Stability analysis for the Lienard equation with discontinuous coefficients

A nonlinear mechanical system, whose dynamics is described by a vector ordinary differential equation of the Lienard type, is considered. It is assumed that the coefficients of the equation can switch from one set of constant values to another, and the total number of these sets is, in general, infinite. Thus, piecewise constant functions with infinite number of break points on the entire time axis, are used to set the coefficients of the equation. A method for constructing a discontinuous Lyapunov function is proposed, which is applied to obtain sufficient conditions of the asymptotic stability of the zero equilibrium position of the equation studied. The results found are generalized to the case of a nonstationary Lienard equation with discontinuous coefficients of a more general form. As an auxiliary result of the work, some methods for analyzing the question of sign-definiteness and approaches to obtaining estimates for algebraic expressions, that represent the sum of power-type terms with non-stationary coefficients, are developed. The key feature of the study is the absence of assumptions about the boundedness of these non-stationary coefficients or their separateness from zero. Some examples are given to illustrate the established results.

Monotonicity of a Quadratic Lienard Equation

We study the period function of the quadratic Lienard equation of a certain type in order to give necessary and sufficient conditions for monotonicity and isochronicty of the period function. We apply this result to identify the region of monotonicity of the period function of particular cases.

Bifurcations of Critical Periods for a Class of Quintic Liénard Equation

In this paper, we investigate a quintic Liénard equation which has a center at the origin. We give the conditions for the parameters for the isochronous centers and weak centers of exact order. Then, we present the global phase portraits for the system having isochronous centers. Moreover, we prove that at most four critical periods can bifurcate and show with appropriate perturbations that local bifurcation of critical periods occur from the centers.

Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive type

In this paper, the existence of positive periodic solutions is studied for super-linear neutral Liénard equation with a singularity of attractive type $$ \bigl(x(t)-cx(t-\sigma)\bigr)''+f\bigl(x(t) \bigr)x'(t)-\varphi(t)x^{\mu}(t)+ \frac{\alpha(t)}{x^{\gamma}(t)}=e(t), $$ ( x ( t ) − c x ( t − σ ) ) ″ + f ( x ( t ) ) x ′ ( t ) − φ ( t ) x μ ( t ) + α ( t ) x γ ( t ) = e ( t ) , where $f:(0,+\infty)\rightarrow R$ f : ( 0 , + ∞ ) → R , $\varphi(t)>0$ φ ( t ) > 0 and $\alpha(t)>0$ α ( t ) > 0 are continuous functions with T-periodicity in the t variable, c, γ are constants with $|c|<1$ | c | < 1 , $\gamma\geq1$ γ ≥ 1 . Many authors obtained the existence of periodic solutions under the condition $0<\mu\leq1$ 0 < μ ≤ 1 , and we extend the result to $\mu>1$ μ > 1 by using Mawhin’s continuation theorem as well as the techniques of a priori estimates. At last, an example is given to show applications of the theorem.