Melnikov theory for nonlinear implicit ODEs

Abstract We apply dynamical system methods and Melnikov theory to study small amplitude perturbation of some implicit differential equations. In particular we show persistence of such orbits connecting singularities in finite time provided a Melnikov like condition holds

Nonnegative solutions of nonlinear integral equations

summary:Existence results of nonnegative solutions of asymptotically linear, nonlinear integral equations are studied

Melnikov theory for weakly coupled nonlinear RLC circuits

We apply dynamical system methods and Melnikov theory to study small amplitude perturbation of coupled nonlinear RLC systems. In particular we show persistence of such orbits connecting singularities in finite time provided a Melnikov like condition holds

Periodic and bounded solutions of functional differential equations with small delays

We study existence and local uniqueness of periodic solutions of nonlinear functional differential equations of first order with small delays. Bifurcations of periodic and bounded solutions of particular periodically forced second-order equations with small delays are investigated as well

Periodic solutions for a class of differential equation with delays depending on state

In this paper, we use Schauder and Banach fixed point theorems to study the existence, uniqueness and stability of periodic solutions of a class of iterative differential equation $$x\u27(t)=sum_{m=1}^ksum_{l=1}^infty C_{l, m}(t)(x^{[m]}(t))^l+G(t),$$ where $x^{[m]}(t)$ denotes $m$th iterate of $x(t)$ for $m=1,2, ldots, k.$

Transversal homoclinics in nonlinear systems of ordinary differential equations

Bifurcation of transversal homoclinics is studied for a pair of ordinary differential equations with periodic perturbations when the first unperturbed equation has a manifold of homoclinic solutions and the second unperturbed equation is vanishing. Such ordinary differential equations often arise in perturbed autonomous Hamiltonian systems

Periodic solutions for a class of differential equation with delays depending on state

In this paper, we use Schauder and Banach fixed point theorems to study the existence, uniqueness and stability of periodic solutions of a class of iterative differential equation $$x\u27(t)=sum_{m=1}^ksum_{l=1}^infty C_{l, m}(t)(x^{[m]}(t))^l+G(t),$$ where $x^{[m]}(t)$ denotes $m$th iterate of $x(t)$ for $m=1,2, ldots, k.$

Explicit solution and dynamical properties of atmospheric Ekman flows with boundary conditions

In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with the constant eddy viscosity. Different from the previous work, we modify the boundary conditions and derive the explicit solution by using the notation of matrix cosine and matrix sine. For the arbitrary height-dependent eddy viscosity, we get the solution of the classical problem with zero velocity and acceleration at the bottom of the layer. In addition, uniqueness is shown and dynamical properties of solution are characterized