Limit cycles of a class of Liénard systems derived from state-dependent impulses

Discontinuous phenomena, in which objects may behave continuously and sometimes discretely are not only found in nature and under laboratory conditions but also in simple, familiar contexts. For example, this phenomenon is skillfully incorporated into the internal structure of mechanical wristwatches. Unless an extremely small amount of state-dependent impulse is applied intermittently, the reciprocating rotational movement of the balance and hairspring, which is the heart of the mechanical wristwatch, cannot be maintained. The small amount of state-dependent impulse, which is often overlooked, can make a significant difference; however, very few studies have examined this subject. This study assumes the underlying cause of discontinuous behaviors as impulses generated when an object reaches a particular state, assuming that the continuous behavior follows the Liénard system, which is widely studied in the field of electrical circuits. The main theorem provides the conditions under which the effect of the impulses causes a stable limit cycle in the Liénard system, even if no limit cycle exists when there are no impulses. The Poincaré–Bendixson theorem for discontinuous dynamical systems and phase plane analysis are used to prove the main theorem. Several examples and their simulations are provided to illustrate the main theorem.     

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 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.     

The applied geometry of a general Liénard polynomial system

In this work, applying a canonical system with field rotation parameters and using geometric properties of the spirals filling the interior and exterior domains of limit cycles, we solve the limit cycle problem for a general Liénard polynomial system with an arbitrary (but finite) number of singular points.     

Periodic solutions of Liénard equation with a singularity and a deviating argument

In this paper, we study the existence of periodic solutions of the Liénard equation with a singularity and a deviating argument $x^{″}+f\left(x\right)x^{\prime }+g(t,x(t-\sigma ))=0\text{.}$ When $g$ has a strong singularity at $x=0$ and satisfies a new small force condition at $x=\infty$, we prove that the given equation has at least one positive $T$-periodic solution.     

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.

     

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.     

Integrability conditions of a resonant saddle in Liénard-like complex systems

We consider a complex differential system with a resonant saddle that remind the classical Liénard systems in the real plane. For such systems we determine the conditions of analytic integrability of the resonant saddle.     

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

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Generalized theorems of Liénard and Shepherd

The paper considers a real polynomial p(x)=₀+₁x+...+_nxⁿ(₀ > 0) for which there hold inequalities ₁>0, ₃>0, ... or ₂>0, ₄>0, ..., where ₁, ₂, ..., _jn are the Hurwitz determinants for polynomial p(x). It is proven that polynomial p(x) can have, in the right half-plane, only real roots, where the quantity of positive roots of polynomial p(x) equals the quantity of changes of sign in the system of coefficients a₀, a₂, ..., a_n–2, a_n, when n is even, and ₀, a₂,..., a_n–1, a_n, when n is odd. From the proven theorem, in particular, there follows the Liénard and Shepherd criterion of stability.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 13–21, July, 1977.