Theoretical Design and Circuit Implementation of Multidirectional Multi-Torus Chaotic Attractors

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper introduces a novel fourth-order double-torus chaotic circuit. Based on this basic circuit, a systematic theoretical design approach is proposed for generating 1-D <formula formulatype="inline"><tex>$n$</tex></formula> torus, 2-D <formula formulatype="inline"><tex>$n\times m$</tex></formula>–torus, 3-D <formula formulatype="inline"><tex>$n\times m\times l$</tex></formula> torus, and 4-D <formula formulatype="inline"><tex>$n \times m \times l \times p$</tex> </formula> torus chaotic attractors. This is the first autonomous circuit reported in the literature for generating multidirectional multi-torus (MDMT) chaotic attractors. The dynamical behaviors of these MDMT chaotic systems are further investigated, including equilibrium points, bifurcations, Lyapunov exponents, and PoincarÉ maps. Theoretical analysis shows that the MDMT chaotic attractors can be generated by switching and displacing a basic linear circuit. Finally, a block circuit diagram is designed for hardware implementation of the MDMT chaotic attractors. This is also the first time in the literature to experimentally verify a maximal 1-D 20–torus, a maximal 2-D 5<formula formulatype="inline"><tex>$\,\times \,$</tex></formula>5 torus, and a maximal 4-D 5<formula formulatype="inline"><tex>$\,\times \,$</tex></formula>5<formula formulatype="inline"><tex>$\,\times \,$</tex></formula>3<formula formulatype="inline"> <tex>$\,\times \,$</tex></formula>3 torus chaotic attractors. </para>