Lorenz Equations

1
Lorenz Equations
3 state variables ? 3dimension system 3
parameters seemingly simple equations note
only 2 nonlinear terms but incredibly rich
nonlinear behavior in the system
2
fixed points
0 lt r lt 1
(x,y,z)1 (0,0,0) (x,y,z)1 (0,0,0) (x,
y,z)2 (x,y,z)3
r 1
C
C-
the origin is always a fixed point
The existence of C and C- depends only on r, not
b or ?
3
stability of the origin
saddle node
stable node
4
Example for ? 1 r 4
r gt 1 saddle node at the origin
y
unstable manifold
?1 1, v1 (1,2,0)
?z -b, vz (0,0,z)
stable manifold
x
stable manifold
?2 -3, v2 (1,-2,0)
z
b does not affect the stabilty. b only affects
the rate of decay in the z eigendirection
5
Summary of Bifurcation at r 1
0lt r lt 1 r gt 1 stable node saddle
node new fixed point, C new fixed
point, C-
The origin looses stability and 2 new symmetric
fixed points emerge. What type of bifurcation
does this sound like? What is the classification
of the new fixed fixed points?
6
Plotting the location of the fixed points as a
function of r
x
example for b1 other b values would
look qualitatively the same
origin stable
origin unstable
r
Stability of the symmetric fixed points?
Looking like a supercritical pitchfork
7
stability of C and C-
need to find eigenvalues to classify
8
eigenvalues of a 3x3 matrix
in general eigenvalues are found by solving
the characteristic equation
for a 3x3 matrix
result is the characteristic polynomial with 3
roots ?1, ?2, ?3
9
Remember for 2x2 2D systems (I.e. 2 state
variables)
Characteristic equation
Characteristic polynomial
Tip can use mathematica to find a characteristic
polynomial of a matrix
2nd order polynomial for a 2x2 matrix The
eigenvalues are the roots of the characteristic
polynomial Therefore 2 eigenvalues for a 2x2
matrix of a 2 dimension system
10
eigenvalues of a 3x3 matrix
In general The determinent of a 3x3 matrix can
be found by hand by
So the characteristic equation becomes
11
Det of A
Trace of A
Characteristic Polynomial
12
Homework problem
Due Monday Problem 9.2.1 Parameter value where
the Hopf bifurcation occurs
13
C and C- are stable for r gt 1 but less than the
next critical parameter value
2D unstable manifold
unstable limit cycle
1D stable manifold
C is locally stable because all trajectories
near stay near and approach C as time goes to
infinity
14
Supercritical pitchfork at r1
x
r