lorenz equations 3 state variables 3dimension system 3 parameters seemingly simple equations note...
TRANSCRIPT
![Page 1: Lorenz Equations 3 state variables 3dimension system 3 parameters seemingly simple equations note only 2 nonlinear terms but incredibly rich nonlinear](https://reader036.vdocument.in/reader036/viewer/2022083008/56649ea25503460f94ba6869/html5/thumbnails/1.jpg)
Lorenz Equations
dx
dt=σ(y−x)
dydt
=rx−y−xz
dzdt
=xy−bz
3 state variables 3dimension system
3 parameters
seemingly simple equations
note only 2 nonlinear terms
but incredibly rich nonlinearbehavior in the system
σ > 0
r > 0
b > 0
![Page 2: Lorenz Equations 3 state variables 3dimension system 3 parameters seemingly simple equations note only 2 nonlinear terms but incredibly rich nonlinear](https://reader036.vdocument.in/reader036/viewer/2022083008/56649ea25503460f94ba6869/html5/thumbnails/2.jpg)
fixed points
(x*,y*,z*)1 (0,0,0)
(x*,y*,z*)1 (0,0,0)
(x*,y*,z*)2
(x*,y*,z*)3
0 < r < 1
+ b(r −1),+ b(r −1),r −1( )
− b(r −1),− b(r −1),r −1( )
r ≥ 1
C+
C-
the origin is always a fixed point
The existence of C+ and C- depends only on r, not b or σ
![Page 3: Lorenz Equations 3 state variables 3dimension system 3 parameters seemingly simple equations note only 2 nonlinear terms but incredibly rich nonlinear](https://reader036.vdocument.in/reader036/viewer/2022083008/56649ea25503460f94ba6869/html5/thumbnails/3.jpg)
stability of the origin
Δ =σ (1− r)
τ = −σ −1
τ 2 − 4Δ = (σ −1)2 + 4σ r
δ fδ x
δ f
δ y
δg
δ x
δg
δ y
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟(0,0,0)
=−σ σ
r −1
⎛⎝⎜
⎞⎠⎟
when r >1⇒ Δ < 0 ⇒
τ < 0 when σ > −1 ⇒ always true
τ < 0 for all parameter values
τ 2 − 4Δ > 0 for all parameter values
when 0< r <1⇒ Δ > 0 ⇒ have to look at τ and τ 2 −4Δ
stable node
saddle node
![Page 4: Lorenz Equations 3 state variables 3dimension system 3 parameters seemingly simple equations note only 2 nonlinear terms but incredibly rich nonlinear](https://reader036.vdocument.in/reader036/viewer/2022083008/56649ea25503460f94ba6869/html5/thumbnails/4.jpg)
y
x
z
r > 1
saddle nodeat the origin
z= -b, vz = (0,0,z)
1= 1, v1 = (1,2,0)
Example for σ = 1r = 4
2= -3, v2 = (1,-2,0)
unstable manifold
stable manifold
stable manifold
b does not affect the stabilty.b only affects the rate of decay in the z eigendirection
![Page 5: Lorenz Equations 3 state variables 3dimension system 3 parameters seemingly simple equations note only 2 nonlinear terms but incredibly rich nonlinear](https://reader036.vdocument.in/reader036/viewer/2022083008/56649ea25503460f94ba6869/html5/thumbnails/5.jpg)
Summary of Bifurcation at r = 1
0< r < 1 r > 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?
![Page 6: Lorenz Equations 3 state variables 3dimension system 3 parameters seemingly simple equations note only 2 nonlinear terms but incredibly rich nonlinear](https://reader036.vdocument.in/reader036/viewer/2022083008/56649ea25503460f94ba6869/html5/thumbnails/6.jpg)
origin stable origin unstable
Stability of the symmetric fixed points?
x
r
example for b=1
other b values would lookqualitatively the same
Plotting the location of the fixed points as a function of r
Looking like a supercritical pitchfork
![Page 7: Lorenz Equations 3 state variables 3dimension system 3 parameters seemingly simple equations note only 2 nonlinear terms but incredibly rich nonlinear](https://reader036.vdocument.in/reader036/viewer/2022083008/56649ea25503460f94ba6869/html5/thumbnails/7.jpg)
stability of C+ and C-
δ fδ x
δ f
δ y
δ f
δ z
δ g
δ x
δg
δ y
δg
δ z
δh
δ x
δh
δ y
δh
δ z
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟C+
=
−σ σ 0
r − z −1 −x
y x −b
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟C+
=
−σ σ 0
1 −1 − b(r − 1)
b(r − 1) b(r − 1) −b
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
C+ =(x* ,y* ,z* ) =( b(r −1), b(r −1),r −1)
need to findeigenvalues to classify
![Page 8: Lorenz Equations 3 state variables 3dimension system 3 parameters seemingly simple equations note only 2 nonlinear terms but incredibly rich nonlinear](https://reader036.vdocument.in/reader036/viewer/2022083008/56649ea25503460f94ba6869/html5/thumbnails/8.jpg)
eigenvalues of a 3x3 matrix
det(A−I ) =0
det(A−I ) =a11 − a12 a13
a21 a22 − a23
a31 a32 a33 −=0
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
![Page 9: Lorenz Equations 3 state variables 3dimension system 3 parameters seemingly simple equations note only 2 nonlinear terms but incredibly rich nonlinear](https://reader036.vdocument.in/reader036/viewer/2022083008/56649ea25503460f94ba6869/html5/thumbnails/9.jpg)
Remember for 2x22D systems (I.e. 2 state variables)
Tip: can use mathematica to find a characteristic polynomial of a matrix
A =a bc d
⎛⎝⎜
⎞⎠⎟
det(A−I ) =deta− b
c d−⎛⎝⎜
⎞⎠⎟=0Characteristic equation
Characteristic polynomial (a−)(d−)−bc=0
2 −(a+d) + ad−bd=0
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
![Page 10: Lorenz Equations 3 state variables 3dimension system 3 parameters seemingly simple equations note only 2 nonlinear terms but incredibly rich nonlinear](https://reader036.vdocument.in/reader036/viewer/2022083008/56649ea25503460f94ba6869/html5/thumbnails/10.jpg)
eigenvalues of a 3x3 matrix
a11 − a12 a13
a21 a22 − a23
a31 a32 a33 −=0
det
a11 a12 a13
a21 a22 a23
a31 a32 a33
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟=
a11 a12 a13
a21 a22 a23
a31 a32 a33
=a11
a22 a23
a32 a33
−a12
a21 a23
a31 a33
+ a13
a21 a22
a31 a32
In general: The determinent of a 3x3 matrix can be found by hand by :
(a11 −)a22 − a23
a32 a33 −−a12
a21 a23
a31 a33 −+ a13
a21 a22 −a31 a32
=0
So the characteristic equation becomes:
![Page 11: Lorenz Equations 3 state variables 3dimension system 3 parameters seemingly simple equations note only 2 nonlinear terms but incredibly rich nonlinear](https://reader036.vdocument.in/reader036/viewer/2022083008/56649ea25503460f94ba6869/html5/thumbnails/11.jpg)
− 3 + (a11 + a22 + a33)λ 2 + (a12a21 − a11a22 + a13a31 + a23a32 − a11a33 − a22a33)λ
+(-a13 a22 a31 + a12 a23 a31 + a13 a21 a32 - a11 a23 a32 - a12 a21 a33 + a11 a22 a33) = 0
Characteristic Polynomial
− 3 + Tr(A)λ 2 + (a12a21 − a11a22 + a13a31 + a23a32 − a11a33 − a22a33)λ + det(A) = 0
Trace of ADet of A
![Page 12: Lorenz Equations 3 state variables 3dimension system 3 parameters seemingly simple equations note only 2 nonlinear terms but incredibly rich nonlinear](https://reader036.vdocument.in/reader036/viewer/2022083008/56649ea25503460f94ba6869/html5/thumbnails/12.jpg)
Homework problem
Due Monday
Problem 9.2.1
Parameter value where the Hopf bifurcation occurs
![Page 13: Lorenz Equations 3 state variables 3dimension system 3 parameters seemingly simple equations note only 2 nonlinear terms but incredibly rich nonlinear](https://reader036.vdocument.in/reader036/viewer/2022083008/56649ea25503460f94ba6869/html5/thumbnails/13.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
C+ and C- are stable for r > 1 but less than the next critical parameter value
1< r < rH
where rH =σ(σ +b+ 3)σ −b−1
and σ −b−1> 0unstable limit cycle
1D stable manifold
2D unstable manifold
C+ is locally stable because all trajectories near stay near and approach C+ as time goes to infinity
![Page 14: Lorenz Equations 3 state variables 3dimension system 3 parameters seemingly simple equations note only 2 nonlinear terms but incredibly rich nonlinear](https://reader036.vdocument.in/reader036/viewer/2022083008/56649ea25503460f94ba6869/html5/thumbnails/14.jpg)
Supercritical pitchfork at r=1
x*
r