Chaotic systems based on higher-order oscillatory equations

Abstract

This paper discusses the design process toward new lumped chaotic systems that originates in higherorder ordinary differential equations commonly used as description of ideal oscillators. In investigated thirdorder case, two chaotic oscillators were constructed. These systems are dual in the sense of vector field geometry local to fixed points. The existence of robust chaos was proved by both standard routines of numerical analysis and practical measurement. For the fourthorder oscillatory equation, the concept based on interaction between superinductor and supercapacitor was examined in detail. Since both “superelements” are active, the nonlinearity essential to the evolution of chaos is fully passive. It is demonstrated that complex motion is robust and does not represent long transient behavior or numerical artefact. The existence of chaos was verified using standard quantifiers of the flow, such as the largest Lyapunov exponents, recurrence plots, approximate entropy and sensitivity calculation. A good final agreement between theoretical assumptions and practical results will be concluded, on a visual comparison basis.This paper discusses the design process toward new lumped chaotic systems that originates in higherorder ordinary differential equations commonly used as description of ideal oscillators. In investigated thirdorder case, two chaotic oscillators were constructed. These systems are dual in the sense of vector field geometry local to fixed points. The existence of robust chaos was proved by both standard routines of numerical analysis and practical measurement. For the fourthorder oscillatory equation, the concept based on interaction between superinductor and supercapacitor was examined in detail. Since both “superelements” are active, the nonlinearity essential to the evolution of chaos is fully passive. It is demonstrated that complex motion is robust and does not represent long transient behavior or numerical artefact. The existence of chaos was verified using standard quantifiers of the flow, such as the largest Lyapunov exponents, recurrence plots, approximate entropy and sensitivity calculation. A good final agreement between theoretical assumptions and practical results will be concluded, on a visual comparison basis.