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Chaotic differential equations

The study of nonlinear systems of chaotic differential equations is of considerable research interest However, the subject will not be pursued furflier here but the reader is referred to the literature (or Web sites) for more details. For flie purpose here, one of the major observations is flie very sensitive nature of flie solution to the initial conditions which is one of the distinguishing features of such chaotic systems of nonlinear equations. This is sometimes referred to as flie butterfly effect where it has been observed that if the weaflier equations for our world are a set of chaotic equations then flie flap of a butterfly s wings in one part of flie world... [Pg.571]

The chaotic nature of individual MD trajectories has been well appreciated. A small change in initial conditions (e.g., a fraction of an Angstrom difference in Cartesian coordinates) can lead to exponentially-diverging trajectories in a relatively short time. The larger the initial difference and/or the timestep, the more rapid this Lyapunov instability. Fig. 1 reports observed behavior for the dynamics of a butane molecule. The governing Newtonian model is the following set of two first-order differential equations ... [Pg.228]

Turbulence is generally understood to refer to a state of spatiotemporal chaos that is to say, a state in which chaos exists on all spatial and temporal scales. If the reader is unsatisfied with this description, it is perhaps because one of the many important open questions is how to rigorously define such a state. Much of our current understanding actually comes from hints obtained through the study of simpler dynamical systems, such as ordinary differential equations and discrete mappings (see chapter 4), which exhibit only temporal chaosJ The assumption has been that, at least for scenarios in which the velocity field fluctuates chaotically in time but remains relatively smooth in space, the underlying mechanisms for the onset of chaos in the simpler systems and the onset of the temporal turbulence in fluids are fundamentally the same. [Pg.470]

Walas (1991) has shown that the following set of ordinary differential equations gives random chaotic behaviour. [Pg.659]

The most important quantitative measure for the degree of chaotic-ity is provided by the Lyapunov exponents (LE) (Eckmann and Ru-elle, 1985 Wolf et. al., 1985). The LE defines the rate of exponential divergence of initially nearby trajectories, i.e. the sensitivity of the system to small changes in initial conditions. A practical way for calculating the LE is given by Meyer (Meyer, 1986). This method is based on the Taylor-expansion method for solving differential equations. This method is applicable for systems whose equations of motion are very simple and higher-order derivatives can be determined analytically (Schweizer et.al., 1988). [Pg.186]

In the 1960 s the meteorologist Edward Lorenz worked on systems of differential equations describing weather patters, and found something utterly different. The smallest modification in the initial conditions can have a dramatic effect, resulting in a completely different outcome after a certain time. Such behaviour is called chaotic. The sets of differential equations initially were rather complex but later he developed a simpler set which shows the same effect. [Pg.98]

The occurrence of kinetic instabilities as well as oscillatory and even chaotic temporal behavior of a catalytic reaction under steady-state flow conditions can be traced back to the nonlinear character of the differential equations describing the kinetics coupled to transport processes (diffusion and heat conductance). Studies with single crystal surfaces revealed the formation of a large wealth of concentration patterns of the adsorbates on mesoscopic (say pm) length scales which can be studied experimentally by suitable tools and theoretically within the framework of nonlinear dynamics. [31]... [Pg.66]

At the beginning of the twentiest century, the French mathematician, Henri Poincare, found that the solution of certain coupled nonlinear differential equations exhibits chaotic behavior although the underlying laws were fully deterministic. He pointed out that two systems starting with slightly different initial conditions would, after some time, move into very different directions. Since empirical observations are never exact in the mathematical sense but bear a finite error of measurement, the behavior of such systems could not be predicted beyond a certain point these systems seem to be of random nature. A random process - in contrast to a deterministic process - is characterized by From A follows B with probability pB, C with probability pc etc. [Pg.782]

Arrowsmith, D. K., and C. M. Place, Dynamical Systems - Differential Equations, Maps and Chaotic Behaviour, Chapman and Hall, London, 1992. [Pg.1215]

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). [Pg.229]

This set of equations is a nonlinear eigenvalue time delay differential equation. Such equations, even for one variable, often have periodic or chaotic solutions and, from the physics of the problem are also certain of having pulse-like solutions in some systems. [Pg.198]

So, apart from the regular behavior, which is either steady-state, periodic, or quasi-periodic behavior (trajectory on a torus, Figure 3.2), some dynamic systems exhibit chaotic behavior, i.e., trajectories follow complicated aperiodic patterns that resemble randomness. Necessary but not sufficient conditions in order for chaotic behavior to take place in a system described by differential equations are that it must have dimension at least 3, and it must contain nonlinear terms. However, a system of three nonlinear differential equations need not exhibit chaotic behavior. This kind of behavior may not take place at all, and when it does, it usually occurs only for a specific range of the system s control parameters 9. [Pg.49]

Undoubtedly, the most promising modehng of the cardiac dynamics is associated with the study of the spatial evolution of the cardiac electrical activity. The cardiac tissue is considered to be an excitable medium whose the electrical activity is described both in time and space by reaction-diffusion partial differential equations [519]. This kind of system is able to produce spiral waves, which are the precursors of chaotic behavior. This consideration explains the transition from normal heart rate to tachycardia, which corresponds to the appearance of spiral waves, and the fohowing transition to fibrillation, which corresponds to the chaotic regime after the breaking up of the spiral waves, Figure 11.17. The transition from the spiral waves to chaos is often characterized as electrical turbulence due to its resemblance to the equivalent hydrodynamic phenomenon. [Pg.349]

On the basis of the rate equations, differential equations can be set up. Due to the adsorption/chemisorption and feedback steps the solution of this set of differential equations is periodic or chaotic. Usually the occurrence of adsorption/chemisorption leads to nonlinearity and negative impedance. [Pg.191]

Nearly all models discussed so far have one feature in common they are not distributed models and can describe only spatially uniform systems. Many of the mathematical models use ordinary differential equations, and the resultant time series are nearly always simply periodic. This approach, however, describes only part of the experimentally observed behavior there is a great deal of experimental evidence for spatial heterogeneity and chaotic oscillatory behavior in heterogeneously catalyzed systems. [Pg.105]

The main problem in the solution of non-linear ordinary and partial differential equations in combustion is the calculation of their trajectories at long times. By long times we mean reaction times greater than the time-scales of intermediate species. This problem is especially difficult for partial differential equations (pdes) since they involve solving many dimensional sets of equations. However, for dissipative systems, which include most applications in combustion, the long-time behaviour can be described by a finite dimensional attractor of lower dimension than the full composition space. All trajectories eventually tend to such an attractor which could be a simple equilibrium point, a limit cycle for oscillatory systems or even a chaotic attractor. The attractor need not be smooth (e.g., a fractal attractor in a chaotic system) and is in some cases difficult to compute. However, the attractor is contained in a low-dimensional, invariant, smooth manifold called the inertial manifold M which locally attracts all trajectories exponentially and is easier to find [134,135]. It is this manifold that we wish to investigate since the dynamics of the original system, when restricted to the manifold, reduce to a lower dimensional set of equations (the inertial form). The inertial manifold is, therefore, a useful notion in the field of mechanism reduction. [Pg.364]

There are two main types of dynamical systems differential equations and iterated nuips (also known as difference equations). Differential equations describe the evolution of systems in continuous time, whereas iterated maps arise in problems where time is discrete. Differential equations are used much more widely in science and engineering, and we shall therefore concentrate on them. Later in the book we will see that iterated maps can also be very useful, both for providing simple examples of chaos, and also as tools for analyzing periodic or chaotic solutions of differential equati ons. [Pg.7]

Our work in the previous three chapters has revealed quite a bit about chaotic systems, but something important is missing intuition. We know what happens but not why it happens. For instance, we don t know what causes sensitive dependence on initial conditions, nor how a differential equation can generate a fractal attractor. Our first goal is to understand such things in a simple, geometric way. [Pg.423]

Figure 6.1 Average plankton population density as a function of the Damkohler number for logistic growth with non-uniform carrying capacity of the form K(x,y) = Kq + (5sin(27rx) sin(27ry) and chaotic mixing in the time-periodic sine-flow of Eq. (2.66). The continuous line represents results from the solution of the full partial differential equation with diffusion (Pe 104) and stars ( ) show the time-averaged plankton populations calculated from the non-diffusive Lagrangian representation. Figure 6.1 Average plankton population density as a function of the Damkohler number for logistic growth with non-uniform carrying capacity of the form K(x,y) = Kq + (5sin(27rx) sin(27ry) and chaotic mixing in the time-periodic sine-flow of Eq. (2.66). The continuous line represents results from the solution of the full partial differential equation with diffusion (Pe 104) and stars ( ) show the time-averaged plankton populations calculated from the non-diffusive Lagrangian representation.
Chaotic behaviour can arise in any system whose motion is described by a nonlinear differential equation. Whether or not it is prevalent depends on the details of the problem, but it is a general theorem that any system described by a nonlinear differential equation possesses some chaotic regime. [Pg.363]

Poincar6 was the first to assume the possibility of "irregular" or "chaotic" behavior of solutions of differential equations. From very slightly different initial conditions, resulting fix>m errors in experimental measurements for example, the solutions can exponentially deviate over time resulting in extreme sensitivity to the initial conditions. The state of a system becomes effectively impossible to predict or "chaotic". This was the result observed in 1963 by Lorenz. [Pg.15]

We now know that there are processes, which are not stochastic, whose output mimics stochastic behavior. This phenomenon is now called chaos. Chaos is a jargon word that means that a system has certain mathematical properties. It should not be confused with its nontechnical homonym that means confusion or disorder. A chaotic system can be described by a set of nonlinear difference or differential equations that have a small number of independent variables. Because these equations can be integrated in time, the future values of the variables are completely determined by their past values. [Pg.368]


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