Energy function of 2D and 3D dynamical systems

It is far well-known that energy function of a two-dimensional autonomous dynamical system can be simply obtained by multiplying its corresponding second-order ordinary differential equation, i.e., its equation of motion by the first time derivative of its state variable. In the nineties, one of us (J.C.S.) stated that a three-dimensional autonomous dynamical system can be also transformed into a third-order ordinary differential equation of motion todays known as jerk equation. Although a method has been developed during these last decades to provide the energy function of such three-dimensional autonomous dynamical systems, the question arose to determine by which type of term, i.e., by the first or second time derivative of their state variable, the corresponding jerk equation of these systems should be multiplied to deduce their energy function. We prove in this work that the jerk equation of such systems must be multiplied by the second time derivative of the state variable and not by the first like in dimension two. We then provide an interpretation of the new term appearing in the energy function and called jerk energy. We thus established that it is possible to obtain the energy function of a three-dimensional dynamical system directly from its corresponding jerk equation. Two and three-dimensional Van der Pol models are then used to exemplify these main results. Applications to Lorenz and Chua's models confirms their validity.