Steady states and stability in dynamical systems
August 2018
Assume there is a simple differential equation, say dy/dx = xy. It is separable, and we can arrive at the (general) solution fairly easily, which is y = Ce^(1/2 * x^2), where C is a constant. What does this mean?

In essence, the initial differential equation postulates a requirement that a function must satisfy. In this case, the derivative of the function must at all times be equal to the function itself, scaled by the current value of x.

This means that we are actually in search of functions and not any particular values of variables. Any function that satisfies the relation presented by the DE is then a valid solution. How many solutions do we have? Our solution includes a constant multiplying the exponential function, and the range of the constant is not defined. This means that any constant will suffice, and consequently we have infinite number of solutions. At this point, we are more interested in the nature of the solution function and less in where it intercepts the y-axis.

The interesting thing we can do at this point is use the general solution in plotting a slope field:

The correctness of the slope field can be confirmed by plotting the function for several different constants:

The convenient thing about this type of system is that each possible trajectory is completely determined by all the points that belong to it. This means that if we know just a single point that our process goes through, we can determine its whole trajectory. This point is often called the initial condition of the DE. Combining the general solution with an initial condition we can solve for the constant and thus obtain a specific and complete solution.

Now, often differential equations come in a group, and we end up having a system of differential equations. Say we have two equations

p' = 3p + 4q
q' = p + 2q

This type of system is called coupled because each function affects the set of trajectories for the other. Now, as with other systems of equations, we can represent the system in terms of linear algebra. Define v' = [p'; q'], v = [p; q] and L = [3 4; 1 2]. Then

[p'; q'] = [3 4; 1 2][p; q]
v' = Lv
Lv = v'

where L now maps vector v to its derivative vector. Now, L being in essence the function of the equation accepting input v and outputting its derivative, the properties of L will turn out to be central in understanding the behavior of the system.