The transversal homoclinic solutions and chaos for stochastic ordinary differential equations

We consider the persistence of a transversal homoclinic solution and chaotic motion for ordinary differential equations with a homoclinic solution to a hyperbolic equilibrium under an unbounded random forcing driven by a Brownian force. By Lyapunov–Schmidt reduction, the persistence of transversal homoclinic solution is reduced to find the zeros of some bifurcation functions defined between two finite spaces. It is shown that, for almost all sample paths of the Brownian motion, the perturbed system exhibits chaos.     

Remarks on homoclinic solutions for semilinear fourth-order ordinary differential equations without periodicity

This paper mainly discusses the existence of nontrivial homoclinic solutions for nonperiodic semilinear fourth-order ordinary differential equation
u^（4）＋pu″＋a（x）u-b（x）u^2=c（x）u^3=3
arising in the study of pattern formation by means of Mountain Pass Lemma.     

Multiple homoclinic solutions for singular differential equations

The homoclinic bifurcations of ordinary differential equation under singular perturbations are considered. We use exponential dichotomy, Fredholm alternative and scales of Banach spaces to obtain various bifurcation manifolds with finite codimension in an appropriate infinite-dimensional space. When the perturbative term is taken from these bifurcation manifolds, the perturbed system has various coexistence of homoclinic solutions which are linearly independent.     

Existence of positive homoclinic solutions for damped differential equations

This paper is concerned with the existence of positive homoclinic solutions for the second-order differential equation

$$\begin{aligned} u^{\prime \prime }+cu^{\prime }-a(t)u+f(t,u)=0, \end{aligned}$$

where \(c\ge 0\) is a constant and the functions a and f are continuous and not necessarily periodic in t. Under other suitable assumptions on a and f, we obtain the existence of positive homoclinic solutions in both cases sub-quadratic and super-quadratic by using critical point theorems.

     

A method for solution of nonlinear ordinary differential equations

We present a method for the solution of nonlinear second-order differential equations by using a system of Fredholm equations of the second kind.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1254–1260, September, 1995.     

Decay of solutions of ordinary differential equations

We investigate the rate of decay of eigenfunctions of Schrödinger equations using a perturbation method which consists of making a perturbation B of the operator L of the form B[y]=L[y]−(g⁻¹Lg)[y], where g is an appropriately chosen function. In our theory we allow B to be either relatively compact or satisfy a certain boundedness condition. We give some examples which apply the results of our main theorems coupled with recent work on the relative boundedness and compactness of differential operators.     

Periodic solutions of nonautonomous ordinary differential equations

For higher order ordinary differential equations, new sufficient conditions on the existence and uniqueness of periodic solutions are established. Results obtained cover the case when the right-hand side of the equation is not of a constant sign with respect to an independent variable.     

Periodic solutions of nonautonomous ordinary differential equations

For higher order ordinary differential equations, new sufficient conditions on the existence and uniqueness of periodic solutions are established. Results obtained cover the case when the right-hand side of the equation is not of a constant sign with respect to an independent variable.     

Error bounds for numerical solutions of ordinary differential equations

Summary An a posteriori error bound, for an approximate solution of a system of ordinary differential equations, is derived as the solution of a Riccati equation. The coefficients of the Riccati equation depend on an eigenvalue of a matrix related to a Jacobian matrix, on a Lipschitz constant for the Jacobian matrix, and on the approximation defect. An upper bound is computable as the formal solution of a sequence of Riccati equations with constant coefficients. This upper bound may sometimes be used to control step length in a numerical method.     

A preprocessing method for parameter estimation in ordinary differential equations

Parameter estimation for nonlinear differential equations is notoriously difficult because of poor or even no convergence of the nonlinear fit algorithm due to the lack of appropriate initial parameter values. This paper presents a method to gather such initial values by a simple estimation procedure. The method first determines the tangent slope and coordinates for a given solution of the ordinary differential equation (ODE) at randomly selected points in time. With these values the ODE is transformed into a system of equations, which is linear for linear appearance of the parameters in the ODE. For numerically generated data of the Lorenz attractor good estimates are obtained even at large noise levels. The method can be generalized to nonlinear parameter dependency. This case is illustrated using numerical data for a biological example. The typical problems of the method as well as their possible mitigation are discussed. Since a rigorous failure criterion of the method is missing, its results must be checked with a nonlinear fit algorithm. Therefore the method may serve as a preprocessing algorithm for nonlinear parameter fit algorithms. It can improve the convergence of the fit by providing initial parameter estimates close to optimal ones.     

A criterion for the oscillation of solutions of second-order ordinary differential equations

A generalization is obtained of A. Wintner's oscillation theorem for second-order linear equations.Translated from Matematicheskie Zametki, Vol. 8, No. 6, pp. 773–776, December, 1970.