Nonlinear dissipative systems

One can see that the roll solutions within the interval 1/3 < K < 1 are unstable with respect to disturbances with k / 0, ky = 0, i.e. to longitudinal modulations. This kind of instability in nonlinear dissipative systems was discovered by Eckhaus [48] and is called the Eckhaus instability. 

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. 

Yamada, T., Kuramoto, Y. (1976a) Spiral waves in a nonlinear dissipative system. Prog. Theor. Phys. 55, 2035... 

When nonlinear, dissipative systems are driven far from equilibrium, they often exhibit multistability, whereby there exist two or many coexisting steady states and/or oscillating states. Through internal or external noise, such systems may be driven to undergo transitions between these states. Moreover, the noise itself may create new states and transitions between them [9]. A wide variety of systems chosen from the fields of chemistry [10], optics [11] and biology [12] display such a behavior. 

The typical strategy employed in studying the behavior of nonlinear dissipative dynamical systems consists of first identifying all of the periodic solutions of the system, followed by a detailed characterization of the chaotic motion on the attractors. 

Nonlinear Systems. Nonlinearities play a dominant role in physics and many other disciplines. For example, all material laws have a nonlinear characteristic. In many cases, the usually applied linearization procedures are suitable and well established methods, which might lead to satisfying results. However, nonlinear systems can exhibit a behaviour, which is completely absent in the regime of linear dynamics. Both, the development and the maintenance of such a behaviour seem to be provided by a general mechanism nonlinear dissipation. 

Now definitions or frameworks of modem thermodynamics in a broad sense, of classical thermodynamics, and of modem thermodynamics in a narrow sense are very clear. Modern thermodynamics in a broad sense includes all fields of thermodynamics (both classical thermodynamics and modem thermodynamics in a narrow sense) for any macroscopic system, but modem thermodynamics in a narrow sense includes only three fields of thermodynamics, i.e., nonequilibrium nondissipative thermodynamics, linear dissipative thermodynamics and nonlinear dissipative thermodynamics. The modem thermodynamics in a narrow sense should not be called nonequilibrium thermodynamics, because the classical nonequilibrium thermodynamics is not included. Meanwhile, the classical thermodynamics should only be applied to simpler systems without reaction coupling. That is, the application of classical thermodynamics to some modem inorganic syntheses and to the life science may be not suitable. Without the self-consistent classification of modem thermodynamics it was very difficult to really accept the term of modem thermodynamics even only for teaching courses. 

On the theoretical physics side, the Kolmogorov-Arnold-Moser (KAM) theory for conservative dynamical systems describes how the continuous trajectories of a particle break up into a chaotic sea of randomly disconnected points. Furthermore, the strange attractors of dissipative dynamical systems have a fractal dimension in phase space. Both these developments in classical dynamics—KAM theory and strange attractors—emphasize the importance of nonanalytic functions in the description of the evolution of deterministic nonlinear dynamical systems. We do not discuss the details of such dynamical systems herein, but refer the reader to a number of excellent books on the... 

This distinction between a < and a = exemplifies a broader theme in nonlinear dynamics. In general, if a map or flow contracts volumes in phase space, it is called dissipative. Dissipative systems commonly arise as models of physical situations involving friction, viscosity, or some other process that dissipates energy. In contrast, area-preserving maps are associated with conservative systems, particularly with the Hamiltonian systems of classical mechanics. 

It is obvious that in the real physical situations we are not able to avoid dissipation processes. For dissipative systems, we cannot take an external excitation too weak (the parameter e cannot be too small) since the field interacting with the nonlinear oscillator could be completely damped and hence, our model could become completely unrealistic. Moreover, the dissipation in the system leads to a mixture of the quantum states instead of their coherent superpositions. Therefore, we should determine the influence of the damping processes on the systems discussed here. To investigate such processes we can utilize various methods. For instance, the quantum jumps simulations [38] and quantum state diffusion method [39] can be used. Description of these two methods can be found in Ref. 40, where they were discussed and compared. Another way to investigate the damping processes is to apply the approach based on the density matrix formalism. Here, we shall concentrate on this method [12,41,42]. 

Chapters 3 and 4 introduce the concepts, techniques, and computational tools of nonlinear dynamics. Professor Robert Q. Topper focuses on applications to conservative systems, while Professors Raima Larter and Kenneth Showalter cover the uses of nonlinear dynamics in dissipative systems (they dissipate rather than conserve energy). The major application to chemistry of conservative nonlinear dynamics is in the realm of molecular dynamics, whereas chemical kinetics is the main focus of the chapter on dissipative systems. In both of these chapters, a geometrical approach to understanding dynamics is introduced. As Dr. Topper clearly explains in Chapter 3, much work has gone into developing a better understanding of molecular dynamics by... 

Many theories on the nonlinear dynamics of dissipative systems are based on the first-order ordinary differential equations... 

This ratio is vahd when the system operates close to thermodynamic equihbrium. It is, however, typical for heterogeneous catalysis to occur far from equihbrium in an open, nonlinear, dissipative, distributed, and multiparametric medium. Thus heterogeneous catalytic reactions exhibit diverse nonlinear phenomena the multiplicity of steady states (stable and unstable) hysteresis phenomena the ignition and extinction of the process critical phenomena phase transitions a high sensitivity of the process to changes in the parameters oscillations and wave phenomena chaotic regimes the formation of dissipative structures and seh organization phenomena. 

From the standpoint of the classical (analytical) theory with which we were concerned in this review, the situation is obviously absurd since each of these two equations is linear and of a dissipative type (since h > 0) trajectories of both of these equations are convergent spirals tending to approach a stable focus. However, if one carries out a simple analysis (see Reference 6, p. 608), one finds that change of equations for = 0, results in the change of the focus in a quasi-discontinuous manner, so that the trajectory can still be closed owing to the existence of two nonanalytic points on the -axis. If, however, the trajectory is closed, this means that there exists a stationary oscillation and in such a case the system (6-197) is nonlinear, although, from the standpoint of the differential equations, it is linear everywhere except at the two points at which the analyticity is lost. 

The present analysis shows that when a thermodynamic gradient is first applied to a system, there is a transient regime in which dynamic order is induced and in which the dynamic order increases over time. The driving force for this is the dissipation of first entropy (i.e., reduction in the gradient), and what opposes it is the cost of the dynamic order. The second entropy provides a quantitative expression for these processes. In the nonlinear regime, the fluxes couple to the static structure, and structural order can be induced as well. The nature of this combined order is to dissipate first entropy, and in the transient regime the rate of dissipation increases with the evolution of the system over time. 

This is the same equation of motion that is satisfied by the original coordinate qa(t), except that the stochastic driving term is absent. The relative dynamics is therefore deterministic. We have chosen the notation accordingly and left out the index a in the definition (41) of Aq (although, of course, we cannot expect the relative dynamics to remain noiseless in the full nonlinear system). Although noiseless, the relative dynamics is still dissipative because Eq. (43) retains the damping term. 

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