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Abstract
Nonlinear dynamical system often exhibits chaos, which is characterized by sensitive dependence on initial condition or more precisely by a positive Lyapunov exponent. The Lyapunov exponents quantify the exponential divergence or growth of initially nearby
trajectories.
chaotic behavior study is conducted three steps. The first step is detecting the chaotic behavior via largest Lyapunov exponent. The second step is study of the control and synchronization of chaotic system. The third is the study of the chaotic behavior in parametric space defined by the controlling parameter of the system.
Recognizing and quantifying chaotic behavior in nonlinear dynamical systems’represent an important step toward understanding the nature of chaotic behavior. The thesis will focus on the problem of quantifying the chaotic behavior in nonlinear dynamical using largest Lyapunov exponents. Positive values of the largest Lyapunov exponent indicate chaos.
There are many methods for computing the Lyapunov exponents. All of these methods need rescaling and reorthogonatization due to the lack of numerical stability and the round off errors of the numerical algorithms used in nonlinear system.
In this thesis, a new method for computing the largest Lyapunov exponent of n-dimensional dynamical system is presented. The method relies on the owing of normal forms. This method does not need rescaling or reorthogonatization in the numerical computation. Therefore, it does not suffer from the underflow, overflow or other numerical disabilities associated with
the standard techniques.