Geometry of non-standard Hamiltonian structures of Liénard equations and contact structure

We start with a self-contained brief review of the construction of non-standard Lagrangian and Hamiltonian structures for the Liénard equations satisfying Chiellini condition, we apply it to a special polynomial class of Liénard equation and explore the integrability structure. Then after a brief exposition of the contact geometry and its connection with the non-standard Hamiltonian structures, we describe the time evolution of the contact Hamiltonian and Liénard equation. We present the formulation of the Liénard equation in terms of General Equation for the Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) method and also study the gradient-type flow. Finally, we describe the generalized Liénard equation by using the conformal Hamiltonian mechanics and illustrate our construction using spatio-temporal and autocatalysis systems.