Unstable Fixed Points Research Articles

Control of Discrete Time Chaotic Systems via Combination of Linear and Nonlinear Dynamic Programming

In this article by introducing and subsequently applying the Min–Max method, chaos has been suppressed in discrete time systems. By using this nonlinear technique, the chaotic behavior of Behrens–Feichtinger model is stabilized on its first and second-order unstable fixed points (UFP) in presence and absence of noise signal. In this step, a comparison has also been carried out among the proposed Min–Max controller and the Pyragas delayed feedback control method. Next, to reduce the computation required for controller design, the clustering method has been introduced as a quantization method in the Min–Max control approach. To improve the performance of the acquired controller through clustering method obtained with the Min–Max method, a linear optimal controller is also introduced and combined with the previously discussed nonlinear control law. The resultant combined controller has been applied on the Henon map and through comparison with both Pyragas controller, and the linear optimal controller alone, its advantages are discussed.

Non-classical large deviations for a noisy system with non-isolated attractors

We study the large deviations of a simple noise-perturbed dynamical system havingcontinuous sets of steady states, which mimic those found in some partial differentialequations related, for example, to turbulence problems. The system is a two-dimensionalnonlinear Langevin equation involving a dissipative, non-potential force, which has theessential effect of creating a line of stable fixed points (attracting line) touching a line ofunstable fixed points (repelling line). Using different analytical and numericaltechniques, we show that the stationary distribution of this system satisfies, inthe low-noise limit, a large deviation principle containing two competing terms:(i) a ‘classical’ but sub-dominant large deviation term, which can be derivedfrom the Freidlin–Wentzell theory of large deviations by studying the fluctuationpaths or instantons of the system near the attracting line, and (ii) a dominantlarge deviation term, which does not follow from the Freidlin–Wentzell theory, asit is related to fluctuation paths of zero action, referred to as sub-instantons,emanating from the repelling line. We discuss the nature of these sub-instantons, andshow how they arise from the connection between the attracting and repellinglines. We also discuss in a more general way how we expect these to arise in moregeneral stochastic systems having connected sets of stable and unstable fixedpoints, and how they should determine the large deviation properties of thesesystems.

Implementation of dynamic programming for chaos control in discrete systems

In this paper the control of discrete chaotic systems by designing linear feedback controllers is presented. The linear feedback control problem for nonlinear systems has been formulated under the viewpoint of dynamic programming. For suppressing chaos with minimum control effort, the system is stabilized on its first order unstable fixed point (UFP). The presented method also could be employed to make any desired n th order fixed point of the system, stable. Two different methods for higher order UFPs stabilization are suggested. Afterwards, these methods are applied to two well-known chaotic discrete systems: the Logistic and the Henon Maps. For each of them, the first and second UFPs in their chaotic regions are stabilized and simulation results are provided for the demonstration of performance.

Mirror symmetry breaking as a problem in dynamic critical phenomena

The critical properties of the Frank model of spontaneous chiral synthesis are discussed by applying results from the field theoretic renormalization group (RG). The long time and long wavelength features of this microscopic reaction scheme belong to the same universality class as multicolored directed percolation processes. Thus the following RG fixed points (FPs) govern the critical dynamics of the Frank model for d<4: one unstable FP that corresponds to complete decoupling between the two enantiomers, a saddle point that corresponds to symmetric interspecies coupling, and two stable FPs that individually correspond to unidirectional couplings between the two chiral molecules. These latter two FPs are associated with the breakdown of mirror or chiral symmetry. In this simplified model of molecular synthesis, homochirality is a natural consequence of the intrinsic reaction noise in the critical regime, which corresponds to extremely dilute chemical systems.

A full delayed feedback controller design method for time-delay chaotic systems

A full delayed feedback controller (FDFC) design method is proposed to stabilize unstable fixed points (UFPs) for a class of continuous time-delay chaotic systems. Unlike the existing methods where the controller delay is usually determined by trial-and-error, it is shown in FDFC that both the upper bound of the controller delay and the controller gain can be obtained computationally. The simulation result demonstrates the effectiveness of the proposed FDFC design method.

A stabilization method of chaotic systems based on full delayed feedback controller design

A full delayed feedback controller (FDFC) design method is proposed to stabilize the unstable fixed points (UFPs) for continuous chaotic systems. It is shown that the two parameters of a FDFC, i.e., the controller gain and the upper bound of controller time delay, can be simultaneously obtained by solving some linear matrix inequalities. Two examples are presented to show the effectiveness of proposed method.

Another limitation of DFC when stabilizing unstable fixed points of continuous chaotic systems

Using stability theory of delayed differential equation (DDE), we show that there exists another limitation of delayed feedback control (DFC) with arbitrary delayed time when stabilizing unstable fixed points (UFPs) of continuous chaotic systems. This limitation is called by zero real part limitation, that is, if Jacobian matrix at a UFP has a characteristic exponent with zero real part, the UFP cannot be stabilized by linear DFC with arbitrary delayed time.

On the scale-invariant distribution of the diffusion coefficient for classical particles diffusing in disordered media

The scaling form of the whole distribution P(D) of the random diffusion coefficient D(x) in a model of classically diffusing particles is investigated. The renormalization group approach above the lower critical dimension d=0 is applied to the distribution P(D) using the n-replica approach. In the annealed approximation (n=1), the inverse gaussian distribution is found to be the stable one under rescaling. This identification is made based on symmetry arguments and subtle relations between this model and that of fluc- tuating interfaces studied by Wallace and Zia. The renormalization-group flow for the ratios between consecutive cumulants shows a regime of pure diffusion for small disorder, in which P(D) goes to delta(D-<D>), and a regime of strong disorder where the cumulants grow infinitely large and the diffusion process is ill defined. The boundary between these two regimes is associated with an unstable fixed-point and a subdiffusive behavior: <x**2>=Ct**(1-d/2). For the quenched case (n goes to 0) we find that unphysical operators are generated raisng doubts on the renormalizability of this model. Implications to other random systems near their lower critical dimension are discussed.