Definition 11.1.1.
A sequence of vectors \(\{v_k\}_{k=1}^\infty\) which satisfy the matrix equation \(v_{k+1} = A v_k\) is called a (discrete) linear dynamical system.
Section 11.1 Dynamical Systems
Subsection 11.1.1 Theory of Dynamical Systems
Consider an applied mathematics problem where a vector \(v
\in \RR^n\) somehow represents the state of the situation under consideration. This state could be a position, keeping with the geometry of \(\RR^n\text{,}\) but typically it is just a list of numbers with some other interpretation. They could be the statistics of a countries economy, the concentration of a number of different chemicals in the same solutions, or any other list of quantities that are part of the same system. The state of the system isn’t static: it changes over time. Unlike the continuous systems in calculus, sometimes it is convenient to measure the system after some fixed time intervals. This interval is called a timestep. Finally, since the state is a vector in \(\RR^n\text{,}\) the change over one timestep is a sequence of vectors. If the system is in state \(v_k\text{,}\) then after the next second/minute/day/year/timestep, in the next state \(v_{k+1}\text{.}\) The model is essentially a sequence \(\{v_k\}_{k=1}^\infty\) of states in \(\RR^n\text{.}\)
How does the model move from one state to the next? If the process is linear, its progression can be described by a matrix action. That is, there exists a matrix \(A\) such that \(v_{k+1} = A v_k\) for all \(k \in \NN\text{.}\)
Dynamical systems are powerful and important mathematical models. However, they have some limitations, which are good to state up front. First, obviously, the relationship is linear. Linearity is a good starting point for models, but not all processes in the world are linear. It may be that the relationship between one state and the next state is non-linear, in which case, a matrix would not be able to calculate the state transition.
Another limitation is that each state of the dynamical system comes directly from the state before it by the matrix action. The matrix is the same for each transition, so the only thing that affects any particular state is the state before it. A dynamical system has no memory: states other than the directly previous state can have no effect on the next state. This again is good for some models, but many systems also build up some kind of memory that needs to be taken into account.
Subsection 11.1.2 Long Term Behaviour of Dynamical Systems
When I build a dynamical system to model some situation in the world, it’s natural to ask about the long-term behaviour of the system. From a certain starting value, where will the dynamical system go? This question is mostly answered by eigenvectors and eigenvalues.
Consider a dynamical system described by a matrix \(A\text{.}\) If \(v\) is an eigenvector with eigenvalue \(\lambda\text{,}\) then \(Av = \lambda v\text{.}\) The matrix action is a change of state over a timestep. If the state is an eigenvector, then the only effect of the timestep is to multiply all the quantities in the state vector by the same constant. What does this mean? First, it means that the ratio between the state quantities is fixed. The values are changed, but if the state is an eigenvector, the relative values of the quantities are fixed. This is a kind of stabilization. Second, the long term values are determined by \(\lambda\) in six cases.
- If \(\lambda = 0\text{,}\) then this is a collapsing state and all future states are simply the zero vector.
- If \(0 \lt |\lambda| \lt 1\text{,}\) then the sequence of states displays exponential decay. The long term behaviour of \(A^k v\) is an exponential decay of the original vector.
- If \(\lambda = -1\text{,}\) then there is a 2-period oscillating state. The sequence begins with the state \(v\) and jumps back and forth between \(v\) and \(-v\text{.}\)
- If \(\lambda = 1\text{,}\) then \(v\) is a steady state: the sequence never changes. These steady states are often very important in modelling.
- If \(\lambda > 1\text{,}\) then the states display exponential growth. The long term behaviour of \(A^k v\) is exponential growth of the original vector.
- If \(\lambda \lt -1\text{,}\) then the states display exponential growth with an oscillation. The sign of the vector with flip back and forth while the absolute value of the state quantities will grow exponentially.
As an aside, I could ask what happens for complex eigenvalues, since those will naturally also occur. Since the characteristic polynomial will have real coefficients, these complex eigenvalues will come in conjugate pairs. As you might learn in other courses, such pairs and the exponential behaviour (\(\lambda^n\)) give rise to sinusoidal behaviour after clever linear combinations. The reasons for this will have to wait for another course.
This is all well for eigenvectors, but what if the starting state isn’t an eigenvector? Well, the eigenvectors still end up controlling the long term behaviour, but it can get a bit more complicated. Ideally, there a full set of eigenvectors and I can write any starting state \(v\) as a linear combination of eigenvectors \(v = a_1 v_1 + a_2 v_2 + \ldots + a_n v_n\text{,}\) where each eigenvector \(v_i\) has eigenvalue \(\lambda_i\text{.}\) Then, the long term behaviour of the system with the starting state \(v\) is indeed given by the eigenvalues.
\begin{equation*}
A^k v = A^k (a_1 v_1 + a_2 v_2 + \ldots + a_n v_n) = \lambda_1^k
a_1 v_1 + \lambda_2^k a_2 v_2 + \ldots + \lambda_n^k a_n v_n
\end{equation*}
If any of the \(\lambda_i\) have an absolute value less than one, those terms will decay away. The eigenvalues which have an absolute value greater than one will grow. The largest eigenvalue (in absolute value) will lead to the largest growth, so it will dominate.
This can still be quite complicated, particularly if there are many eigenvalues with absolute values greater than one. For a completely arbitrary matrix \(A\text{,}\) it’s pretty difficult to use this to understand the system. However, the types of matrices which show up in dynamical systems usually have special properties, which make for more predictable and understandable systems.
In many models, the coefficients of the matrix will be probabilities, transition terms, growth rates or other positive real numbers. The matrix \(A\) will very often be a matrix with all non-negative entries. There is a powerful theorem that tells us what to expect for the eigenvalues/eigenvectors of such a matrix: the Perron-Frobenius theorem. There are some technical details in the assumptions for the theorem; I’ll state a weak version first.
Theorem 11.1.2.
Let \(A\) be an \(n\times n\) matrix with non-negative coefficients. Then there is a largest non-negative eigenvalue \(\lambda_1\) with an eigenvector that has all non-negative entries. All other eigenvalues \(\lambda\) satisfy \(|\lambda| \leq |\lambda_1|\text{.}\)
The stronger version of the theorem needs an new definition.
Definition 11.1.3.
Let \(A\) be an \(n \times n\) matrix. Then \(A\) is called irreducible if for all \(i\) and \(j\text{,}\) there exists a positive integer \(m\) such that the \(ij\)th entry of \(A^m\) is non-zero.
This definition roughly captures the idea that all the coefficients in the states are somehow related. Using the definition, here is a stronger version of the Perron-Frobenius theorem.
Theorem 11.1.4.
Let \(A\) be an \(n \times n\) irreducible matrix with non-negative coefficients. Then there is a unique largest positive eigenvalue \(\lambda_1\) with a 1-dimensional eigenspace and an eigenvector that has all positive entries. All other eigenvalues \(\lambda\) satisfy \(|\lambda| \lt |\lambda_1|\text{.}\) Moreover, if \(r_i\) is the sum of the entries of \(A\) in the \(i\)th row, then \(\lambda\) is bounded above and below by the largest and the smallest of the \(|r_i|\text{.}\)
For dynamical systems with irreducible non-negative matrices, the long term behaviour is really controlled and understood by this unique largest positive eigenvalue and the corresponding eigenvector. You will see this clearly in the next section, where I develop the theory of Leslie matrices, a nice example of a dynamical system.
