Consider this equation

.

Let us assume the solution to be of the form .

We see that the solution is given as the Lambert W function (which was defined in Maple in its early years).  Assume , we find the approximate value of  to be

The graph of the solution curve is shown below.

In applied mathematics and science, few differential equations admit closed form solutions, and often we need to employ numerical methods.  The next example demonstrates the usage of the dsolve command for a delay differential equation.   

Suarez and Schopf proposed a simple nonlinear model for the El Niño phenomenon.  Their model relies on the existence of a strong positive feedback in the coupled ocean-atmosphere system and on some unspecified nonlinear mechanism invoked to limit the growth of unstable perturbations.  Its key element is the inclusion of the effects of equatorially trapped oceanic waves propagating in a closed basin through a time delayed term.  Let  denote the temperature anomaly, that is, the deviation from a suitably defined long term average temperature, the time derivative is

,

where  is the nondimensional delay (wave transit time) and  measures the influence of the returning signal relative to that of the local feedback.  

Let us assign the values for  and  to be 0.7 and 3, respectively.  

Using an initial condition of , we solve the differential equation.

We try a different value for the delay.

We observe that solutions of the delay differential equation are self-sustained oscillations.  The reader can verify the conclusion of Suarez and Schopf that "the period of the oscillation is at least twice as long as the assumed delay."  However, when the delay drops to a certain value, the oscillation ceases, as seen in the following.

The delay is the time required for equatorially trapped oceanic waves to propagating across the Pacific Ocean.  The existence of a threshold delay time offers a possible explanation for the lack of El Niño phenomenon in smaller water bodies such as the Atlantic and Indian Oceans.  

If we consider the effect of global warming, with a rate of , the modified delay differential equation becomes:

Let us try .

The above plot shows the comparison between the model with global warming (the blue curve, with ) and the original model.  We see no noticeable difference in the steady state solutions.  However, when the global warming rate gets too large, the oscillatory solution disappears, as seen below.  

Drastic qualitative changes in the solution curves as one smoothly varies the parameter is called "bifurcation."  In conclusion, Maple's newly added option to solve delay differential equations numerically allows users to investigate the delayed action oscillator for El Niño and investigate bifurcation due to the time delay and the global warming rate, and possibly other factors.    

1. Ian Boutle, Richard H. S. Taylor, and Rudolf A. Römer, El Niño and the delay action oscillator, American Journal of Physics, 75, 15--24 (2007).

2. Nicholas J. Higham, The Princeton Companion to Applied Mathematics, Princeton University Press, 2015.

3. Max J. Suarez and Paul S. Schopf, A Delayed Action Oscillator for ENSO, Journal of the Atmospheric Sciences, 45, 3283--3287 (1988).