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Lorenz Equations

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. fixed points. 0 < r < 1. (x*,y*,z*) 1 (0,0,0) (x*,y*,z*) 1 (0,0,0) (x*,y*,z*) 2 (x*,y*,z*) 3 .

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Lorenz Equations

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  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 < r < 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 > 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 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?

  6. Plotting the location of the fixed points as a function of r x example for b=1 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 > 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 r=1 x* r

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