Bicurcation
The bifurcation on the standard form mu - x^2
Saddle node
The closed orbit
Limit cycle
The fixpoint with pure real eigenvalues of opposite sign
Saddle point
The first definition of chaos in a system.
Sensitive dependence on initial conditions.
The conditions for a Liapunov function
V(x0) = 0
V(x) > 0 for x0 =/ x
V'(x) < 0 for x0 =/ x
The bifurcation on the standard form x mu - x^2
Transcritical
The necessary condition for a system to be a gradient system.
partial xdot / partial y = partial ydot / partial x
The stability of a fixpoint in the map p where |p'(x)|<=1
stable
The name of lambda
The Lyapunov coefficient
The formular for eigenvalues as a function of trace and determinant
( tau +/- sqrt( tau^2 - 4 delta) ) / 2
The bifurcation with this standard form:
mu x - x^3
Super critical pitchfork bifurcation
Can a fixpoint exist on a limit cycle?
No
The critical points that has no eigenvalues with zero real part.
Hyperbolic fix points
Can chaos happen in a system of dimension 1? (in continuous case)
No!
The equations for a Hamiltonian system.
xdot = partial H / partial y
ydot = - partial H / partial x
The bifurcation with Liapunov number < 0
super critical hopf bifurcation
Cant a limit cycle happen in a 1D system?
no
The line fixpoints are found on given a map.
The behaviour of orbits in chaotic systems.
aperiodic
What rotation direction (in relation the the clock) gets an index of 1?
Counter clockwise.
The system that undergoes a Hopf bifurcation at (0,0) with mu = 0
xdot = mu x - y + p(x,y)
ydot = x + mu y + q(x,y)
The boundary separating to modes of behaviour
The separatrix
The index of a limit cycle
-1
The formula for sensitivity in initial conditions
|delta_n| = |delta_0| exp(lambda n)
The shift to polar coordinates on matrix form
[cos(Theta), sin(Theta) ;
-1/r sin(Theta), 1/r cos(Theta)]
*
[f(r cos(Theta),r sin(Theta) ;
f(r cos(Theta),r sin(Theta) ]