Dynamical Systems

2nd order characteristics equation

\[\begin{multline*} \Delta_A(\lambda)=\det (\lambda I -A)= \det \begin{bmatrix} \lambda- a_{11} & -a_{12}\\ -a_{21} & \lambda -a_{22} \end{bmatrix}=\\= % \lambda^2-(a_{11}+a_{22})\lambda+(a_{11}a_{22}- a_{12}a_{21})=0 \end{multline*}\] Where the eigenvalues are \[\lambda_{1,2}=\frac{(a_{11}+a_{22})\pm \sqrt{(a_{11}+a_{22})^2-4 (a_{11}a_{22}-a_{12}a_{21})}}{2} \]

In practice

The eigenvectors corresponding to each eigenvalue can be found by solving for the components of \(v\) in the equation \[ (A-\lambda I)v=0\]

Ten Possibilities

All the cases can be shown with the following graphics \(\mbox{tr}(A),\det A\) with the parabola \(-4 \det A+\mbox{tr}(A)^2\).

Stable node

let \(\lambda_1<\lambda_2<0\).

Two trajectories are oriented with the eigenvector \(v_1\) and two with \(v_2\).

Other trajectories are oriented toward \(v_1\) for \(t\rightarrow -\infty\) and toward \(v_2\) for \(t\rightarrow \infty\).

Stable node (2)

\[ A= \begin{bmatrix} -2 & 1\\ 1 & -2 \end{bmatrix} \Rightarrow \lambda_2=-3, \]

\[ v_2=[1,-1] \qquad \lambda_1=-1, v_1=[1,1]^T\]

Stable node: interpretation

In terms of population dynamics:

• \(a_{11}<0\) and \(a_{22}<0\)
  competition between individuals in both population
• \(a_{12}>0\) et \(a_{21}>0\)
  collaboration between both populations
• yet \(\lambda_1 \lambda_2 >0 \Rightarrow a_{11} a_{22} > a_{12} a_{21}\)
  competition impact bigger than the cooperation one
• The system brings the extinction of both species independently of the initial state.

Unstable node

Let \(\lambda_1>\lambda_2>0\).

Two trajectories are oriented as the eigenvector \(v_1\) and two along \(v_2\). The speed along those trajectories depends on the absolute value of \(\lambda_i, i=1,2\), hence \(v_1\) is faster than \(v_2\)

Other trajectories are oriented toward \(v_1\) for \(t\rightarrow \infty\) and toward \(v_2\) for \(t\rightarrow -\infty\)

Unstable node (2)

\(A= \begin{bmatrix} 2 & 1\\ 2 & 3 \end{bmatrix}\)

\(\Rightarrow \lambda_1=1, v_1=[-1,1]^T, \qquad \lambda_2=4, v_2=[1,2]\)

Unstable node: interpretation

For population dynamics:

• \(a_{11}>0\) and \(a_{22}>0\) means self-sustained growth

• \(a_{12}>0\) et \(a_{21}>0\) means collaboration between them

• without brakes the two populations explode independently of the initial state.

Saddle

Let \(\lambda_1<0<\lambda_2\).

Two trajectories are oriented as the eigenvector \(v_1\) and two along \(v_2\).

Other trajectories are oriented along \(v_1\) for \(t\rightarrow -\infty\) and along \(v_2\) for \(t\rightarrow \infty\)

Saddle (2)

\[ A= \begin{bmatrix} 5 & 9\\ 6 & 2 \end{bmatrix} \Rightarrow \lambda_1=11, v_1=[3/2,1]^T, \qquad \lambda_2=-4, v_2=[-1,1] \]

Saddle (3)

Saddle: interpretation

For population dynamics:

• \(a_{11}>0\) and \(a_{22}>0\) means self-sustained growth

• \(a_{12}>0\) et \(a_{21}>0\) means collaboration between them

• However, here \[ \det A =\lambda_1 \lambda_2= a_{11} a_{22} -a_{12}a_{21}<0 \Rightarrow a_{11} a_{22} <a_{12}a_{21}\]

  In other words, the cooperation effect is bigger than the self-sustained effect

Complex conjugate eigenvalues

If \(\lambda_1=a+ib, \lambda_2=a-ib\).
Trajectories can spiral in one of the two opposite directions

• [\(a<0\):] The system is asymptotically stable and trajectories spiral towards the origin. The origin is called a stable focus.
• [\(a>0\):] Unstable system with trajectories spiraling out of the origin. The origin is an unstable focus.
• [\(a=0\):] Trajectories are closed ellipse with period \(T=\frac{2 \pi}{b}\). The origin is a center.

Stable focus

Stable focus (2)

\[ A= \begin{bmatrix} 1/3 & -2\\ 3 & -1 \end{bmatrix} \Rightarrow \lambda_{1,2}=-1/3 \pm 5/3 \sqrt{2} i, \]

Stable focus: interpretation

• In this case the internal competition in population \(2\) is more important than the self-sustaining of population \(1\)

• Both populations become extinct after some oscillations

Unstable focus

\[ A= \begin{bmatrix} 2 & -2\\ 2 & 2 \end{bmatrix} \Rightarrow \lambda_{1,2}=2 \pm 2i, \]

Centre

\[ A= \begin{bmatrix} 1 & -2\\ 3 & -1 \end{bmatrix} \Rightarrow \lambda_{1,2}=\pm \sqrt{5} i, \]

Center: interpretation

If complex conjugate solutions \(\mbox{Re}(\lambda_1)+\mbox{Re}(\lambda_2)=0\) and \(\lambda_1 \lambda_2 = b^2 > 0\). Then

• \(a_{11}=-a_{22}\), self sustaining of population \(1\) is equal and opposite to the one of population \(2\)

• \(a_{12}a_{21}=a_{11}a_{22}-\lambda_1 \lambda_2<0\), i.e. \(a_{12}\) et \(a_{21}\) have opposite signs.

• Two possibilities (\(a_{11}>0\)), predation or parasitism \[ A= \begin{bmatrix} a_{11} & <0\\ >0 & -a_{11} \end{bmatrix}, \qquad \begin{bmatrix} a_{11} & >0\\ <0 & -a_{11} \end{bmatrix}\]

Center: interpretation (II)

• In case of predation of \(2\) on \(1\), too much population \(2\) leads to too few from population \(1\) which in turn impact population \(2\).
• Similarly too much from population \(1\) leads to too much from population \(2\) which impact negatively population \(1\).
• Cyclical behavior.

Non-Simple Systems

If \(\det A =\lambda_1 \lambda_2 =0\) and the eigenvalue \(\lambda_2 \neq 0\).

• \(\lambda_1=0\): all the states belonging to the line \(a_{11} x_1+a_{12} x_2=0\) are equilibriums. All the trajectories are parallel to that line \(v_2\). Hence we have only two possibilities:

• \(\lambda_2 < \lambda_1=0\): equilibriums are stable.

• \(0=\lambda_1 <\lambda_2\): equilibriums are unstable.

Non-simple Systems: Case 1

\(\lambda_2 < \lambda_1=0\)

Non-simple Systems: Case 1

\(\lambda_2 < \lambda_1=0\)

\[ A= \begin{bmatrix} 0 & 1\\ 0 & -1 \end{bmatrix} \Rightarrow \lambda_1=0, , \qquad \lambda_2=-1, v_2=[-1,1] \] Stable equilibriums are on the line \(x_2=0\).

Non-simple Systems: Case 2

\(0=\lambda_1 <\lambda_2\)

\[ A= \begin{bmatrix} 0 & 1\\ 0 & 1 \end{bmatrix} \Rightarrow \lambda_1=0, \qquad \lambda_2=1, v_2=[1,1] \] Unstable equilibriums are on the line \(x_2=0\).

Eigen values equal and real

Let \(\lambda_1=\lambda_2=\lambda \neq 0\). Two possibilities:

• [Matrix \(A\) diagonalisable:] Infinity of eigenvectors, every straight line going through the origin is a trajectory. if \(\lambda<0 (>0)\) we have asymptotic (un)stability. The origin is a singular node.
• [Matrix \(A\) non-diagonalisable:] Only one eigenvector and one line having a trajectory. The origin is a degenerate node.
• If \(\lambda_1=\lambda_2=0\) there is an infinite unstable equilibrium points, all the trajectories are on parallel lines .

Singular node

Every line going through the origin is a trajectory.

Singular node (2)

\[ A= \begin{bmatrix} -1 & 0\\ 0 & -1 \end{bmatrix} \Rightarrow \lambda_{1,2}=-1, \qquad A v= \lambda v \mbox { pour tout } v \]

Degenerate node

\(\lambda_1=\lambda_2<0\)

Only one eigenvector and thus one straight line as a trajectory.

Degenerate node (2)

\[ A= \begin{bmatrix} 3 & -4\\ 1 & -1 \end{bmatrix} \Rightarrow \lambda_{1,2}=1, \qquad v_1=v_2=[2,1] \] Matrix \(A\) not diagonalisable.

Degenerate node (3)

\(\lambda_1=\lambda_2=0\)

\[ A= \begin{bmatrix} 2 & -4\\ 1 & -2 \end{bmatrix} \Rightarrow \lambda_{1,2}=0, \qquad v_1=v_2=[2,1] \]

Conclusion

The scientific approach, especially in physics, i.e. Newton, Lagrange,… and up to the beginning of the 20th century tend to be:

• reductionist: the global system can be explained by combining behaviors of sub-systems,
• reversible: time can be reversed, models are bidirectional along the arrow of time,
• deterministic: once you know the laws and the initial conditions, the solution is unique.

However: non-linear

Most models are non-linear (like the Lokta-Volterra)

but can be linearized locally (i.e. multiples focus and nodes).

However: hyper-dimensional

Most models have more than two populations

Hence, a small difference in the input can have a huge impact on the outcome (chaos theory)

Hence,

All this shows that:

To study chaos, mathematics, computer simulation and even experimentation have to be combined… _{[Ref: G. Bontempi (ULB, INFO0305)]}