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.