A new complex Duffing oscillator used in complex signal detection

Abstract

In view of the fact that complex signals are often used in the digital processing of certain systems such as digital communication and radar systems, a new complex Duffing equation is proposed. In addition, the dynamical behaviors are analyzed. By calculating the maximal Lyapunov exponent and power spectrum, we prove that the proposed complex differential equation has a chaotic solution or a large-scale periodic one depending on different parameters. Based on the proposed equation, we present a complex chaotic oscillator detection system of the Duffing type. Such a dynamic system is sensitive to the initial conditions and highly immune to complex white Gaussian noise, so it can be used to detect a weak complex signal against a background of strong noise. Results of the Monte-Carlo simulation show that the proposed detection system can effectively detect complex single frequency signals and linear frequency modulation signals with a guaranteed low false alarm rate. Chaos theory is one of the most significant achievements of nonlinear science. Many scholars have researched the basic theoretical issues [1][2][3][4][5][6][7] and practical applications of chaos. Hitherto, chaos theory has been widely applied in the control [8], synchronization [9], prediction [10], communication [11,12], and detection fields, among others. In signal detection, detection of weak signals using a chaotic oscillator in the real domain [13][14][15][16][17][18][19] has been studied by some researchers, with most researchers using a Duffing oscillator or modified version thereof. The basic idea is that a chaotic system is sensitive to the initial conditions and less influenced by noise. Nowadays complex signal processing is being used in many fields of science and engineering including digital communication systems, radar systems, antenna beamforming applications, coherent pulse measurement systems, and so on. As such, it is very meaningful to study how to detect a complex signal against a noisy background.Most of the recent works on complex chaos have focused on solving complex nonlinear differential equations, complex chaos control and synchronization, and so on. For example, Cveticanin [20] developed an approximate analytic approach for solving strong nonlinear differential equations of the Duffing type with a complex-valued function. Furthermore, excellent agreement is found between the analytic results and numerical ones. Li et al.[21] developed a stochastic averaging method for a quasi-Hamilton system to study the stationary solution in a nonlinear stochastically complex dynamical system. Rauh et al. [22] proved by means of a quadratic Lyapunov function that the complex Lorenz model is globally stable and presented an analytic expression for the upper bound on the magnitude of the time-dependent electric field. Mahmoud and Aly [23] illustrated the existence of periodic attractors of complex damped nonlinear systems by constructing Poincaré plots, and investigated the stability properties of the solutions of a complex nonlinear equati...

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