x-nullcline: The
set of points where the vector field changes its horizontal direction is called
x-nullcline, defined by equation f(x,y)=0. At such point x neither increases
nor decreases because d(x)/dx = 0. X-nullcline divides the plane into 2 planes
where x moves in opposite direction.
y-nullcline: The set of points where the vector field changes its vertical directions is called y-nullcline, defined by equation g(x,y)=0. At such point y neither decreases nor increases. These 2 nullclines partitions the phase plane into 4 different regions
- X and y increase
- X decreases y increases
- X and y decreases
- X increases and y decreases
Equilibrium point
: intersection of nullclines is an equilibrium point.
Trajectories : A solution to the Two dimensional system is a curve (x(t),y(t)) on the
phase plane which is tangent to the vector field is called a trajectory or orbit.
Periodic Orbits: A trajectory that forms a closed loop s called periodic orbit, periodic trajectory or limit
cycle. Periodic orbits cannot occur in one dimensional system. If the
initial point is on periodic orbit the solution remains on the orbit forever
and the system exhibits periodic behavior
x(t)= x(t+T) and y(t)= y(t+T)
The minimal value of T for which above holds is called the
period or the periodic orbit.
Asymptotically stable
periodic orbit: A periodic order is said to be asymptotically stable if any
trajectory with the initial point sufficiently near the orbit approaches to the
orbit as t-> infinity.
Limit Cycle Attractor:
Asymptotically stable periodic orbits are often called limit cycle attractor,
since they attract all nearby trajectories.
Limit Cycle Repeller: The unstable periodic orbit are often called
a repeller, since it repels all near
by trajectories
Note: There is always at least one equilibrium inside any periodic orbit
on a plane.
Equilibria: Equlibria is a point where
f(x,y)=0 and g(x,y)=0. Geometrically equlibria are intersection of nullclines.
If the initial point (x0,y0) is an equilibrium, then trajectory
stays at equilibrium. If the initial point is near the equilibrium, then the trajectory
may converge to or diverge from the equilibrium depending on its stability.
Stability
Stability of Equilibrium : An equilibrium is stable if any trajectory starting sufficiently close
to the equilibrium remains near it for all t>=0.
Asymptotically stable equilibrium: If any trajectories starting
sufficiently near the equilibrium converge to the equilibrium as t->
infinity.
Exponentially stable: If
convergence rate is exponentially fast then equilibrium is said to be
exponentially stable.
Note: stability doesn’t imply asymptotically stable
Neutrally Stable: Some trajectories
neither converge nor diverge from equilibrium
Unstable Equilibrium: An equilibrium is called unstable, if it is
not stable i.e if all nearby trajectories diverge from the equilibrium.
