There exists an embedding theorem by Takens [Takens 1981] which states that a strange attractor can be reconstructed from a single scalar time series. Phase space and attractor reconstruction can therefore be achieved by using a time delay embedding. This embedding technique creates N-dimensional state vectors by integrating measurements of the time series at time t and measurements of the series delayed by time t-tau:
x(t)=[x(t),x(t-tau),x(t-2tau),....,x(t-(N-1)tau)]
This animation presents a time delay embedding of the genus-1 Lorenz attractor in R^3, where the x variable is measured as the scalar time series. The animation loops over values of tau ranging from 0 to 40, effectively varying the amount of information shared between the delay coordinates.
If tau is too small, the delay coordinates will not be fully independent of each other. Information becomes redundant and the attractor does not completely unfold, resulting in an accumulation of points around the bisectrix of the embedding space. This simply means that there is not enough spacing between the delay coordinates for them to explore a substantial portion of phase space.
If tau is chosen to be too large, the delay coordinates become uncorrelated due to sensitivity in initial conditions. This sensitivity is caused by the presence of a positive lyapunov exponent which is also responsible for the rapid decay of the autocorrelation function. The resulting attractor in this case is completely uncorrelated and contains self-intersections which violate the uniqueness theorem.
Bifurcations of the Lorenz Attractor- (UNDER CONSTRUCTION)
One of the goals of dynamical systems theory is understand the spectrum of changes that a dynamical system can undergo when control parameters are varied. Some of the more well known changes involve bifurcations of fixed points and periodic orbits. Fixed point bifurcations are described by the theory of singularities while periodic orbit bifurcations consist of period-doubling and saddle-node bifurcations. These types of bifurcations are well known and have been studied extensively.
Strange attractors can also undergo bifurcations when control parameters are varied. Such bifurcations are referred to as perestroikas. Thanks to the recent work of T.D. Tsankov and R. Gilmore [2003], there now exist a set of tools by which to study these perestroikas. The newly developed tools classify strange attractors in three dimensions and provide insight into the basic stretching and squeezing mechanisms responsible for chaotic behavior.
The figure below shows the bifurcation diagram for the Lorenz attractor which is created using the non-symmetric image for simplicity. Chaotic regimes are identified by dense sets of points or "dark regions" which indicate that a trajectory encounters a large number of states. Within these chaotic regimes, unseen bifurcations can occur in the bounding tori. These are objects used to identify the large scale structure of a strange attractor. Bifurcations of bounding tori result in a fundamental change of the stretching and squeezing mechanisms causing the chaotic behavior. In the figure below, the bifurcations of the attractor are denoted by the vertical dashed lines.
The animations below show the evolution of the Lorenz attractor as the control parameters are scaled by rho. The evolution clearly shows the transition of the attractor from the well known genus-3 state (three holes in the bounding torus) to the less known genus-1 state (one hole in the bounding torus). To find out why strange attractors undergo bifurcations click here.
Lorenz- xy plane Lorenz- xz plane
The Rossler Poincare Section-
Poincare sections in the Rossler attractor are taken at 10 degree increments. The animation clearly shows the stretching, folding and squeezing mechanisms which generate chaotic behavior in strange attractors of genus-1.
Perestroika for the non-symmetric Lorenz attractor
