Dynamical Systems

Introduction

Prof. Patrick E. Meyer

Introduction

Differential equations allows to describe and forecast the evolution of a dynamical system by connecting input variables and state variable, i.e. derivatives.
Trajectories can then be obtained by integrating those differential equations
Unfortunately it is often difficult to compute analytically when models are non-linear hyperdimensional

Evolution of fluxes

Equations: \[ \begin{cases}\dot{x}(t)=c u(t)\\ y(t)=x(t) \end{cases} \]
In function of an input flux \(u\) it describes (a) the evolution of \(x\) which could be some merchandise in a warehouse, (b) the remaining volume of a tank, (c) concentration of a chemical product in the milieu.
Assumptions: no feedback, change can be both positive or negative, the state of the system is observable

Exponential growth

Equations: \[\begin{equation*} \begin{cases} \dot{x}(t)=cx(t) \\ y(t)=x(t) \end{cases} \end{equation*}\]
with solution \(x(t)=x(0) \exp^{ct}\)
there is a feedback because the rate of change depends on the state of the system.
population evolution, economic growth, radioactive decay,…
Assumptions: no exogenous impact

Logistic growth

Equations \[\begin{equation*} \begin{cases} \dot{x}(t)=cx(t)(1-\frac{x(t)}{k}) \\ y(t)=x(t) \end{cases} \end{equation*}\] small \(x\) means \(\dot{x}(t)=cx(t)\) and if \(x>k\) then \(\dot{x}<0\).
this corrects an infinite growth of the previous model
the growth rate can be negative when the pop. is big
fit very well the growth of simple organisms like bacteria/yeast

Lotka Volterra Systems

\[ \begin{cases} \dot{x}_1=a_{11} x_1- a_{12} x_1 x_2 \qquad \mbox{(proie)}\\ \dot{x}_2=a_{21} x_1 x_2- a_{22} x_2 \qquad \mbox{(predateur)} \end{cases} \]

proposed independently by Alfred J. Lotka and Vito Volterra in the 1920’s in order to model prey-predator dynamics
A classical validation of the model is the hare-lynx dynamics in the Hudson bay in the XIX century.

Lotka Volterra: model

The systems models the rate of growth of two populations \(\dot{x}_1\) and \(\dot{x}_2\) that interact in a closed ecosystem, where \(x_1\) and \(x_2\) denotes respectively the preys and the predators

\(a_{11}\) is the birth rate of the preys
\(a_{12}\) is the death rate through predation
\(a_{21}\) is the reproduction rate of predators by captured prey
\(a_{22}\) is the death rate for the predators

Lotka Volterra: results

Lotka Volterra: assumptions

preys have infinite ressource to reproduce (no competition)
predators need preys to reproduce and as a results are in competition
preys grow exponentially without predators
predators decays exponentially without preys

the romeo-juliette model

\[ \begin{cases} \dot{r}(t)=-a j(t)\\ \dot{j}(t)=b r(t) \end{cases} \qquad \mbox{with } a>0, b>0 \]

\(r(t)\) denote the love of Romeo for Juliette and \(j(t)\) the love of Juliette for Romeo

Romeo is attracted by Juliette when she gets cold but get repelled when she is getting too much in love
Juliette tend to mirror Romeo’s love, high when he is in love but cold when he is cold.
Can you model the evolution of their story?

Fuis-moi je te suis, suis-moi je te fuis

4th turning variant

Hard times generate strong men, strong men generate good times, good times generate weak men, weak men generate hard times

Second Order Autonomous System

\[\begin{equation*} \begin{cases} \dot{x}_1=a_{11} x_1+ a_{12} x_2\\ \dot{x}_2=a_{21}x_1+a_{22}x_2 \end{cases} \Leftrightarrow \begin{bmatrix} \dot{x}_1\\ \dot{x}_2\\ \end{bmatrix}= \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix} \begin{bmatrix} {x}_1\\ {x}_2\\ \end{bmatrix} \end{equation*}\]

\(\Leftrightarrow \dot{x}=Ax\)
With two dimensions
In the phase space, trajectories are curves/lines
The system is said not simple if \(\det A=0\). Equilibrium points not at the origin
The system is said simple if \(\det A \neq 0\). where \((0,0)\) is the only equilibrium point of the autonomous system.

2nd order Autonomous System Coefficients

\(a_{ii}\) is the contribution to its own growth.
\(a_{ii}>0\) the population tend to grow naturally and \(a_{ii}<0\) means the population tends to shrink without external help (case of competition for preys).
Coefficients \(a_{ij}\): the impact of pop. \(j\) on the pop. \(i\)
\(a_{ij}>0\) means population \(j\) contributes to the growth of \(i\) (through predation of \(i\) or symbiosis)
\(a_{ij}<0\) means population \(j\) contributes to the shrinkage of population \(i\) (through predation or parasitism of \(j\))