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For other initial conditions, the equation of motion is given by the exponential of a matrix : for an initial point x 0 ,.

1. Background

From the eigenvalues and the eigenvectors of A it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the necessary but not sufficient conditions for chaotic behavior. A discrete-time , affine dynamical system has the form of a matrix difference equation :.

The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map. As in the continuous case, the eigenvalues and eigenvectors of A determine the structure of phase space. Points in this straight line run into the fixed point.

There are also many other discrete dynamical systems. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates usually unspecified, but computable that makes the dynamical system as simple as possible. A flow in most small patches of the phase space can be made very simple.

This is known as the rectification theorem. The rectification theorem says that away from singular points the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space M the dynamical system is integrable. In most cases the patch cannot be extended to the entire phase space. The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one.

If the next time the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches. In general, in the neighborhood of a periodic orbit the rectification theorem cannot be used. This is known as the conjugation equation.

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Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. The results on the existence of a solution to the conjugation equation depend on the eigenvalues of J and the degree of smoothness required from h. As J does not need to have any special symmetries, its eigenvalues will typically be complex numbers. When the eigenvalues of J are not in the unit circle, the dynamics near the fixed point x 0 of F is called hyperbolic and when the eigenvalues are on the unit circle and complex, the dynamics is called elliptic. The hyperbolic case is also structurally stable.

At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems.

For a flow, it will occur when there are eigenvalues on the imaginary axis. For more information, see the main article on Bifurcation theory. Some bifurcations can lead to very complicated structures in phase space. For example, the Ruelle—Takens scenario describes how a periodic orbit bifurcates into a torus and the torus into a strange attractor. In another example, Feigenbaum period-doubling describes how a stable periodic orbit goes through a series of period-doubling bifurcations.

In the Hamiltonian formalism , given a coordinate it is possible to derive the appropriate generalized momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the Liouville measure. In a Hamiltonian system, not all possible configurations of position and momentum can be reached from an initial condition.

Because of energy conservation, only the states with the same energy as the initial condition are accessible. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution. Then almost every point of A returns to A infinitely often. One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the ergodic hypothesis. The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems.

Koopman approached the study of ergodic systems by the use of functional analysis. An observable a is a function that to each point of the phase space associates a number say instantaneous pressure, or average height. This idea has been generalized by Sinai, Bowen, and Ruelle SRB to a larger class of dynamical systems that includes dissipative systems.

SRB measures replace the Boltzmann factor and they are defined on attractors of chaotic systems. Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This seemingly unpredictable behavior has been called chaos.

Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit the stable manifold and another of the points that diverge from the orbit the unstable manifold. This branch of mathematics deals with the long-term qualitative behavior of dynamical systems.

Here, the focus is not on finding precise solutions to the equations defining the dynamical system which is often hopeless , but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible attractors? Note that the chaotic behavior of complex systems is not the issue.

Meteorology has been known for years to involve complex—even chaotic—behavior. Chaos theory has been so surprising because chaos can be found within almost trivial systems. The logistic map is only a second-degree polynomial; the horseshoe map is piecewise linear. Dynamical systems are defined over a single independent variable, usually thought of as time.

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A more general class of systems are defined over multiple independent variables and are therefore called multidimensional systems. Such systems are useful for modeling, for example, image processing. From Wikipedia, the free encyclopedia. This article is about the general aspects of dynamical systems. For technical details, see Dynamical system definition. For the study, see Dynamical systems theory.

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Stability Analysis of Continuous Systems with Structural Perturbations

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