MATH383: Dynamical systems general concepts

Overview

• Overview: nonlinear systems of differential equations

• Flow maps, stable and unstable equilibria

• Poincaré sections

• Examples:

  • Logistic map

  • Duffing oscillator

  • Lorenz system

  • Van der Pol oscillator

Overview

• Nonlinear systems of differential equations ${𝒚}^{\text{'}}=𝒇(t,𝒚)$, $𝒚:ℝ\to {ℝ}^n$, $𝒇:ℝ\times {ℝ}^n\to {ℝ}^n$

• Very few nonlinear systems can be solved analytically

• Solutions can be found by numerical approximation (e.g., Euler, Runge-Kutta)

• Numerical solutions can be combined with analysis of qualitative behavior

• Tools:

  • $𝒇$ continuous $\Rightarrow$ solutions are unique, trajectories from different initial values do not cross

  • Effect of system parameters $𝒑$, ${𝒚}^{\text{'}}=𝒇(t,𝒚;𝒑)$, is crucial in understanding behavior, e.g.

    ${\displaystyle m{y}^{\text{'}\text{'}}+c{y}^{\text{'}}+ky=f,𝒑=\left(\begin{array}{lll}m& c& k\end{array}\right)}$

  • Flow map: ${\Phi }_t({𝒚}_0;𝒑)=𝒚(t;{𝒚}_0;𝒑)$ is the family of trajectories that start from initial condition $(0,{𝒚}_0)$. Interest is to determine how families of trajectories change when varying the system parameters $𝒑$

Logistic map

• Many of the features of dynamical system analysis are exhibited by a simple model, the logistic map describing the population $\stackrel{\sim }{y}\left(\stackrel{\sim }{t}\right)$ of a species.

• Consider first the Malthusian model: population increase is proportional to current population

  ${\displaystyle {\stackrel{\sim }{y}}^{\text{'}}=r\stackrel{\sim }{y}\Rightarrow \stackrel{\sim }{y}\left(\stackrel{\sim }{t}\right)=e^{rt}{\stackrel{\sim }{y}}_0,}$

  that predicts (unbounded) exponential growth starting from initial population ${\stackrel{\sim }{y}}_0$.

• Modify the above model to include the effect of diminshing resources by modifying the growth rate to become $r(1-\stackrel{\sim }{y}/K)$. When the population $\stackrel{\sim }{y}$ reaches the carrying capacity $K$, the population growth rate becomes zero

  ${\displaystyle {\stackrel{\sim }{y}}^{\text{'}}=r\stackrel{\sim }{y}(1-\frac{\stackrel{\sim }{y}}{K})\Rightarrow \stackrel{\sim }{y}\left(\stackrel{\sim }{t}\right)=\frac{K{e}^{r\stackrel{\sim }{t}}}{K+y_0(e^{r\stackrel{\sim }{t}}-1)}{\stackrel{\sim }{y}}_0}$

• Are two parameters needed? No, rescaling time $t=\stackrel{\sim }{t}/r$, and population $y=\stackrel{\sim }{y}/K$ leads to

  ${\displaystyle y^{\text{'}}=y(1-y),y\left(t\right)=\frac{e^t}{1+y_0(e^t-1)}{y}_0}$

  $$a differential equation with no parameters.

Equilibria

• A first concept in dynamical systems analysis is that of an equilibrium point, a state of the system that does not change. For the logistic map $y^{\text{'}}=y(1-y)$ there are two equilibria

  ${\displaystyle y=0y=1.}$

• A second concept is that of equilibrium point stability, if the system is slightly displaced from an equilibrium point, does it return to its previous state or does it evolve to a different equilibrium?

• Consider $y^{\text{'}}=f\left(y\right)$. Equilibria are determined by roots of $f$, $f\left(y\right)=0$. Denote a root by $y^{\ast }$.

• For the logistic equation the equilibria are $y_1^{\ast }=0$ and $y_2^{\ast }=1$

• Asymptotic behavior of solution to logistic equation

  ${\displaystyle {lim}_{t\to \infty }y\left(t\right)={lim}_{t\to \infty }\frac{e^t}{1+y_0(e^t-1)}{y}_0=1\mathrm{if}{y}_0\ne 0}$

  • $y_1^{\ast }=0$ is an unstable equilibrium point, after small perturbation $y\to 1\ne y_1^{\ast }$

  • $y_2^{\ast }=1$ is a stable equilibrium point, after small perturbation, $y\to 1=y_2^{\ast }$

Insufficiency of phase portraits for systems with $n>2$ dependent variables

• Consider unforced harmonic oscillator $y^{\text{'}\text{'}}+\gamma y^{\text{'}}+\kappa y=0$, $y^{\text{'}}=v$, $v^{\text{'}}=-\gamma v-\kappa y$

• Phase portrait $(y^{\text{'}},v^{\text{'}})$ can readily be represented and the flow map ${\Phi }_t({𝒚}_0;𝒑)=𝒚(t;{𝒚}_0;𝒑)$ is known analytically, e.g., ${\Phi }_t({𝒚}_0;\gamma =0,\kappa )=(\cos \sqrt{\kappa }t,\sin \sqrt{\kappa }t)$.

(%i82) 

plotdf([v,-gamma*v-kappa*y],[y,v],
[trajectory_at,.5,.5],
[y,-1,1],[v,-1,1],
[direction,forward],
[parameters,"gamma=0,kappa=1"],
[sliders,"gamma=0:2,kappa=0.5:2.0"])$

(%i83) 

Poincaré sections

• The stroboscopic effect allows visualization of rotational or oscillatory motion

A Poincaré section is the projection onto the $y$-axis of points sampled at periodic intervals

Duffing oscillator

• Consider the forced, non-linear oscillator, $y^{\text{'}\text{'}}+\delta y^{\text{'}}+\alpha y+\beta y^3=\gamma \sin (\omega t)$, (Duffing)

(%i86) 

plotdf([v,-v/10+y-beta*y^3],[y,v],
[trajectory_at,.5,.5],[nsteps,10000],
[y,-5,5],[v,-5,5],
[direction,forward],
[parameters,"beta=0.25"],
[sliders,"beta=0.1:0.4"])$

(%i87) 

Lorenz system Poincaré sections

• Successive construction of Poincaré sections, $m=2500,5000,10000,25000$ samples

Van der Pol oscillator

• $x^{\text{'}\text{'}}-\mu (1-x^2)x^{\text{'}}+x=0\Rightarrow$

  ${\displaystyle x^{\text{'}}=y,y^{\text{'}}=\mu (1-x^2)y-x}$

(%i4) 

plotdf([y,mu*(1-x^2)*y-x],[x,y],
[trajectory_at,1,1],
[x,-5,5],[y,-5,5],
[direction,forward],
[parameters,"mu=1"],[sliders,"mu=-1:1"],[versus_t,1])$

(%i5)