DIFFERENTIAL EQUATIONS 08
Systems of Differential Equations
A system of differential equations describes several quantities that change together and influence one another.
The main idea
Many models involve more than one changing quantity.
Instead of one equation such as
\[ \frac{dy}{dt}=f(y), \]we may have several equations:
The two variables evolve together.
1. Why a system is needed
Suppose \(x(t)\) affects \(y(t)\), and \(y(t)\) also affects \(x(t)\).
One equation alone cannot fully describe the interaction.
We therefore use one differential equation for each changing quantity.
2. A simple system
Consider
\[ \frac{dx}{dt}=y, \] \[ \frac{dy}{dt}=-x. \]The rate of change of \(x\) depends on \(y\), while the rate of change of \(y\) depends on \(x\).
3. Connection to a second-order equation
From
\[ \frac{dx}{dt}=y, \]differentiate again:
\[ \frac{d^2x}{dt^2} = \frac{dy}{dt}. \]But
\[ \frac{dy}{dt}=-x. \]Therefore,
So this two-equation system is equivalent to a second-order equation.
4. Converting second-order equations into systems
The reverse process is also important.
Suppose
\[ y''=-y. \]Define
\[ v=y'. \]Then
\[ y'=v \]and
\[ v'=-y. \]So the second-order equation becomes the first-order system
This idea is fundamental because higher-order ODEs can be rewritten as systems of first-order ODEs.
5. Vector notation
A system can be written compactly using a vector.
Define
\[ \mathbf{x}(t) = \begin{pmatrix} x(t)\\ y(t) \end{pmatrix}. \]Then
\[ \frac{d\mathbf{x}}{dt} = \begin{pmatrix} dx/dt\\ dy/dt \end{pmatrix}. \]So the system
\[ \frac{dx}{dt}=f(x,y), \qquad \frac{dy}{dt}=g(x,y) \]can be written as
6. Linear systems
A linear two-dimensional system can be written as
\[ \frac{dx}{dt} = ax+by, \] \[ \frac{dy}{dt} = cx+dy. \]In matrix form,
7. What the matrix means
The matrix contains the coefficients that describe how the variables affect each other's rates of change.
For example,
\[ \frac{dx}{dt}=2x-y \]and
\[ \frac{dy}{dt}=3x+4y \]correspond to
\[ \begin{pmatrix} 2 & -1\\ 3 & 4 \end{pmatrix}. \]8. Initial conditions
A system also needs initial values.
For example,
\[ x(0)=x_0, \qquad y(0)=y_0. \]Together, these specify the starting state
\[ \boxed{ \mathbf{x}(0) = \begin{pmatrix} x_0\\ y_0 \end{pmatrix} } \]9. State space
For a two-variable system, the pair
\[ (x,y) \]represents the state of the system.
All possible states form the state space.
For two variables, this can be visualised as the \(xy\)-plane.
10. Trajectories in the phase plane
As time changes, the point
\[ (x(t),y(t)) \]moves through the state space.
The path it follows is called a trajectory.
11. Phase-plane diagram
12. Direction field for a system
At each point \((x,y)\), the system gives
\[ \left( \frac{dx}{dt}, \frac{dy}{dt} \right). \]This vector tells us the instantaneous direction of movement in the phase plane.
13. Equilibrium points
An equilibrium occurs where all rates of change are zero.
For
\[ x'=f(x,y), \qquad y'=g(x,y), \]an equilibrium \((x^*,y^*)\) satisfies
If the system starts there, it remains there.
14. Example of an equilibrium
Consider
\[ x'=x-y, \] \[ y'=x+y. \]At equilibrium,
\[ x-y=0 \]and
\[ x+y=0. \]Adding the equations gives
\[ 2x=0, \]so
\[ x=0. \]Then
\[ y=0. \]Therefore,
\[ \boxed{ (0,0) } \]is the equilibrium.
15. Stability in systems
As in one-dimensional autonomous equations, we can ask whether trajectories move toward or away from an equilibrium.
If nearby trajectories approach the equilibrium, it is stable.
If they move away, it is unstable.
16. Eigenvalues and stability
For a linear system
\[ \mathbf{x}'=A\mathbf{x}, \]the eigenvalues of the matrix \(A\) help determine the behaviour near the equilibrium.
Very roughly:
\[ \text{negative real parts} \Rightarrow \text{decay toward equilibrium}, \] \[ \text{positive real parts} \Rightarrow \text{growth away from equilibrium}. \]This will become especially useful in multivariable modelling.
17. Why eigenvectors matter
If
\[ A\mathbf{v} = \lambda\mathbf{v}, \]then the matrix changes the size of the eigenvector but not its direction.
A solution along this direction often has the form
\[ \mathbf{x}(t) = e^{\lambda t}\mathbf{v}. \]So eigenvectors give important directions of motion, while eigenvalues determine growth or decay along those directions.
18. Population example
Suppose two populations interact:
\[ \frac{dx}{dt}=ax-by, \] \[ \frac{dy}{dt}=cx-dy. \]The growth rate of each population depends on both populations.
This type of coupled structure appears throughout population dynamics and mathematical biology.
19. Epidemic model example
The SIR epidemic model is a system of differential equations:
The susceptible, infectious and recovered populations change together.
A change in one compartment affects the others.
20. Conservation in the SIR system
Add the three equations:
\[ \frac{dS}{dt} + \frac{dI}{dt} + \frac{dR}{dt}. \]Substituting,
\[ -\beta\frac{SI}{N} + \beta\frac{SI}{N} - \gamma I + \gamma I = 0. \]Therefore,
So
\[ S+I+R \]remains constant when the model assumes a closed population.
21. Why systems matter
Real models often contain several interacting quantities.
Systems of ODEs allow us to represent these interactions directly.
22. The central idea
The vector \(\mathbf{x}\) describes the current state. The vector field \(\mathbf{F}\) gives the rate of change of that state.