The article concerns the symmetries of differential equations with short digressions to the underdetermined case and the relevant differential equations with delay. It may be regarded as an introduction into the method of moving frames relieved of the geometrical aspects: the stress is made on the technique of calculations employing only the most fundamental properties of differential forms. The present Part I is devoted to a single ordinary differential equation subjected to the change of the independent...

Continuing the idea of Part I, we deal with more involved pseudogroup of transformations $\overline{x}=\varphi \left(x\right)$, $\overline{y}=L\left(x\right)y$, $\overline{z}=M\left(x\right)z,...$ applied to the first order differential equations including the underdetermined case (i.e. the Monge equation $y^{\text{'}}=f(x,y,z,z^{\text{'}})$) and certain differential equations with deviation (if $z=y\left(\xi \right(x\left)\right)$ is substituted). Our aim is to determine complete families of invariants resolving the equivalence problem and to clarify the largest possible symmetries. Together with Part I, this article may be regarded as an introduction into the...

We describe the nonlinear limit-point/limit-circle problem for the $n$-th order differential equation $$y^{\left(n\right)}+r\left(t\right)f(y,y^{\text{'}},\cdots ,y^{(n-1)})=0.$$ The results are then applied to higher order linear and nonlinear equations. A discussion of fourth order equations is included, and some directions for further research are indicated.

We study the large-time behaviour of the nonlinear oscillator$$m{x}^{\text{'}\text{'}}+f\left(x^{\text{'}}\right)+kx=0,$$where $m,k\>0$ and $f$ is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case $f\left(x^{\text{'}}\right)=A{\left|x^{\text{'}}\right|}^{\alpha -1}{x}^{\text{'}}$ with $\alpha$ real, $A\>0$. We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.

We study the large-time behaviour of the nonlinear oscillator $$m{x}^{\text{'}\text{'}}+f\left(x^{\text{'}}\right)+kx=0,$$ where m, k>0 and f is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case  $f\left(x^{\text{'}}\right)=A{\left|x^{\text{'}}\right|}^{\alpha -1}{x}^{\text{'}}$  with α real, A>0. We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.

We consider numerical approximation to the solution of non-autonomous evolution equations. The order of convergence of the simplest possible Magnus method is investigated.

Let (P,Q) be a C1 vector field defined in a open subset U ⊂ R2. We call a null divergence factor a C1 solution V (x, y) of the equation P ∂V/∂x + Q ∂V/ ∂y = ( ∂P/∂x + ∂Q/∂y ) V. In previous works it has been shown that this function plays a fundamental role in the problem of the center and in the determination of the limit cycles. In this paper we show how to construct systems with a given null divergence factor. The method presented in this paper is a generalization of the classical Darboux method...

We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift $c$. At the point $xgt;0$, we add an absorbing barrier, i.e. individuals touching the barrier are instantly killed without producing offspring. It is known that there is a critical drift $c_0$, such that this process becomes extinct almost surely if and only if $c\ge c_0$. In this case, if $Z_x$ denotes the number of individuals absorbed at the barrier, we give an asymptotic for $P(Z_x=n)$ as $n$ goes to infinity. If $c=c_0$...

A numerical method for the solution of a second order boundary value problem for differential equation with state dependent deviating argument is studied. Second-order convergence is established and a theorem about the asymptotic expansion of global discretization error is given. This theorem makes it possible to improve the accuracy of the numerical solution by using Richardson extrapolation which results in a convergent method of order three. This is in contrast to boundary value problems for...

This paper considers modified second derivative BDF (MSD-BDF) for the numerical solution of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The methods are A$\left(\alpha \right)$-stable for step length $k\le 7$.