On Massera's theorem concerning the uniqueness of a periodic solution for the Liénard equation. When does such a periodic solution actually exist?

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On the Qualitative Behavior of a Class of Generalized Li\ue9nard Planar Systems

We study the problem of existence/nonexistence of limit cycles for a class of Lienard generalized differential systems in which, differently from the most investigated case, the function F depends not only on x but also on the y-variable. In this framework, some new results are presented, starting from a case study which, actually, already exhibits the most significant properties. In particular, the so-called "superlinear case" presents some new phenomena of escaping orbits which will be discussed in detail

Oscillation of solutions of second-order nonlinear differential equations of Euler type

AbstractWe consider the nonlinear Euler differential equation t2x″+g(x)=0. Here g(x) satisfies xg(x)>0 for x≠0, but is not assumed to be sublinear or superlinear. We present implicit necessary and sufficient condition for all nontrivial solutions of this system to be oscillatory or nonoscillatory. Also we prove that solutions of this system are all oscillatory or all nonoscillatory and cannot be both. We derive explicit conditions and improve the results presented in the previous literature. We extend our results to the extended equation t2x″+a(t)g(x)=0

On the uniqueness of limit cycle for certain Liénard systems without symmetry

The problem of the uniqueness of limit cycles for Liénard systems is investigated in connection with the properties of the function F(x). When α and β (α < 0 < β) are the unique nontrivial solutions of the equation F(x) = 0, necessary and sufficient conditions in order that all the possible limit cycles of the system intersect the lines x = α and x = β are given. Therefore, in view of classical results, the limit cycle is unique. Some examples are presented to show the applicability of our results in situations with lack of symmetry

On the uniqueness of limit cycle for certain Liénard systems without symmetry

The problem of the uniqueness of limit cycles for Liénard systems is investigated in connection with the properties of the function $F(x)$. When $\alpha$ and $\beta$~$(\alpha<0<\beta)$ are the unique nontrivial solutions of the equation $F(x)=0$, necessary and sufficient conditions in order that all the possible limit cycles of the system intersect the lines $x=\alpha$ and $x=\beta$ are given. Therefore, in view of classical results, the limit cycle is unique. Some examples are presented to show the applicability of our results in situations with lack of symmetry

On the uniqueness of limit cycle for certain Liénard systems without symmetry

The problem of the uniqueness of limit cycles for Liénard systems is investigated in connection with the properties of the function F(x). When α and β (α < 0 < β) are the unique nontrivial solutions of the equation F(x) = 0, necessary and sufficient conditions in order that all the possible limit cycles of the system intersect the lines x = α and x = β are given. Therefore, in view of classical results, the limit cycle is unique. Some examples are presented to show the applicability of our results in situations with lack of symmetry

On Certain Relaxation Oscillations: Confining Regions

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryContract DA-28-043-AMC-00073(E