the past three decades deterministic classical systems with chaotic dynamics have been the subject of extensive study (Chirikov, 1979)-(Sagdeev et. al., 1988). Dynamical chaos is a phenomenon peculiar to the deterministic systems, i.e. the systems whose motion in some state space is completely determined by a given interaction and the initial conditions. Under certain initial conditions the behaviour of these systems is unpredictable. 

The established tools of nonlinear dynamics provide an elaborate and versatile mathematical framework to examine the dynamic properties of metabolic systems. In this context, the metabolic balance equation (Eq. 5) constitutes a deterministic nonlinear dynamic system, amenable to systematic formal analysis. We are interested in the asymptotic, the linear stability of metabolic states, and transitions between different dynamic regimes (bifurcations). For a more detailed account, see also the monographs of Strogatz [290], Kaplan and Glass [18], as well as several related works on the topic [291 293],... 

There exists a class of dynamical systems, whose distribution function p obeys the (deterministic) Liouville equation, for which one can prove that, as a result of a... 

It is important to stress the fact that in the proof of the MPC theorem, the laws of classical dynamics are never violated. One could summarize the signihcance of the MPC theorem by saying that, for a well-defined class of dynamical systems, the new formulation lays bare the arrow of time that is hidden in the illusorily deterministic formulation of these unstable systems. 

A dynamical system is called intrinsically stochastic if it satisfies the conditions required for the existence of a A operator, and whose deterministic... 

When applied to spatially extended dynamical systems, the PoUicott-Ruelle resonances give the dispersion relations of the hydrodynamic and kinetic modes of relaxation toward the equilibrium state. This can be illustrated in models of deterministic diffusion such as the multibaker map, the hard-disk Lorentz gas, or the Yukawa-potential Lorentz gas [1, 23]. These systems are spatially periodic. Their time evolution Frobenius-Perron operator... 

This formula has been verified for the following dynamical systems sustaining deterministic diffusion [23]. 

Thermostated dynamical systems are deterministic systems with non-Hamiltonian forces modeling the dissipation of energy toward a thermostat [48]. The non-HamUtonian forces are chosen in such a way that the equations of... 

We can understand better this asymptotics by using the Markov chain language. For nonseparated constants a particle in has nonzero probability to reach and nonzero probability to reach A, . The zero-one law in this simplest case means that the dynamics of the particle becomes deterministic with probability one it chooses to go to one of vertices A, A3 and to avoid another. Instead of branching, A2 A and A2 A3, we select only one way either A2 A] or A2 A3. Graphs without branching represent discrete dynamical systems. 

We note that for p =0 and /u 0 the dynamics of (37) reduces to the deterministic dynamics of the original system (35) in the absence of control (u(t) = 0). So we begin our analysis by considering some relevant properties of the deterministic dynamics of a periodically driven nonlinear oscillator. 

A dynamic system is a deterministic system whose state is defined at any time by the values of several variables y(t), the so-called states of the system, and its evolution in time is determined by a set of rules. These rules, given a set of initial conditions y(0), determine the time evolution of the system in a unique way. This set of rules can be either... 

For most macroscopic dynamic systems, the neglect of correlations and fluctuations is a legitimate approximation [383]. For these cases the deterministic and stochastic approaches are essentially equivalent, and one is free to use whichever approach turns out to be more convenient or efficient. If an analytical solution is required, then the deterministic approach will always be much easier than the stochastic approach. For systems that are driven to conditions of instability, correlations and fluctuations will give rise to transitions between nonequihbrium steady states and the usual deterministic approach is incapable of accurately describing the time behavior. On the other hand, the stochastic simulation algorithm is directly applicable to these studies. 

Deterministic dynamics of biochemical reaction systems can be visualized as the trajectory of (ci(t), c2(t), , c v(0) in a space of concentrations, where d(t) is the concentration of ith species changing with time. This mental picture of path traced out in the N-dimensional concentration space by deterministic systems may prove a useful reference when we deal with stochastic chemical dynamics. In stochastic systems, one no longer thinks in terms of definite concentrations at time t rather, one deals with the probability of the concentrations being xu x2, , Wy at time t ... 

Note, however, that the concept of the entropy production rate is of critical importance for analyzing the evolution of systems that are close to equilibrium rather than of dynamic systems, which are described by rigid kinetic schemes with time deterministic behavior ( dynamic machines ). 

The kicked rotor (or the standard map) is one of famous models in chaotic dynamical systems, and it has been studied in various situations [17]. One feature of its chaotic dynamics is the deterministic diffusion along the momentum direction. It is also well known that if we quantize this system, this diffusion is... 

At the microscopic level, chemical reactions are dynamical phenomena in which nonlinear vibrational motions are strongly coupled with each other. Therefore, deterministic chaos in dynamical systems plays a crucial role in understanding chemical reactions. In particular, the dynamical origin of statistical behavior and the possibility of controlling reactions require analyses of chaotic behavior in multidimensional phase space. 

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