Daolin Xu

Manipulating the scaling factor of projective synchronization in three-dimensional chaotic systems

Scaling factor characterizes the synchronized dynamics of projective synchronization in partially linear chaotic systems but it is difficult to be estimated. To manipulate projective synchronization of chaotic systems in a favored way, a control algorithm is introduced to direct the scaling factor onto a desired value. The control approach is derived from the Lyapunov stability theory. It allows us to arbitrarily amplify or reduce the scale of the response of the slave system via a feedback control on the master system. In numerical experiments, we illustrate the application to the Lorenz system. (c) 2001 American Institute of Physics.

Stability criterion for projective synchronization in three-dimensional chaotic systems

Abstract Projective synchronization, in which the state vectors synchronize up to a scaling factor, has recently been observed in coupled partially linear chaotic systems (Lorenz system) under certain conditions. In this Letter, we present a stability criterion that guarantees the occurrence of the projective synchronization in three-dimensional systems. By applying the criterion to two typical partially linear systems (Lorenz and disk dynamo), it shows that only some parameters play the key role in influencing the stability. Projective synchronization only happens when σ>−1 for the Lorenz and μ>0 for the disk dynamo.

Criteria for the Occurrence of Projective Synchronization in Chaotic Systems of Arbitrary Dimension

Abstract The conditions of projective synchronization in partially linear chaotic systems of arbitrary dimension are studied. We find that, for any bounded Jacobian matrix M , projective synchronization happens when all the Lyapunov exponents are nonpositive. Several simple criteria are explored for matrix M with the properties of mik=mki and mik=−mki, which enable us to assess the possibility of the synchronization by the eigenvalues and the diagonal elements of the Jacobian. All conditions are globally stable. Examples are provided to illustrate projective synchronization in high dimension.

On creation of Hopf bifurcations in discrete-time nonlinear systems

Bifurcation characteristics of a nonlinear system can be manipulated by small controls. In this paper, we present a control method to create Hopf bifurcations in discrete-time nonlinear systems. The critical conditions for the Hopf bifurcations are discussed. The center manifold method, normal form technique and the Ioosss Hopf bifurcation theory are employed in the derivation of the control gain. Numerical demonstration is provided. (c) 2002 American Institute of Physics.

Control of degenerate Hopf bifurcations in three-dimensional maps.

A feedback control method is proposed to create a degenerate Hopf bifurcation in three-dimensional maps at a desired parameter point. The particularity of this bifurcation is that the system admits a stable fixed point inside a stable Hopf circle, between which an unstable Hopf circle resides. The interest of this solution structure is that the asymptotic behavior of the system can be switched between stationary and quasi-periodic motions by only tuning the initial state conditions. A set of critical and stability conditions for the degenerate Hopf bifurcation are discussed. The washout-filter-based controller with a polynomial control law is utilized. The control gains are derived from the theory of Chenciners degenerate Hopf bifurcation with the aid of the center manifold reduction and the normal form evolution.

Onset of degenerate Hopf bifurcation of a vibro-impact oscillator

An analytical method of the degenerate Hopf bifurcation is proposed for vibro-impact systems. The phenomenon of the bifurcation and its complicated dynamics are observed. This type of bifurcation originates multi-coexisting solutions dependent of the initial state of the system.

Estimation of initial conditions and parameters of a chaotic evolution process from a short time series

Tracing back to the initial state of a time-evolutionary process using a segment of historical time series may lead to many meaningful applications. In this paper, we present an estimation method that can detect the initial conditions, unobserved time-varying states and parameters of a dynamical (chaotic) system using a short scalar time series that may be contaminated by noise. The technique based on the Newton-Raphson method and the least-squares algorithm is tolerant to large mismatch between the initial guess and actual values. The feasibility and robustness of this method are illustrated via the numerical examples based on the Lorenz system and Rossler system corrupted with Gaussian noise.

Estimation of periodic-like motions of chaotic evolutions using detected unstable periodic patterns

We introduce an approach to detect cyclical patterns embedded within chaotic data and make use of the detected patterns to estimate periodic-like motions in a chaotic process. A chaotic attractor contains many unstable periodic orbits (UPOs). The UPOs are hidden cyclical patterns that dominate the dynamical evolution of the system. Knowledge of UPOs can be used for estimating the trends of chaotic evolutions. A numerical experiment is conducted to illustrate an application on the business cycle detection.

Modeling global vector fields of chaotic systems from noisy time series with the aid of structure-selection techniques.

We address the problem of reconstructing a set of nonlinear differential equations from chaotic time series. A method that combines the implicit Adams integration and the structure-selection technique of an error reduction ratio is proposed for system identification and corresponding parameter estimation of the model. The structure-selection technique identifies the significant terms from a pool of candidates of functional basis and determines the optimal model through orthogonal characteristics on data. The technique with the Adams integration algorithm makes the reconstruction available to data sampled with large time intervals. Numerical experiment on Lorenz and Rossler systems shows that the proposed strategy is effective in global vector field reconstruction from noisy time series.

Identification of Spring-Force Factors of Suspension Systems Using Progressive Neural Network on a Validated Computer Model

The spring-force factor is a multiple scalar that amplifies (or reduces) the spring force of a road-wheel suspension. The setting of spring-force factors for individual suspensions can change the stiffness property of an entire suspension system of a tracked vehicle. In engineering practice, these factors may not be accurately known. In this article, a novel progressive Neural Network (NN) technique is suggested to determine the suspension factors based on the dynamic response (displacement) of the road-wheels. A rope-length sensor can easily measure the displacement. The NN model is established and trained off-line using initial training data including a set of assumed spring-force factors and the displacements of road-wheels computed from a validated model built by the tracked vehicle toolkits (ATV) of ADAMS (a multi-body dynamics simulation tool). The trained NN model can inversely determine the suspension factors by feeding in the displacements of the road-wheels. The identified factors are then used to update the factors in the ATV model from which the new displacements of the road-wheels can be generated. A progressive retraining process of the NN model is continuously conducted until the errors between the calculated displacements and the measured ones decrease to an acceptable level. The procedures are examined for the determination of suspension factors of an in-service tracked vehicle. It is found that the present technique is very robust for the determination of the suspension factors of the tracked vehicle.