- Normal forms of Hopf bifurcations
- important figs
- Systems of Linear ODEs
- Discrete Nonlinear Dynamical Systems
- Conservative Systems
- Ref
- pitchfork:
- 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
- phase space - has coordinates $x_1,…,x_n$
- phase portrait - variable x-axis, derivative y-axis
- 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 $trainfty$
- 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
- 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 = $frac{dRe(alpha)}{dlambda}$
- 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.)
- $mddot{x}+frac{dV}{dx}=0$, multiply by $dot{x} to frac{d}{dt}[frac{1}{2}mdot{x}^2+V(x)]=0$
- so total energy $E=frac{1}{2}mdot{x}^2+V(x)$
- motion of pendulum: $frac{d^2theta}{dt^2}+frac{g}{L}sintheta=0$
- nondimensionalize with $omega=sqrt{g/L}, tau=omega t to ddot{theta}+sintheta =0$
- can multiply this by $dot{theta}$
- $omega$-limit $t to infty$
- $alpha$-limit $t to -infty$
- libration - small orbit surrounding center
- system: $dot{theta}=nu$, $dot{nu} = -sintheta$
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 $x_2(x_1) with frac{dx_1}{dx_2} = frac{dot{x_1}}{dot{x_2}}$
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