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.
Highlights
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
Detection of weak signals using a chaotic oscillator in the real domain [13–19] has been studied by some researchers, with most researchers using a Duffing oscillator or modified version thereof
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
Summary
The most common form of the real Duffing equation [14] is:. Eq (3) represents a system of two coupled nonlinear second-order differential equations (for the sake of readability, t denotes ): x k. We study the nonlinear dynamical behavior of the new complex Duffing equation. For the case of 0 , the equations in eq (5) are completely symmetrical; that is, if one starts from initial conditions (x0 , x 0 ) = ( y0 , y 0 ) , the x and y variables behave identically. We plot only the phase plane trajectory of (x, x ) for the different initial conditions since the equations in eq (5) are symmetrical, and the same trajectory for ( y, y ) can be obtained. From the zoom-in plot within = [0.7, 0.8], we can obtain the critical value c (0.72, 0.73) at which the system is in a critical state (chaotic, but on the verge of changing to a large-scale periodic state)
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