A minimal three-term chaotic flow with coexisting routes to chaos, multiple solutions, and its analog circuit realization

Abstract

We introduce a unique chaotic flow with three terms expressed by $$ \dddot x + abx = d \sinh \left( {\dot{x} + c\ddot{x}} \right) $$. This minimal Jerk model is presented as a novel 3-D chaotic system consisting uniquely of two linear terms and single hyperbolic sine nonlinearity. The presence of coexisting routes to chaos in the numerical studies justifies the appearance of multiple solutions in some ranges of parameters. For instance, up to six different coexisting solutions (a pair of period-5 limit cycles and two pairs of chaotic attractors) have been tracked. Furthermore, the generalized form of the introduced Jerk system is synthesized with a parametric nonlinearity of the form $$ \phi_{k} \left( X \right) = 0.5\left( {exp\left( {k\left( {\dot{x} + c\ddot{x}} \right)} \right) - exp\left( { - \left( {\dot{x} + c\ddot{x}} \right)} \right)} \right) $$. Based on this generalized form, and judiciously adjusting the parameter $$ k $$, an in-depth description of the dynamics of the system at the symmetry boundaries is carried out. Hysteresis and parallel bifurcation behaviors reflect the presence of multiple asymmetric solutions (e.g. the coexistence of four asymmetric attractors) in the new model. It should be mention that the presence of six coexisting attractors reported in this work is rarely shown in third-order systems and therefore represents an enriching contribution to the study of dynamic systems in general. Finally, some PSpice simulations and laboratory tests of the proposed circuit are included.