The classical framework for studying the equations governing the motion of lumped parameter systems presumes one can provide expressions for the forces in terms of kinematical quantities for the individual constituents. This is not possible for a very large class of problems where one can only provide implicit relations between the forces and the kinematical quantities. In certain special cases, one can provide non-invertible expressions for a kinematical quantity in terms of the force, which then...

We are interested in conditions under which the two-dimensional autonomous system ẋ = y, ẏ = -g(x) - f(x)y, has a local center with monotonic period function. When f and g are (non-odd) analytic functions, Christopher and Devlin [C-D] gave a simple necessary and sufficient condition for the period to be constant. We propose a simple proof of their result. Moreover, in the case when f and g are of class C³, the Liénard systems can have a monotonic period function...

We obtain monotonicity results concerning the oscillatory solutions of the differential equation $\left(a\left(t\right)\right|y^{\text{'}}{{|}^{p-1}{y}^{\text{'}})}^{\text{'}}+f(t,y,y^{\text{'}})=0$. The obtained results generalize the results given by the first author in [1] (1976). We also give some results concerning a special case of the above differential equation.

In this paper we consider the equation $$y^{\text{'}\text{'}\text{'}}+q\left(t\right){y^{\text{'}}}^{\alpha }+p\left(t\right)h\left(y\right)=0,$$ where $p,q$ are real valued continuous functions on $[0,\infty )$ such that $q\left(t\right)\ge 0$, $p\left(t\right)\ge 0$ and $h\left(y\right)$ is continuous in $(-\infty ,\infty )$ such that $h\left(y\right)y>0$ for $y\ne 0$. We obtain sufficient conditions for solutions of the considered equation to be nonoscillatory. Furthermore, the asymptotic behaviour of these nonoscillatory solutions is studied.

We consider the problem of state and parameter estimation for a class of nonlinear oscillators defined as a system of coupled nonlinear ordinary differential equations. Observable variables are limited to a few components of state vector and an input signal. This class of systems describes a set of canonic models governing the dynamics of evoked potential in neural membranes, including Hodgkin-Huxley, Hindmarsh-Rose, FitzHugh-Nagumo, and Morris-Lecar...

In this paper we will study some asymptotic properties of a nonlinear third order differential equation viewed as a perturbation of a simpler nonlinear equation investigated recently by the authors in [4].

For the equation $$y^{\left(n\right)}+{\left|y\right|}^k\mathrm{sgn}y=0,k>1,n=3,4,$$ existence of oscillatory solutions $$y={(x^{*}-x)}^{-\alpha }h(log(x^{*}-x)),\alpha =\frac{n}{k-1},x<x^{*},$$ is proved, where $x^{*}$ is an arbitrary point and $h$ is a periodic non-constant function on $ℝ$. The result on existence of such solutions with a positive periodic non-constant function $h$ on $ℝ$ is formulated for the equation $$y^{\left(n\right)}={\left|y\right|}^k\mathrm{sgn}y,k>1,n=12,13,14.$$