The Generalized OGY Method

RUTCOR Research Report RRR16-1998, 1998

Intuitively the chaos is an infinitively sensitive dynamical system. Basically there are two different types of control problems of chaotic systems. The first one is to stabilize the system at an unstable fixed point. In the second one there are given two similar but non-identical systems and the problem is to synchronize the two systems. This report is devoted to the first problem. One of the most successful ways to solve it is the well-known OGY method. It is based on the first order approximation of the system and can be switched on only if the trajectory is close enough to the unstable fixed point. It in this paper an extension of the OGY method is suggested which enlarges the region of convergence and in this way shortens the time necessary to the stabilization.

Astronomy and Astrophysics, 2002

This paper deals with the analytic continuation of periodic orbits of conservative dynamical systems with three degrees of freedom. For variations of any parameter (or integral), it relies on numerical analysis in order to implement a predictor-corrector algorithm to compute the initial conditions of the periodic orbits pertaining to the family. The method proposed here is not restricted to symmetric problems and, since the procedure involves the computation of the variational equations, a side effect is the trivial computation of the linear stability of the periodic orbits. As an illustration of the robustness of the method, several families of periodic orbits of the Restricted Three-Body Problem are computed.

International Journal of Engineering, Technology and Natural Sciences, 2020

In this paper we study the stability of one of a non linear autoregressive model with trigonometric term by using local linearization method proposed by Tuhro Ozaki .We find the singular point ,the stability of the singular point and the limit cycle. We conclude that the proposed model under certain conditions have a non-zero singular point which is a asymptotically salable (when → t N 0) and have an orbitaly stable limit cycle. Also we give some examples in order to explain the method.

Advanced Courses in Mathematics - CRM Barcelona, 2015

Chaos, Solitons & Fractals, 1995

AbsWact-A numerical method is presented for the efficient computation and continuation of periodic solutions of large systems of ordinary differential equations. The method is particularly useful when a shooting approach based on full Newton or quasi-Newton iteration is prohibitively expensive. Both stable and unstable solutions can be computed. The basic idea is to split the eigenspace of the monodromy matrix around the periodic orbit into two orthogonal components which contain, respectively, all eigenvalues with norm less than or greater than some threshold value less than unity. In the former 'stable' subspace one performs time integration, which is equivalent to a Picard iteration; in the much smaller, 'unstable' subspace a Newton step is carried out. A strategy for computing the stable and unstable subspaces without forming the full monodromy matrix is briefly discussed. Numerical results are presented for the one-dimensional Brusselator model. The Newton-Picard method with an appropriately chosen threshold value proves to be accurate and much more efficient than shooting based on full Newton iteration.

Physics Letters A, 1994

Based on a simple geometrical construction, an algorithm is given for stabilizing hyperbolic periodic orbits of two-dimensional maps. The method does not require analytical knowledge of the system's dynamics, only a rough location of the linearized region around the periodic orbit to be stabilized (the target region). This construction can thus be used as a chaos controlling method of feedback type, accessible to experiments. A novel method to find the target regions is also given. 0375-9601/94/$07.00 ~) 1994 Elsevier Science B.V. All fights reserved SSDI0375-9601 (94)00365-V

NATO ASI Series, 1994

The accurate computation of periodic orbits and the precise knowledge of their properties are very important for studying the behavior of many dynamical systems of physical interest. In this paper, we present an efficient numerical method for computing to any desired accuracy periodic orbits (stable, unstable and complex) of any period. This method always converges rapidly to a periodic orbit independently of the initial guess, which is important when the mapping has many periodic orbits, stable and unstable close to each other, as is the case with conservative systems. We illustrate this method first, on the 2-Dimensional quadratic Henon's mapping, by computing rapidly and accurately several periodic orbits of high period. We also apply our method here to a 3-D conservative mapping as well as a 4-D complex version of Henon's map.

Celestial Mechanics, 1981

The usual description of motion near a periodic orbit as a solution to a linear time-periodic system (a Floquet problem) can be formally extended to higher orders of approximation. Each subsequent order problem is a linear, time-multiply periodic system. In formulating the second order problem the order of the Hamiltonian can be locally reduced by one degree of freedom for every exact integral of the motion present in the original problem. As in the Floquet problem, the system's state transition matrix at each order can be formally decomposed into the product of a bounded multiply-periodic matrix and the exponential of a constant matrix times the time. This yields stability information for the second and higher order approximations, analogous to Poincar6 exponents in the Floquet problem. However, second and higher order stability exponents are functions of the displacement from the original periodic orbit, so this method may be able to predict the size of the region of stable motion surrounding a periodic orbit.

Numerical Continuation Methods for Dynamical Systems, 2007

Continuation methods utilizing boundary value solvers are an effective tool for computing unstable periodic orbits of dynamical systems. Auto [1] is the standard implementation of these procedures. Unfortunately, the collocation methods used in Auto often require very fine meshes for convergence on problems with multiple time scales. This inconvenience prompts the search for alternative methods for computing such periodic orbits; we introduce here new multiple-shooting algorithms based on geometric singular perturbation theory.

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2011