Nonautonomous ODE Research Articles

Delayed Loss of Stability in Nonautonomous Differential Equations With Retarded Argument

Assume that zero is a stable equilibrium of an ODE $\dot x=f(x,\lambda)$ for parameter values $\lambda < \lambda_0$ and becomes unstable for $\lambda > \lambda_0$. If we suppose that $\lambda(t)$ varies slowly with t, then, under some conditions, the trajectories of the nonautonomous ODE $\dot x=f(x,\lambda(t))$ stay close to zero even long after $\lambda(t)$ has crossed the value $\lambda_0$. This phenomenon is called "delayed loss of stability" and is well known for ODEs. In this paper, we describe an analogous phenomenon for delay equations of the form $\dot{x}(t)=f(t,x(t-1))$. We study an example which requires combining linearization at zero with estimates on the nonlinear behavior away from zero, and where we obtain an explicit estimate on the time until the growth of |x(t)| becomes "visible."

Differential equations with bounded positive Green’s functions and generalized Aizerman’s hypothesis

We derive conditions for the positivity and boundedness of the Green functions of the higher order linear nonautonomous ODE. By virtue of these conditions, the existence of positive solutions for a class of nonlinear equations is proved. In addition, upper and lower estimates for the Green functions are established. Moreover, it is shown that nonlinear equations, having separated nonautonomous linear parts, satisfy the generalized Aizerman hypothesis on absolute stability, if they have the positive Green functions.

Isolating chains and chaotic dynamics in planar nonautonomous ODEs

We present a topological method for detecting complicated dynamics in nonautonomous periodic ODE's. The method is based on the notion of isolating chain. As an application we show the chaotic behaviour of processes generated by the equations in the complex plane z ̇ = z ̄ k+ e iφt z ̄ , where k>3 is an even number and φ is close to 0.

Isolating Segments, Fixed Point Index, and Symbolic Dynamics II. Homoclinic Solutions

We present a topological method for detecting homoclinic solutions in nonautonomous ODE's. As an application we show that there exist infinitely many (geometrically distinct) homoclinic solutions to the trivial 0 solution in the planar systemż=z(1+eiκt|z|2)for 0<κ⩽0.495.

On Existence of Infinitely Many Homoclinic Solutions

In this note we prove some sufficient condition for the existence of homoclinic solutions in nonautonomous ODE’s. As an application we show that there exist infinitely many (geometrically distinct) homoclinic solutions to the trivial 0 solution in the planar system $$$$

Identifying attractors in a ferroresonant circuit: a simulated perturbation approach

Abstract In a previous paper (S.K. Chakravarthy, Nonlinear oscillations due to spurious energisation of transformers, Proc IEE, EPA, Vol. 145, No. 6, Nov. 1998, 585–592) we have demonstrated the nature of oscillations that can occur in a spuriously energised transformer. The system, in that case, was a third order nonlinear (non-autonomous) ode from which bifurcation equations were determined by the method of multiple scales. The computational difficulty became awesome when analysing oscillations due to spurious energisation of power transformers. The order of the system increased from a third order to a fourth order system in this case. We further demonstrated that nonlinear interaction in power system involves dynamic phenomena of widely different speeds. Using the ‘brute force’ of simulation to search the phase space for the attractors of the system is not feasible. In this paper, we hence study higher order nonlinear systems by an approach similar to the method of perturbations. We propose to use numerical simulation as the basic tool for assessing the effect of perturbations and have coined the term ‘simulated perturbation’. In this approach, the starting point is the conservative autonomous system for which all possible oscillatory modes can be determined. We then introduce other variables in the equations one parameter at a time and then, study the changes in the nature of the oscillations. As we shall show, the method allows one to study a complex nonlinear ode with relative ease and yet obtain a feel for the factors behind the behaviour observed.

Dynamics and Bifurcations of a Planar Map Modeling Dispersion Managed Breathers

We study a nonautonomous ODE with piecewise-constant coefficients and its associated two-dimensional Poincare mapping. The ODE models variations in amplitude and phase of a pulse propagating in a lossless optical fiber with periodically varying dispersion. We derive semiexplicit exact solutions and use them to locate fixed points and to describe their bifurcations and stability types. We also discuss the global structure of the Poincare map and interpret our results for modulated pulse propagation.

Approximate analysis for the formation of adiabatic shear bands

Parametric solutions are given for the formation of adiabatic shear bands in the context of the onedimensional nonlinear theory where inertia and elasticity are ignored. When heat conduction is also ignored, the exact solution reduces completely to a sequence of quadratures. For a perfectly plastic material with heat conduction, an implicit parametric solution is also constructed. This is similar to the previous one in many ways, but now it involves two quadratures, a single nonautonomous first-order ODE. and two functions that obey heat equations. This solution appears to be very accurate (compared to the full finite element solution) until the time of stress collapse. Results indicate that for weak rate hardening of the power law type, intense localization depends strongly on the initial characteristics of thermal softening and not at all on the high temperature characteristics. Within the context of rigid/perfect plasticity, a scaling law for the critical strain is given, and a figure of merit is defined that ranks materials according to their tendency to form adiabatic shear bands.