Stabilizing the unstable periodic orbits of a chaotic system using model independent adaptive time-delayed controller

In this paper we deal with the control of chaotic systems. Knowing that a chaotic attractor contains a myriad of unstable periodic orbits (UPO’s), the aim of our work is to stabilize some of the UPO’s embedded in the chaotic attractor and which have interesting characteristics. First, using the input-to-state linearization method in conjunction with a time-delayed state feedback, we design a control signal that can achieve stabilization. Next, an adaptive time-delayed state feedback is proposed which shows at once efficiency and simplicity and circumvents the construction complexity of the first controller. Finally, we propose a reduced order sliding mode observer to estimate the necessary states for the design of an adaptive time delayed state feedback controller. This last controller has one main advantage, it in fact achieves UPO stabilization without using the system model. The efficacy of the proposed methods is illustrated by numerical simulations onto Chua’s system.     

Investigating the nature of motion in 3D perturbed elliptic oscillators displaying exact periodic orbits

We study the nature of motion in a 3D potential composed of perturbed elliptic oscillators. Our technique is to use the results obtained from the 2D potential in order to find the initial conditions generating regular or chaotic orbits in the 3D potential. Both 2D and 3D potentials display exact periodic orbits together with extended chaotic regions. Numerical experiments suggest that the degree of chaos increases rapidly as the energy of the test particle increases. About 97?% of the phase plane of the 2D system is covered by chaotic orbits for large energies. The regular or chaotic character of the 2D orbits is checked using the S(c) dynamical spectrum, while for the 3D potential we use the S(c) spectrum, along with the P(f) spectral method. Comparison with other dynamical indicators shows that the S(c) spectrum gives fast and reliable information about the character of motion.     

A semi-numerical method for periodic orbits in a bisymmetrical potential

We use a semi-numerical method to find the position and period of periodic orbits in a bisymmetrical potential, made up of a two dimensional harmonic oscillator, with an additional term of a Plummer potential, in a number of resonant cases. The results are compared with the outcomes obtained by the numerical integration of the equations of motion and the agreement is good. This indicates that the semi-numerical method gives general and reliable results. Comparison with other methods of locating periodic orbits is also made.     

Global stabilization of periodic orbits using a proportional feedback control with pulses

We investigate the stabilization of periodic orbits of one-dimensional discrete maps by using a proportional feedback method applied in the form of pulses. We determine a range of the parameter μ values representing the strength of the feedback for which all positive solutions of the controlled equation converge to a periodic orbit.     

Fast galerkin method and its application to determine periodic solutions of non-linear oscillators

The Galerkin method is an approximate method which finds wide application in solving differential and integral equations. But a large amount of computation is needed in order to get a high order approximation by using the method. Applying the FFT technique to form a so-called fast Galerkin method, we can reduce the computation work greatly, when taking trigonometric functions as characteristic functions. Taking the periodic solution of non-linear oscillators as an example, we illustrate the procedure and the efficiency of the method. Moreover, with some modifications we extend the applicability of the method, so that not only periodic solutions with known periods, but also those with unknown periods, as well as subharmonics, combination tones, etc., can be treated with the method. Some techniques are described which can be used to simplify the computation.     

Detecting unstable periodic orbits and unstable quasiperiodic orbits in vibro-impact systems

In this paper, unstable dynamics is considered for the models of vibro-impact systems with linear differential equations coupled to an impact map. To provide a skeleton for the organization of chaotic attractors, we propose a method for detecting unstable periodic orbits embedded in chaotic attractors through a combination of unconstrained optimization technique and Poincaré map. Three numerical examples from different vibro-impact models demonstrate that the strategy can efficiently detect unstable periodic orbits in chaotic attractors. In order to explore the mechanism responsible for the creation of multi-dimensional tori attractors, we also present another method to detect unstable quasiperiodic orbits embedded multi-dimensional tori attractors by examining a specially transient time series. The upper bound and lower bound of the transient time series (in the Poincaré map) can be obtained by analyzing transient Lyapunov exponent and transient Lyapunov dimension. Some examples verify the effectiveness of the numerical algorithm.     

A modified incremental harmonic balance method for rotary periodic motions

The determination of periodic solutions is an essential step in the study of dynamic systems. If some of the generalized coordinates describing the configuration of a system are angular positions relative to certain reference axes, the associated periodic motions divide into two types: oscillatory and rotary periodic motions. For an oscillatory periodic motion, all the generalized coordinates are periodic in time. On the other hand, for a rotary periodic motion, some angular coordinates may have unbounded magnitude due to the persistent circulation about their pivots. In this case, although the behaviour of the system is periodic physically, those angular coordinates are not periodic in time. Although various effective methods have been developed for the determination of oscillatory periodic motion, the rotary periodic motion can only be determined by brute force integration. In this paper, the incremental harmonic balance (IHB) method is modified so that rotary periodic motions can be determined as well as oscillatory periodic motions in a unified formulation. This modified IHB method is applied to a practical device, a rotating disk equipped with a ball-type balancer, to show its effectiveness.     

A novel construction of homoclinic and heteroclinic orbits in?nonlinear oscillators by a perturbation-incremental method

A novel construction of homoclinic/heteroclinic orbits (HOs) in nonlinear oscillators is presented in this paper. An accurate analytical solution of a HO for small perturbation can be obtained in terms of trigonometric functions. An advantage of the present construction is that it gives an accurate approximate solution of a HO for large parametric value in relatively few harmonic terms while other analytical methods such as the Lindstedt?CPoincaré method and the multiple scales method fail to do so.     

Utilizing true periodic orbits in chaos-based cryptography

Nonlinear Dynamics - Digital realizations of chaos-based cryptosystems suffer from lack of a reliable method for implementation. The common choice for implementation is to use fixed or...     

Coexisting periodic orbits in vibro-impacting dynamical systems

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Relative periodic orbits in plane Poiseuille flow

A branch of relative periodic orbits is found in plane Poiseuille flow in a periodic domain at Reynolds numbers ranging from Re=3000$\mathit{Re}=3000$ to Re=5000$\mathit{Re}=5000$. These solutions consist in sinuous quasi-streamwise streaks periodically forced by quasi-streamwise vortices in a self-sustained process. The streaks and the vortices are located in the bulk of the flow. Only the amplitude, but not the shape, of the averaged velocity components does change as the Reynolds number is increased from 3000 to 5000. We conjecture that these solutions could therefore be related to large- and very large-scale structures observed in the bulk of fully developed turbulent channel flows.     

Variable-coefficient harmonic balance for periodically forced nonlinear oscillators

This paper explores the application of the method of variable-coefficient harmonic balance to nonautonomous nonlinear equations of the form XsF(X, t:), and in particular, a one-degree-of-freedom nonlinear oscillator equation describing escape from a cubic potential well. Each component of the solution, X(t), is expressed as a truncated Fourier series of superharmonics, subharmonics and ultrasubharmonics. Use is then made of symbolic manipulation in order to arrange the oscillator equation as a Fourier series and its coefficient are evaluated in the traditional way. The time-dependent coefficients permit the construction of a set of amplitude evolution equations with corresponding stability criteria. The technique enables detection of local bifurcations, such as saddle-node folds, period doubling flips, and parts of the Feigenbaum cascade. This representation of the periodic solution leads to local bifurcations being associated with a term in the Fourier series and, in particular, the onset of a period doubled solution can be detected by a series of superharmonics only. Its validity is such that control space bifurcation diagrams can be obtained with reasonable accuracy and large reductions in computational expense.     

Incremental harmonic balance method for periodic forced oscillation of a dielectric elastomer balloon

Dielectric elastomer(DE) is suitable in soft transducers for broad applications,among which many are subjected to dynamic loadings, either mechanical or electrical or both. The tuning behaviors of these DE devices call for an efficient and reliable method to analyze the dynamic response of DE. This remains to be a challenge since the resultant vibration equation of DE, for example, the vibration of a DE balloon considered here is highly nonlinear with higher-order power terms and time-dependent ...     

Bifurcations of periodic orbits in Duffing equation with periodic damping and external excitations

The complex behaviors of Duffing equation with periodic damping and external excitations are investigated. The existence conditions and bifurcations of periodic orbits with three different frequencies resonant conditions are concerned by the second-order averaging method and the Melnikov method. The rich dynamical behaviors are so distinct when different periodic damping excitations are added, including more complicated averaged equations, bifurcation curves, bifurcation conditions, and even chaos. The numerical simulations show the consistence with the theoretical analysis and reveal new complex phenomena which cannot be given by theoretical analysis.     

Stability transitions for periodic orbits in hamiltonian systems

The paper considers one-parameter families of periodic solutions of real analytic Hamiltonian systems with two degrees of freedom, the parameter being the energy h. Conditions are given which guarantee that this family will undergo infinitely many changes in stability status as h tends to some finite value h ₀. First considered is the case of a critical point (with eigenvalues ±, ±i, and >0) of the Hamiltonian at energy h ₀ with the property that the family limits to a homoclinic orbit asymptotic to this point. Some generalizations of this case are given, and applications are made to examples such as the Hénon-Heiles Hamiltonian. We obtain an infinite sequence of distinct energy intervals converging to h ₀ on which the periodic orbits are elliptic. Requirements for the elliptic stability of the orbits are then given. The additional conditions for an infinite sequence of distinct energy intervals converging to h ₀, on which the orbits are hyperbolic, involve the coexistence problem for an associated Hill's equation that appears when the relevant Poincaré maps along the orbits are computed in coordinates. The results are compared to the case where the critical point has eigenvalues (±±i), and >0, investigated by Henrard and Devaney.     

Double impact periodic orbits for an inverted pendulum

There exist many types of possible periodic orbits that impact at the walls for the inverted pendulum impacting between two rigid walls. Previous studies only focused on single impact periodic orbits and symmetric periodic orbits that bounce back and forth between the two walls. They respectively correspond to Types I and II orbits in the Chow, Shaw and Rand classification. In this paper we discuss two types of double impact periodic orbits that have not been studied before. The equations need to be solved for double impact orbits are transcendental and it is very hard to see the structure of the solutions. Consequently the analysis of double impact orbits is much more difficult than that of Types I and II orbits. A combination of analytical and numerical methods is employed to investigate the existence, stability and bifurcations of these orbits. Grazing bifurcations, which do not present for Types I and II orbits, are also observed.