Disorder & Chaos Test #3
1D flows—meaning of derivatives - ANS-growth/decay models
-Dx/dt = +/-kx
-Dx/dt=ax-bx2
-1D flows can only have fixed points bc once you start in one direction you cannot back up as
you would cross yourself
-Only one state variable
-Classic cases are modeling growth and decay systems
-How something changes with time is known as a rate derivative whereas how something
changes with position is known as a gradient
How to identify fixed points and whether they are attracting or repelling - ANS-if the slope on
either side of the fixed point is below the x axis, the arrow goes to the left, if it is above the
x-axis it goes to the right
-if both arrows are pointing toward the fixed point it is an attractor
-if both arrows are pointing away from the fixed point it is a repeller
Dissipative, conservative, & driven systems - ANSDissipative - losing energy J <1
Conservate - no change in energy J = 1
Driven systems - gaining energy J >1
Properties of attractors - ANS-1D: length, fixed point, equilibrium
-2D: area, torus or a limit cycle, periodical
-3D: volume, strange attractor, chaotic, sensitivity to initial conditions
-Zero volume in the embedding phase (Liouville Theorem)
-Invariant under the action of the flow (Becomes visible only when orbits or trajectories go to it)
-Contained in the region of phase space known as a Basin of Attraction which can be defined as
the set of all initial conditions whose trajectories/orbits lead to the attractor. If the Basin of
Attraction has no volume, it is a repeller
How to identify if there is a fixed point at +/- infinity OR finite values - ANS-Dx/dt=ax-bx2
-A and b are parameters each >0
-ON fixed point At x=x*, dx/dt=0 so.. When derivative goes to 0 trajectory has to stop so set it to
zero Has 2 fixed points x*=0 and x*=a/b
-NOT on fixed points Where dx/dt is >0, x will get larger and move to the right, if dx/dt<0 it will
move to the left
Non-crossing rule - ANS-a trajectory cannot cross itself without going into a cycle
-every point on the plane is a state of the system and on a flow you cannot return to the same
state and then do something else, its deterministic
1D flows—meaning of derivatives - ANS-growth/decay models
-Dx/dt = +/-kx
-Dx/dt=ax-bx2
-1D flows can only have fixed points bc once you start in one direction you cannot back up as
you would cross yourself
-Only one state variable
-Classic cases are modeling growth and decay systems
-How something changes with time is known as a rate derivative whereas how something
changes with position is known as a gradient
How to identify fixed points and whether they are attracting or repelling - ANS-if the slope on
either side of the fixed point is below the x axis, the arrow goes to the left, if it is above the
x-axis it goes to the right
-if both arrows are pointing toward the fixed point it is an attractor
-if both arrows are pointing away from the fixed point it is a repeller
Dissipative, conservative, & driven systems - ANSDissipative - losing energy J <1
Conservate - no change in energy J = 1
Driven systems - gaining energy J >1
Properties of attractors - ANS-1D: length, fixed point, equilibrium
-2D: area, torus or a limit cycle, periodical
-3D: volume, strange attractor, chaotic, sensitivity to initial conditions
-Zero volume in the embedding phase (Liouville Theorem)
-Invariant under the action of the flow (Becomes visible only when orbits or trajectories go to it)
-Contained in the region of phase space known as a Basin of Attraction which can be defined as
the set of all initial conditions whose trajectories/orbits lead to the attractor. If the Basin of
Attraction has no volume, it is a repeller
How to identify if there is a fixed point at +/- infinity OR finite values - ANS-Dx/dt=ax-bx2
-A and b are parameters each >0
-ON fixed point At x=x*, dx/dt=0 so.. When derivative goes to 0 trajectory has to stop so set it to
zero Has 2 fixed points x*=0 and x*=a/b
-NOT on fixed points Where dx/dt is >0, x will get larger and move to the right, if dx/dt<0 it will
move to the left
Non-crossing rule - ANS-a trajectory cannot cross itself without going into a cycle
-every point on the plane is a state of the system and on a flow you cannot return to the same
state and then do something else, its deterministic
