Nonlinear Autonomous Ordinary Differential Equations Research Articles

A global annually-averaged climate model with cloud, water vapor and CO2 feedbacks

In consideration of the radiation transfer, latent and sensible heat exchange between oceans and atmosphere, a three-dimensional autonomous nonlinear ordinary differential equation is established by statistical parameterization method. The variables of the model are the mean ocean surface temperatureTs, mean atmospheric temperatureTa and atmospheric relative humidityf, and the feedbacks of clouds, water vapor and CO2 are involved. The steady state corresponding to the present-day climate can be obtained from this model. The analysis of parameter sensibility in the steady state indicates that clouds have considerable negative feedback effects and water vapor may affect the sign of CO2 feedback. The stability analysis of the steady state to small disturbance indicates that with increase of the positive feedback effect of clouds, the steady state goes through such a structural variance series as a stable node→a stable focal point→an unstable focal point→an unstable node, and when the steady state becomes unstable it undergoes a subcritical Hopf bifurcation. When the steady state is at a focal point, the periodic oscillation solutions of damping or amplifying can be obtained with the period being about two years.

Oscillations in a Third Order Differential Equation Modeling a Nuclear Reactor

We consider a third order autonomous nonlinear ordinary differential equation which models a two temperature feedback nuclear reactor. Using geometric methods and the Brouwer fixed point theorem, we show the existence of periodic solutions over a specific range of parameters. In light of these results, conjectures are given discussing the global behavior of families of periodic solutions bifurcating from the steady state.