Qualitative behavior of solutions of Liénard-type systems with state-dependent impulses

State-dependent impulsive differential equations can be used to describe phenomena in which the velocity of an object suddenly changes when the object enters a predetermined state. This study examines the effect of state-dependent impulses on the rotation of all orbits of the Liénard system, which plays an important role in many research areas, including electrical circuits, diverse engineering fields, economics, ecology, and physiology. Determining whether all orbits of the Liénard system other than the origin, which is the only fixed point, rotate around the origin, is the basis of other properties of the orbit and has become a significant research subject; however, very few studies have examined the effects of state-dependent impulses on this behavior, and the main theorem presented in this paper addresses this. This rotation problem is reduced to establishing whether all orbits intersect the vertical isocline, which is discussed in detail. To facilitate the understanding of the proof of the main theorem, an overview is presented before providing the actual proof. The main theorem and some lemmas are proved using phase plane analysis. The application of the main theorem to Euler’s equations is also described.     

Small-amplitude limit cycles of polynomial Liénard systems

In this paper,we study the number of limit cycles appeared in Hopf bifurcations of a Linard system with multiple parameters.As an application to some polynomial Li’enard systems of the form x=y,y=gm(x)-fn(x)y,we obtain a new lower bound of maximal number of limit cycles which appear in Hopf bifurcation for arbitrary degrees m and n.     

On the number of limit cycles of polynomial Liénard systems

Liénard systems are very important mathematical models describing oscillatory processes arising in applied sciences. In this paper, we study polynomial Liénard systems of arbitrary degree on the plane, and develop a new method to obtain a lower bound of the maximal number of limit cycles. Using the method and basing on some known results for lower degree we obtain new estimations of the number of limit cycles in the systems which greatly improve existing results.     

Limit cycle bifurcations of some Liénard systems with a cuspidal loop and a homoclinic loop

In this paper, we study the number of limit cycles of some polynomial Liénard systems with a cuspidal loop and a homoclinic loop, and obtain some new results on the lower bound of the maximal number of limit cycles for these systems.     

Small-amplitude limit cycles of some Liénard-type systems

As we know, the Liénard system and its generalized forms are classical and important models of nonlinear oscillators, and have been widely studied by mathematicians and scientists. The main problem considered by most people is the number of limit cycles. In this paper, we investigate two kinds of Liénard systems and obtain the maximal number (i.e. the least upper bound) of limit cycles appearing in Hopf bifurcations by applying some known bifurcation theorems with technical analysis.     

Integrability and solvability of polynomial Liénard differential systems

We provide the necessary and sufficient conditions of Liouvillian integrability for Liénard differential systems describing nonlinear oscillators with a polynomial damping and a polynomial restoring force. We prove that Liénard differential systems are not Darboux integrable excluding subfamilies with certain restrictions on the degrees of the polynomials arising in the systems. We demonstrate that if the degree of a polynomial responsible for the restoring force is greater than the degree of a polynomial producing the damping, then a generic Liénard differential system is not Liouvillian integrable with the exception of linear Liénard systems. However, for any fixed degrees of the polynomials describing the damping and the restoring force we present subfamilies possessing Liouvillian first integrals. As a by-product of our results, we find a number of novel Liouvillian integrable subfamilies. In addition, we study the existence of nonautonomous Darboux first integrals and nonautonomous Jacobi last multipliers with a time-dependent exponential factor.     

RESEARCH ANNOUNCEMENTS On the Uniqueness of Limit Cycle for a Generalized Liénard System

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Existence of limit cycles for Liénard-type systems with p-Laplacian

The purpose of this paper is to give sufficient conditions under which an equivalent system to the equation has at least one stable limit cycle, where is the one-dimensional p-Laplacian. The main results are proved by means of phase plane analysis with the Poincaré-Bendixson theorem. Sufficient conditions are also given for the origin to be unstable and for all solutions to be bounded in the future. Jitsuro Sugie: Supported in part by Grant-in-Aid for Scientific Research 16540152     

Stability of Liénard Equation

We consider a balance stability problem for the second order nonlinear differential equations of the Liénard type. Investigations are carried out by means of constant-sign Lyapunov functions for problems of stability, asymptotical stability (local and global), and instability. We implicitly formulate a method of construction of constant-sign functions suitable for solving problems of motion stability. Special attention is paid to a problem of non-asymptotical stability, where we demonstrate possibilities of new assertions that rely upon a usage of constant-sign Lyapunov functions.     

Liénard systems for quadratic systems with invariant algebraic curves

We study the character of the friction function f(x) and the restoring force g(x) in the Liénard system to which a quadratic system with an invariant second-order algebraic curve (an ellipse that is a limit cycle, a hyperbola defining two separatrix cycles, or a parabola) or fourth-order algebraic curve with an oval being a limit cycle can be reduced. Invariant curves are constructed for quadratic systems in a five-parameter canonical family, which can readily be reduced to Liénard systems.     

Two Limit Cycles in Liénard Piecewise Linear Differential Systems

Some techniques for studying the existence of limit cycles for smooth differential systems are extended to continuous piecewise linear differential systems. Rigorous new results are provided on the existence of two limit cycles surrounding the equilibrium point at the origin for systems with three zones separated by two parallel straight lines without symmetry. As a relevant application, it is shown the existence of bistable regimes in an asymmetric memristor-based electronic oscillator.