Share Give access Share full text access. Share full text access. Please review our Terms and Conditions of Use and check box below to share full-text version of article. Abstract This paper presents a global bifurcation scenario for chaotic dynamical systems that solve optimization problems.
Citing Literature. Related Information. Close Figure Viewer. Browse All Figures Return to Figure. Previous Figure Next Figure. We have a dedicated site for Germany. Global Analysis of Nonlinear Dynamics collects chapters on recent developments in global analysis of non-linear dynamical systems with a particular emphasis on cell mapping methods developed by Professor C.
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History of dynamical systems - Scholarpedia
This collection of contributions prepared by a diverse group of internationally recognized researchers is intended to stimulate interests in global analysis of complex and high-dimensional nonlinear dynamical systems, whose global properties are largely unexplored at this time. Luo, Southern Illinois University Edwardsville, aluo siue.
Before the advent of computers , finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories.
The difficulties arise because:.
In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions frequency, stability, asymptotic, and so on. Aleksandr Lyapunov developed many important approximation methods. His methods, which he developed in , make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamic system. In , he published his Dynamical Systems Birkhoff's most durable result has been his discovery of what is now called the ergodic theorem.
Combining insights from physics on the ergodic hypothesis with measure theory , this theorem solved, at least in principle, a fundamental problem of statistical mechanics. The ergodic theorem has also had repercussions for dynamics. Stephen Smale made significant advances as well. His first contribution is the Smale horseshoe that jumpstarted significant research in dynamical systems.
He also outlined a research program carried out by many others.
Oleksandr Mykolaiovych Sharkovsky developed Sharkovsky's theorem on the periods of discrete dynamical systems in The notion of smoothness changes with applications and the type of manifold. When T is taken to be the reals, the dynamical system is called a flow ; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow. When T is taken to be the integers, it is a cascade or a map ; and the restriction to the non-negative integers is a semi-cascade.
The vector field v x is a smooth function that at every point of the phase space M provides the velocity vector of the dynamical system at that point. There is no need for higher order derivatives in the equation, nor for time dependence in v x because these can be eliminated by considering systems of higher dimensions. Other types of differential equations can be used to define the evolution rule:. Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally Banach spaces —in which case the differential equations are partial differential equations.
In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity. Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the N -dimensional Euclidean space, so any point in phase space can be represented by a vector with N numbers. The solution to this system can be found by using the superposition principle linearity. For other initial conditions, the equation of motion is given by the exponential of a matrix : for an initial point x 0 ,.
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.
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Points in this straight line run into the fixed point. There are also many other discrete dynamical systems.
Introduction to Stability Analysis of 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. 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.
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