Chaos synchronization and parameter estimation from a scalar output signal

Local observability of state variables and parameters in nonlinear modeling quantified by delay reconstruction

Features of the Jacobian matrix of the delay coordinates map are exploited for quantifying the robustness and reliability of state and parameter estimations for a given dynamical model using a measured time series. Relevant concepts of this approach are introduced and illustrated for discrete and continuous time systems employing a filtered Hénon map and a Rössler system.

State and parameter estimation using unconstrained optimization

We present an efficient method for estimating variables and parameters of a given system of ordinary differential equations by adapting the model output to an observed time series from the (physical) process described by the model. The proposed method is based on (unconstrained) nonlinear optimization exploiting the particular structure of the relevant cost function. To illustrate the features and performance of the method, simulations are presented using chaotic time series generated by the Colpitts oscillator, the three-dimensional Hindmarsh-Rose neuron model, and a nine-dimensional extended Rössler system.

Estimating parameters by anticipating chaotic synchronization

Anticipating chaotic synchronization based parameter estimation of chaotic system is investigated. We propose a method to estimate the unknown parameters of the interested chaotic system even if only a scalar time series is available. Although the Krasovskii-Lyapunov functional method often results in delay-independent stability condition, stability of the synchronization manifold usually relates to time delay. We analyze the stability of anticipating synchronization based parameter estimation numerically. Series of driven systems are used to increase the anticipation time, however, result in longer time to estimate the unknown parameters. These results are also confirmed by numerical simulations.

Estimating parameters of a nonlinear dynamical system

A method based on a modified Newton-Raphson scheme is presented to estimate parameters of a nonlinear dynamical system from the time series data of the variables. The method removes some of the problems associated with the standard synchronization based methods. An important achievement of this method is that it is possible to determine the exact form of dynamical equations for systems with quadratic nonlinearity.

Adaptive scheme for synchronization-based multiparameter estimation from a single chaotic time series and its applications

Chaos-synchronization-based multiparameter estimation of a multiply delayed feedback system is investigated. We propose an adaptive method that can estimate all the parameters of the response system using the driving signal only. In the past few years, various methods have been developed for estimation of multiparameters of a chaotic system but most of them require more than one time series to estimate all the parameters of a chaotic or hyperchaotic system. The proposed method requires only a single chaotic time series to estimate all the parameters. A sufficient condition for synchronization is derived and it is shown that the numerical results well support the analytic calculations. The synchronized system has applications in cryptographic encoding for digital and analog signals, which is shown with an example.

Data assimilation as a nonlinear dynamical systems problem: stability and convergence of the prediction-assimilation system

We study prediction-assimilation systems, which have become routine in meteorology and oceanography and are rapidly spreading to other areas of the geosciences and of continuum physics. The long-term, nonlinear stability of such a system leads to the uniqueness of its sequentially estimated solutions and is required for the convergence of these solutions to the system's true, chaotic evolution. The key ideas of our approach are illustrated for a linearized Lorenz system. Stability of two nonlinear prediction-assimilation systems from dynamic meteorology is studied next via the complete spectrum of their Lyapunov exponents; these two systems are governed by a large set of ordinary and of partial differential equations, respectively. The degree of data-induced stabilization is crucial for the performance of such a system. This degree, in turn, depends on two key ingredients: (i) the observational network, either fixed or data-adaptive, and (ii) the assimilation method.

Design of fast state observers using a backstepping-like approach with application to synchronization of chaotic systems

A simple technique is introduced to build fast state observers for chaotic systems when only a scalar time series of the output is available. This technique relies on using a backstepping-like approach via introducing new virtual states that can be observed using the drive-response synchronization mechanism. The proposed dynamic structure of the virtual states allows for employing control parameters that can adjust the convergence rate of the observed states. In addition, these control parameters can be used to improve the transient performance of the response system to accommodate small and large variations of the initial conditions, thus achieving superior performance to conventional synchronization techniques. Simple Lyapunov functions are used to estimate the range of the control parameters that guarantees stable operation of the proposed technique. Three benchmark chaotic systems are considered for illustration; namely, the Lorenz, Chua, and Rossler systems. The conflict between stability and agility of the states observer is analyzed and a simple tuning mechanism is introduced. Implementation of the proposed technique in both analog and digital forms is also addressed and experimental results are reported ensuring feasibility and real-time applicability. Finally, advantages and limitations are discussed and a comparison with conventional synchronization methods is investigated.

Parameter identification technique for uncertain chaotic systems using state feedback and steady-state analysis

A technique is introduced for identifying uncertain and/or unknown parameters of chaotic dynamical systems via using simple state feedback. The proposed technique is based on bringing the system into a stable steady state and then solving for the unknown parameters using a simple algebraic method that requires access to the complete or partial states of the system depending on the dynamical model of the chaotic system. The choice of the state feedback is optimized in terms of practicality and causality via employing a single feedback signal and tuning the feedback gain to ensure both stability and identifiability. The case when only a single scalar time series of one of the states is available is also considered and it is demonstrated that a synchronization-based state observer can be augmented to the state feedback to address this problem. A detailed case study using the Lorenz system is used to exemplify the suggested technique. In addition, both the Rössler and Chua systems are examined as possible candidates for utilizing the proposed methodology when partial identification of the unknown parameters is considered. Finally, the dependence of the proposed technique on the structure of the chaotic dynamical model and the operating conditions is discussed and its advantages and limitations are highlighted via comparing it with other methods reported in the literature.