chaos

# 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 $\delta z(t)$