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.
We do not know Van der Pol-type equations with nonlinear restoring force having explicitly an exact periodic solution. We present, for the first time, nonpolynomial Van der Pol oscillator equations that do not satisfy the classical existence theorems. We exhibit their exact harmonic and isochronous solutions and prove the existence of limit cycles by using averaging theory. We also present first integrals and exact solutions of polynomial Van der Pol-Duffing equations to show that they do not have any limit cycle. Additionally, we prove that the damped Duffing-type equations are equivalent to the conservative Duffing equations exhibiting nonoscillatory solutions.
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