<abstract><p>In the attractive research field of nonlinear differential equations, there are a few studies devoted to finding exact and explicit harmonic and isochronous periodic solutions and limit cycles. In this contribution, we present some classes of polynomial mixed Lienard-type differential equations that can generate many equations with exact solutions. These classes of equations constitute counterexamples of the classical existence theorems.</p></abstract>

In this paper we study a Lienard equation without restoring force. Although this equation does not satisfy the classical existence theorems, we show, for the first time, that such an equation can exhibit harmonic periodic solutions. As such the usual existence theorems are not entirely adequate and satisfactory to predict the existence of periodic solutions.

In this paper, we present an exceptional Lienard equation consisting of a modified Van der Pol-Helmholtz oscillator equation. The equation, a frequency-dependent damping oscillator, does not satisfy the classical existence theorems but, nevertheless, has an isochronous centre at the origin. We exhibit the exact and explicit general harmonic and isochronous solutions by using the first integral approach. The numerical results match very well analytical solutions.Mathematics Subject Classi cation (2010). 34A05, 34A12, 34A34, 34C25, 34C60.

This paper is devoted to investigating the existence of exact harmonic solutions and limit cycles of certain modified Emden-type equations. The exact and general solutions obtained are in opposition to the predictions of classic existence theorems.

Although Jacobi elliptic functions have been known for almost two centuries, they are still the subject of intensive investigation. In this paper, contrary to the usual definition, we prove that the Jacobi elliptic functions can be defined by using nonconservative equations with limit cycles through existence theorems involving first integrals. This allows extending their validity domains, that is, their range of applications.

Real-world systems, such as physical and living systems, are generally subject to vibrations that can affect their long-term integrity and safety. Thus, the determination of the law that governs the evolution of the oscillatory quantity has become a major topic in modern engineering design. The process often leads to solving nonlinear differential equations. However, one can admit that the main objective of the theory of differential equations to obtain explicit solutions is far from being carried out. If we know how to solve linear systems, the case of systems of nonlinear differential equations is not in general solved. Isochronous nonlinear systems have therefore received particular attention. This chapter is devoted to presenting some recent developments and advances in the theory of isochronous oscillations of nonlinear systems. The harmonic oscillator as a prototype of isochronous systems is investigated to state some useful definitions (section 2), and the existence of second-order isochronous nonlinear systems having explicit elementary first integrals with an exact sinusoidal solution and higher-order autonomous nonlinear systems that reproduce the dynamics of the harmonic oscillator is proven (section 3). Finally, higher-order nonautonomous nonlinear systems that can exhibit isochronous oscillations are shown (section 4), and a conclusion for the chapter is presented.

In this paper we present a general class of differential equations of Ermakov-Pinney type which may serve as truly nonlinear oscillators. We show the existence of periodic solutions by exact integration after the phase plane analysis. The related quadratic Lienard type equations are examined to show for the first time that the Jacobi elliptic functions may be solution of second-order autonomous non-polynomial differential equations.

In this paper, we show the existence of damped Mathieu and periodic Lienard-type equations that can be solved in an explicit way or by quadrature. We prove for the first time an isochronous periodic solution for a Lienard-type equation with periodic coefficients, which does not exhibit parametric resonance.