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 (x*,y*,z*) 1 (0,0,0) (x*,y*,z*) 2 (x*,y*,z*) 3 0 < 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 

stability of the origin stable node saddle node

y x z r > 1 saddle node at the origin z = -b, v z = (0,0,z) 1 = 1, v 1 = (1,2,0) Example for  = 1 r = 4 2 = -3, v 2 = (1,-2,0) unstable manifold stable manifold b does not affect the stabilty. b only affects the rate of decay in the z eigendirection

Summary of Bifurcation at r = stable nodesaddle 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?

origin stableorigin unstable Stability of the symmetric fixed points? x r example for b=1 other b values would look qualitatively the same Plotting the location of the fixed points as a function of r Looking like a supercritical pitchfork

stability of C+ and C- need to find eigenvalues to classify

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

Remember for 2x2 2D systems (I.e. 2 state variables) Tip: can use mathematica to find a characteristic polynomial of a matrix Characteristic equation Characteristic polynomial 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

eigenvalues of a 3x3 matrix In general: The determinent of a 3x3 matrix can be found by hand by : So the characteristic equation becomes:

Characteristic Polynomial Trace of A Det of A

Homework problem Due Monday Problem Parameter value where the Hopf bifurcation occurs

C+ and C- are stable for r > 1 but less than the next critical parameter value unstable 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

Supercritical pitchfork at r=1 x* r