Lorenz This demo has the following directories and subdirectories: - Basic: Computes the basic bifurcation diagram with the zero solution family, the two legs of the bifurcating nonzero stationary family, and the periodic solution families that bifurcate from Hopf bifurcations along the nonzero family, and which end in homoclinic orbits. - Period-Doubling: This demo shows how to start the continuation of a periodic solution family from numerical data. Such data are normally obtained by long-time integration with an initial value solver followed by the extraction of a reasonably accurate periodic orbit. Starting from such data, this demo locates successive period-doubling bifurcations, and switches branches at these. - Heteroclinics This demo follows orbits in the unstable manifold of one of the non-zero equilibria, and detects heteroclinic connections from this equilibrium to the origin. - Manifolds/Origin/Fixed-Length: Computes the stable manifold of the origin (also known as the "Lorenz-manifold"), by using orbit continuation to cover the manifold with orbits of fixed length. - Manifolds/Origin/Sphere: Computes the Lorenz-manifold, where the end points of the orbits lie mostly on the surface of a sphere, but regularly retract into the interior when a heteroclinic connection from the origin to one of the nonzero equilibria is encountered inside the sphere. - Manifolds/Origin/Isolas: This demo is similar to Manifolds/Origin/Sphere, but with detection of manifold orbits that correspond to "tangencies". Subsequently these tangencies are continued in rho to a new target value of rho. Finally, the manifold orbits at the new target value of rho are continued, keeping their end points on a sphere, and rho fixed. The demo shows that this can lead to the detection of "isolas" on the surface of the sphere. - Manifolds/Origin/Plane: This demo is also similar to Manifolds/Origin/Sphere. The difference is that during the manifold orbit continuation, the demo locates and saves orbits whose end point lies in the plane z=rho-1. Moreover, in an additional computation, such orbits are continued with the end point constrained to remain in this plane. - Manifolds/Orbits/Rho21.0: Computes the stable manifold of a primary periodic orbit at rho=21.0. Similar to the demo in Manifolds/Origin/Sphere, the end point of the manifold orbits is kept (mostly) on a sphere. - Manifolds/Orbits/Rho24.3 Computes the stable manifold of a primary periodic orbit at rho=24.3. Similar to the demo in Manifolds/Origin/Sphere, the end point of the manifold orbits is kept (mostly) on a sphere, To reset all directories and subdirectories to their original form, type "auto clean_all.auto". This assumes that no changes have been made other than running the AUTO python script files. References: E. J. Doedel, B. Krauskopf, H. M. Osinga, in preparation, 2013. E. J. Doedel, B. Krauskopf, H. M. Osinga, Global invariant manifolds in the transition to preturbulence in the Lorenz system, Indagationes Mathematicae 22(3-4): 222-240, 2011. P. Aguirre, E. J. Doedel, B. Krauskopf, H. M. Osinga, Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields, Discrete and Continuous Dynamical Systems, Vol. 29 #4, 2011, 1309-1344. E. J. Doedel, B. Krauskopf, H. M. Osinga, Global bifurcations of the Lorenz manifold, Nonlinearity 19, 2006, 2947-2972.