lorenz.m contains the instructions and algorithm to find and compute the aproximate inestable periodic orbit of the Lorenz system for r = 18 and r = 21 (To compute a different r, define r and a good starting point x0).
Complementary functions:
- P.m
Given the vectors X and T that contain the corresponding points and times obtained with ode45 of an orbit, a starting point x and a plane Σ:{z = h}, it finds the exact point in Σ that corresponds to compute a return from the starting point x and its orbital period.
- Q.m
Given a starting point x and a plane Σ:{z = h}, it finds the euclidean distance between the starting point and the one in Σ obtained in P.m.
The main script lorenz.m is divided into different parts:
- It finds for which r the eigenvalues of Df corresponding to the Lorenz system go from being real to complex.
- It computes the pitchfork bifurcation in r = 1.
- Finds the inestable periodic orbit for r = 18.
- Finds the inestable periodic orbit for r = 21.
Figures legend:
- Orbit values of x, y and z along time for x0 defined above for r = 18.
- 3DPlot of the orbit starting at x0 for r=18. The plane Σ is computed in blue (r = 18).
- XY representation of the inestable periodic orbit (r = 18 and r = 21)
- Orbit values of x, y and z along time for a point in the inestable periodic orbit (r = 18).
- 3DPlot of the inestable periodic orbit (green) and the orbit starting at x1 defined above (r = 18).
- Orbit values of x, y and z along time for x0 defined above for r = 21.
- 3DPlot of the orbit starting at x0 for r=18. The plane Σ is computed in blue (r = 21).
- Orbit values of x, y and z along time for a point in the inestable periodic orbit (r = 21).
- 3DPlot of the inestable periodic orbit (green) and the orbit starting at x1 defined above (r = 21).
- Points used for polynomial interpolation in P function.
- Representation of the Pitchfork bifurcacion.
- Plot of the Hopf subcritical bifurcation (r aprox 24.73...) -> x coordinate
- Plot of the Hopf subcritical bifurcation (r aprox 24.73...) -> y coordinate