With algebraic equation, a system of equations is a set of equation (usually more than one, but technically one equation can still be a sort-of boring system). To solve a system of algebraic equations is to try to find values of the variable(s) that satisfy all the equation.
Definition6.1.1.
A system of differential equations is defined in exactly the same way: it is a set of differential equations about the same function or functions. A solution is a set of functions which satisfy all of the differential equations simultaneously.
Many of the definition from Section 2.1 can be directly extended to systems.
Definition6.1.2.
The order of a system of differential equation is the highest derivative involved in any of the equation.
A system of differential equation is an autonomous system if all of the equations in the system are autonomous.
A system of differential equations is a linear system if all of the equations in the system are linear. Otherwise, it is a non-linear system.
A linear system of differential equation is homogeneous if all of the linear equations in the system are homogeneous (all of the ‘forcing terms’ are zero).
Solutions to systems of differential equation typically come in familiar, due to constants of integration. To determine a specific solution requires initial conditions. If there are multiple functions involved, I may need to specify initial conditions for each of the functions.
The difference between linear systems and non-linear systems is immensely important. I’ll demonstrate some exactly solutions for linear system (with constant coefficients) in Section 6.3. However, the key examples to follow will be entirely non-linear. One might wonder why it is still useful to study linear systems after the motivating examples show such linearity. There are, in fact, strong reasons to still understand linear systems. Often, the linear approximation to a non-linear system can demonstrate many of the dynamics of the system. This approximation also is an important first step to understanding the non-linear, introducing the complication as an adjustment to the linear approximation.
Subsection6.1.2Key Examples
There are a number of important motiavting examples. I first look to biology, where the interaction of species produces systems of linear equations. The Lokta-Volterra equations model predator-prey relationships. Let \(p(t)\) be the population of prey and \(q(t)\) be the population of predators. These equations describe the interactions between the two derivatives.
The coefficients \(a,b,c\) and \(d\) are all positive coefficients. \(a\) is a natural birth rate for the prey, \(b\) is the death of prey due to predation, \(c\) is the growth rate of predators due to predation, and \(d\) is the natural death rate of predators. The product \(pq\) represents the number of interactions between predators and prey. The Lokta-Volterra system is a classic system in mathematical biology; however, its non-linearity makes it difficult to solve and the solutions are not expressible by elementary functions. To help understanding the system without having to actually calculate difficult solutions, I will introduce qualitative methods of analysis in Section 6.2.
Mathematical biology has many systems similar to Lokta-Volterra. If \(p\) and \(q\) are two species competing for the same resources, there is a very similar two-equation DE model.
The coefficients \(a_p\) and \(a_q\) are the natural growth rates of each species, \(c_p\) and \(c_q\) are the carrying capacities of the environment for each species, and \(b_p\) and \(b_q\) measure the effects of competition. Like the Lokta-Voltera equations, this is a difficult non-linear system where I should not expect elementary function solutions.
Infection disease models are also systems of differential equations. The SIR model is a good example. In a population exposed to an infection disease, let \(S\) be the susceptible population, \(I\) the infected population, and \(R\) the recovered population. The model is this system of three differential equations.
\begin{align*}
\frac{dS}{dt} \amp = -a I S\\
\frac{dI}{dt} \amp = a I S - b I\\
\frac{dR}{dt} \amp = b I
\end{align*}
Here \(a\) measures the increase in infection due to interactions between the infected and susceptible population and \(b\) measures the natural recovery of the infected population. Many similar models exist to model infection disease.
This table is already pretty complicated, with many linear differential equation. In systems biology, this is only the start. A more complete model of cellular reactions can involve hundreds or thousands of coupled differential equations. These models are almost exclusively linear, simply because systems of this size aren’t at all approachable if they are not linear. The actualy cellular interactions are most likely non-linear, and these models, as complex as they are, remain at best linear approximations of even more complicated situations.
Examples exist in other disciplines as well. Back in the world of physics, I can look at the physics of the coupled spring. In this sytem, there is a mass \(m_1\) on a spring with spring constant \(k_1\) attached to a fixed object. Then there is a second mass \(m_2\) attached to the first by a spring wih spring constant \(k_2\text{.}\) Analyzing the forces and using Newton’s law of motion results in a system of DEs.
