Jerk forms dynamics of a Chua’s family and their new unified circuit implementation

Abstract

Abstract A scheme to implement the Jerk form of the Chua system family using a controllable canonical form applied in linear systems is proposed. The main thought is that the nonlinear function with a single independent variable input can be superposed by a multiple linear function, which is regarded as the input of the system, and its single variable is regarded as the output of the system. Its state space could be reconstructed into the controllable canonical form. The output after transformation is fed back to the nonlinear function as its input independent variable, thus the controllable canonical form is transformed into the Jerk form of the Chua’s chaotic system. All Jerk forms of three‐order Chua system, and part Jerk forms of four‐order Chua system are presented. Analysis of the Lyapunov exponent spectrum, eigenvalues of the original three‐order, four‐order Chua systems and their Jerk forms, and the same values demonstrate the equivalent of the systems for both structures. According to the complexity and National Institute of Standards and Technology (NIST) test results, the pseudo‐random performance of the chaotic sequence brought by the Jerk forms of the Chua system is better. The simplest unified circuit block diagram for the Jerk forms of the Chua’s family is also given. By changing their resistance parameters, the bifurcation diagrams show that Jerk forms’ systems are entering chaos, these circuits implement results of 2–6 scroll attractors. Experimental observations are provided for confirmation.

Keywords

• chaos
• nonlinear dynamical systems
• Chua's circuit
• bifurcation
• eigenvalues and eigenfunctions
• Lyapunov methods