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 $$y^{\left(n\right)}={\left|y\right|}^k\mathrm{sgn}y,k>1,n=12,13,14.$$

The lower bounds of the spacings b-a or a’-a of two consecutive zeros or three consecutive zeros of solutions of third order differential equations of the form y”’ + q(t)y’ + p(t)y = 0 (*) are derived under very general assumptions on p and q. These results are then used to show that $t_{n+1}-tₙ\to \infty$ or $t_{n+2}-tₙ\to \infty$ as n → ∞ under suitable assumptions on p and q, where ⟨tₙ⟩ is a sequence of zeros of an oscillatory solution of (*). The Opial-type inequalities are used to derive lower bounds of the spacings d-a or b-d for...

We discuss Lyapunov stability/instability of both lower and upper equilibria of free damped pendulum with periodically oscillating suspension point. We recall the results of Bogolyubov and Kapitza, provide new effective criteria of stability/instability of the equilibria of pendulum equation, and give the exact and complete proofs. The criteria obtained are formulated in terms of positivity/negativity of Green's functions of the periodic boundary value problems for linearized equations. Furthermore,...

In this paper, necessary and sufficient conditions are obtained for every bounded solution of $${[y\left(t\right)-p\left(t\right)y(t-\tau )]}^{\left(n\right)}+Q\left(t\right)G\left(y(t-\sigma )\right)=f\left(t\right),t\ge 0,(*)$$ to oscillate or tend to zero as $t\to \infty$ for different ranges of $p\left(t\right)$. It is shown, under some stronger conditions, that every solution of $(*)$ oscillates or tends to zero as $t\to \infty$. Our results hold for linear, a class of superlinear and other nonlinear equations and answer a conjecture by Ladas and Sficas, Austral. Math. Soc. Ser. B 27 (1986), 502–511, and generalize some known results.

Sufficient conditions are formulated for existence of non-oscillatory solutions to the equation $$y^{\left(n\right)}+\sum _{j=0}^{n-1}{a}_j\left(x\right)y^{\left(j\right)}+p\left(x\right){\left|y\right|}^k\mathrm{sgn}y=0$$ with $n\ge 1$, real (not necessarily natural) $k>1$, and continuous functions $p\left(x\right)$ and $a_j\left(x\right)$ defined in a neighborhood of $+\infty$. For this equation with positive potential $p\left(x\right)$ a criterion is formulated for existence of non-oscillatory solutions with non-zero limit at infinity. In the case of even order, a criterion is obtained for all solutions of this equation at infinity to be oscillatory. Sufficient conditions are obtained...

Mostramos la existencia de dos curvas de datos iniciales (x0, v0) para las cuales las soluciones x(t) correspondientes del problema de Cauchy asociado a la ecuación xtt + |xt|α-1 xt + x = 0, supuesto α ∈ (0,1), se anulan idénticamente después de un tiempo finito. Mediante métodos asintóticos y argumentos de comparación mostramos que para muchos otros datos iniciales las soluciones decaen a 0, en un tiempo infinito, como t-α / (1-α).