Chandan Singh | chaos

chaos

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Normal forms of Hopf bifurcations

  • pitchfork: \(\dot{x} = \lambda x - x^3\)
  • subcritical pitchfork: $\dot{x} = \lambda x + x^3$
  • saddle node (turning point): $\dot{x} = \lambda - x^2$
  • transcritical: $\dot{x} = \lambda x - x^2$

important figs

  • period-doubling (flip bifurcation) $f = \mu x (1-x) (f = \mu sin (\pi x) $ is similiar)
  • inverse tangent bifurcation - unstable and stable P-3 orbits coalesce, move slightly off bisector and becomes chaotic
  • pendulum
  • energy surface - trajectories run around the surface, not down it
  • Conservative systems: 6.5
    • study Hamiltonian p. 187-188
  • Pendulum: 6.7
  • dynamics - study of things that evolve with time
  • chaos - deterministic, aperiodic, sensitive, long-term prediction impossible
    1. phase space - has coordinates $x_1,…,x_n$
    2. phase portrait - variable x-axis, derivative y-axis
    3. bifurcation diagram - parameter x-axis, steady state y-axis
    • draw separate graphs for these
  • first check - look for fixed points
  • for 1-D, if f’ $<$ 0 then stable
  • stable f.p. = all possible ICs in a.s.b.f.n. result in trajectories that remain in a.s.b.f.n. for all time
  • asymptotically stable f.p. - stable and approaches f.p. as $t\ra\infty$
  • hyperbolic f.p. - eigenvals aren’t strictly imaginary
  • bifurcation point of f.p. - point where num solutions change or phase portraits change significantly
  • globally stable - stable from any ICs
  • autonomous = f is a function of x, not t
  • we can always make a system autonomous by having $x_n$ = t, so $\dot{x_n}$ = 1
  • dimension = number of 1st order ODEs, dimension of phase-space
  • existence and uniqueness thm: if $\dot{x}$ and $\dot{x}’$ are continuous, then there is some unique solution
  • linearization - used to find stability of f.p.s
  • \[\dot{x} = f(x) &\\ \text{define }\delta x = (x-\bar{x}) \\ \dot{\delta x} = \frac{d}{dt}(x-\bar{x}) = \dot{x} = f(x) = f(\bar{x}+\delta x) \\ \dot{\delta x} =\cancelto{0}{f(\bar{x})} + \delta x f'(\bar{x}) + \cancelto{\text{0 by HGT iff f'!=0}}{O(x^2)} \\ \dot{\delta x} = \delta x f'(\bar{x}) \to \text{ now solve FOLDE} \\\]
  • solving Hopf: use polar to get $\dot{\rho}, \dot{\theta}$
  • multiply one thing by cos, one by sin, then add

  • $ \rho = \sqrt{x_1^2 + x_2^2}
    \theta = tan^{-1}(\frac{x_2}{x_1})$
  • Hysterisis curve - S-shaped curve of fixed branches - ruler getting larger - snap bifurcation - both axes are parameters

Systems of Linear ODEs

  • solutions are of the form $\underbar{x}(t) = \underbar{C}_1e^{\alpha_1 t} + \underbar{C}_2e^{\alpha_2 t}$
  • Eigenspaces: $E^S$ (stable), $E^U$ (unstable), $E^C$ (center - real part) - plot eigenvectors
  • how to solve these systems?
    • solve eigenvectors
  • positive real part - goes out
  • negative real part - goes in
  • bifurcation requires 0 as eigenvalue
  • has imaginary component: spiral / focus
  • purely imaginary - center = stable, but not a.s.
  • finite velocity = $\frac{dRe(\alpha)}{d\lambda}$
  • change coordinates to polar
  • for $\lambda \geq 0$, solution is a stable L.C. (from either direction spirals into a circular orbit)
  • attracting - any trajectory that starts within $\delta$ of $\bar{\underbar{x}}$ evolves to $\bar{\underbar{x}}$ as t $\to \infty$ (it doesn’t have to remain within $\delta$ at all times
  • stable (Lyapanov stable) - any trajectory that starts within $\delta$ remains within $\varepsilon$ for all time ($\varepsilon$ is chosen first)
  • asymptotically stable - attracting and stable
  • hyperbolic f.p. - iff all eigenvals of the linearization of the nds about the f.p. have nonzero real parts \

Discrete Nonlinear Dynamical Systems

  • functional iteration: $x_{n+m} = f^m(x_n)$ (apply f m times)
  • fixed point: $f(x^)=x^$
  • f.p. stable if $|\frac{df}{dx}(x^*)|<1$, unstable if $>$ 1
  • check n-orbit by checking nth derivative: $\frac{df^n}{dx}(x_i^) = \prod_{i=1}^{n-1} \frac{df}{dx}(x_i^)$
  • period-doubling bifurcations
  • self-stability - orbit for which the stability-determining derivative is zero. This means that the max of the map and the point at which the max occurs are in the orbit.
  • type I intermittency - exhibited by inverse tangent bifurcation
  • Feigenbaum sequence - period-doubling path to chaos, keep increasing parameter until period is chaotic

\begin{center} \begin{tabular}{ | m{4cm} | m{4cm} | } \hline \multicolumn{2}{|c|}{3D Attractors}
\hline Type of Attractor & Sign of Exponents \ \hline Fixed Point & (-, -, -)\ Limit Cycle & (0, -, -) \ Torus & (0, 0, -)
Strange Attractor & (+, 0, -)
\hline \end{tabular} \end{center}

  • homoclinic orbit - connects unstable manifold of saddle point to its own stable manifold
    • e.g. trajectory that starts and ends at the same fixed point
  • manifolds are denoted by a W (ex. $W^S$ is the stable manifold)
  • heteroclinic orbit - connects unstable manifold of fp to stable manifold of another fp \

Conservative Systems

  • $F(x) = -\frac{dV}{dx}$ (by defn.)
  • $m\ddot{x}+\frac{dV}{dx}=0$, multiply by $\dot{x} \to \frac{d}{dt}[\frac{1}{2}m\dot{x}^2+V(x)]=0$
  • so total energy $E=\frac{1}{2}m\dot{x}^2+V(x)$
  • motion of pendulum: $\frac{d^2\theta}{dt^2}+\frac{g}{L}sin\theta=0$
  • nondimensionalize with $\omega=\sqrt{g/L}, \tau=\omega t \to \ddot{\theta}+sin\theta =0$
  • can multiply this by $\dot{\theta}$
  • $\omega$-limit $t \to \infty$
  • $\alpha$-limit $t \to -\infty$
  • libration - small orbit surrounding center
  • system: $\dot{\theta}=\nu$, $\dot{\nu} = -sin\theta$

Hamiltonian Dynamical System

  • $\dot{\underbar{x}}=\frac{\partial H}{\partial y}(\underbar{x},\underbar{y})$ , $\dot{\underbar{y}}=-\frac{\partial H}{\partial x}(\underbar{x},\underbar{y})$ for some function H called the Hamiltonian
  • we can only have centers (minima in the potential) and saddle points (maxima)
  • separatrix - orbit that separates trapped and passing orbits
  • Poincare Benderson Thm - can’t have chaos in a 2D system

Ref

  • $\frac{\partial}{\partial x}(f_1 * f_2 * f_3) = \frac{\partial f_1}{\partial x} f_2 f_3 + \frac{\partial f_2}{\partial x} f_1 f_3 + \frac{\partial f_3}{\partial x} f_1 f_2$
  • $e^{\mu it} = cos(\mu t)+ isin(\mu t)$
  • $x = A e^{(\lambda + i)t} + B e^{(\lambda - i)t} \implies x = (A’ sin(t) + B’ cos(t)) e^{\lambda t} $ If we have $\dot{x_1},\dot{x_2}$ then we can get $x_2(x_1) with \frac{dx_1}{dx_2} = \frac{\dot{x_1}}{\dot{x_2}}$

Benard Convection

  • The Navier-Stokes Equations
    1. Position
    2. Differential element in configuration space
    3. Time

Lorenz equations

  • Lorenz was studying hurricanes
  • ODEs for 3 coefficients in larger system (others are pretty small
    • use stream functions
  • Navier-Stokes PDEs - conservation of momentum - derive these equations as project?
    • no-flow steady-state $T(y) = T_B + (T_T-T_B)y$
  • Lorenz (ODE’s): x,y,z are coefficients in Fourier expansion, not dimensions \(\dot{x}(t) = -\sigma x+ \sigma y\) \(\dot{y}(t) = (\rho_o)x - y - xz\) \(\dot{z}(t) = -bz + xy\)
  • $p_o$ - proportional to size of system
  • b - aspect ration - related to height:width
  • Navier-Stokes
    • independent variables
      1. Position (vector)
      2. Time (scalar)
    • dependent variables
      1. Density (scalar)
  • fixed points
    • $b>0, \rho_0 > 0, \sigma>0$ - let b,$\sigma$ be fixed
    • fixed point $(\bar{x},\bar{y},\bar{z})$ is constant
    • equations
      • $\dot{\bar{x}}(t) = 0 = -\sigma \bar{x}+ \sigma \bar{y}$
      • $\dot{\bar{y}}(t) = 0 = (\rho_o) \bar{x} - \bar{y} - \bar{x}\bar{z}$
      • $\dot{z}(t) = 0 = -b\bar{z} + \bar{x}\bar{y}$
    • doing some algebra we get
      • $FP_0, \forall \rho_o$
        • $\bar{x} = 0$
        • $\bar{y} = 0$
        • $\bar{z} = 0$
      • $FP_+, \rho_o \geq 1$
        • $\bar{x} = +\sqrt{b (\rho_o-1)} $
        • $\bar{y} = +\sqrt{b (\rho_o-1)}$
        • $\bar{z} = \rho_o-1$
      • $FP_-, \rho_o \geq 1$
        • $\bar{x} = -\sqrt{b (\rho_o-1)} $
        • $\bar{y} = -\sqrt{b (\rho_o-1)}$
        • $\bar{z} = \rho_o-1$
  • these equations have symmetry
    • replace $(x(t),y(t),z(t))$ by $(-x(t),-y(t)),z(t)) \to$ we get back the original equations

$ M= \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \newline 3 & 4 & 5 & 6 & 7 \end{bmatrix} $

Linearization of Lorenz Equations about FPs $\bar{\underline{x}}$

  • $\frac{d}{dt}\delta x(t) = \frac{\partial f}{\partial x} _ \bar{\underline{x}}= -\sigma$
  • $\frac{d}{dt}\delta y(t) = \frac{\partial f_2}{\partial x} _ \bar{\underline{x}}*\delta x(t) + \frac{\partial f_2}{\partial y} _ \bar{\underline{x}}*\delta y(t) + \frac{\partial f_2}{\partial z} _ \bar{\underline{x}}*\delta z(t) +…$ (higher order derivatives)
    • $ = (r-\bar{z})\delta x(t) + -1\delta y(t) + -\bar{x}*\delta z(t) + \delta x \delta z$
  • do the same thing for $\delta z(t)$