Orbits Of Nonlinear Systems Research Articles

Generic stabilizability for time-delayed feedback control.

Time-delayed feedback control is one of the most successful methods to discover dynamically unstable features of a dynamical system in an experiment. This approach feeds back only terms that depend on the difference between the current output and the output from a fixed time T ago. Thus, any periodic orbit of period T in the feedback-controlled system is also a periodic orbit of the uncontrolled system, independent of any modelling assumptions. It has been an open problem whether this approach can be successful in general, that is, under genericity conditions similar to those in linear control theory (controllability), or if there are fundamental restrictions to time-delayed feedback control. We show that, in principle, there are no restrictions. This paper proves the following: for every periodic orbit satisfying a genericity condition slightly stronger than classical linear controllability, one can find control gains that stabilize this orbit with extended time-delayed feedback control. While the paper's techniques are based on linear stability analysis, they exploit the specific properties of linearizations near autonomous periodic orbits in nonlinear systems, and are, thus, mostly relevant for the analysis of nonlinear experiments.

Nonlinear optimal stabilization of unstable periodic orbits

The framework for the optimal stabilization of unstable periodic orbits in nonlinear systems is proposed. First, the stable manifold method for approximately solving Hamilton-Jacobi equations in the optimal control problem is extended to nonlinear periodic systems. Then, this method is applied for optimal stabilization problem of error dynamics around an unstable periodic orbit. The effectiveness of the proposed method is evaluated via numerical simulations for the Rossler chaotic system.

The construction of homo- and heteroclinic orbits in non-linear systems

Pade and quasi-Pade approximants are used to construct homo- and heteroclinic orbits of non-linear systems. By using the convergence condition for Pade approximants and the conditions at infinity the problem can be solved with sufficiently high accuracy. Actual computations are carried out for the non-autonomous Duffing equation, the equations of vibrations of a parametrically driven mathematical pendulum, and the van der Pol-Duffing equation with non-linear elastic characteristic.