Lorenz system
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Unlike the logistic map or the Hénon map, the Lorenz equations represent a time continuous system. Edward N. Lorenz introduced these equations [Lorenz63] as an approximative model of the Rayleigh-Benard-experiment (see there for an explanation of the parameters)

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If two of the system parameters are set to the frequently used values   = 10 and b = 8/3 and the third one, R, is increased the system will exhibit the following route to chaos [Sparrow82], [Jackson85]:

For 0 < R < 1 the origin (0,0,0) is a globally stable fixed point. At R=1 this fixed point becomes unstable and two new stable fixed points C+ = ((b(R-1))1/2, (b(R-1))1/2, R-1) and C- = (-(b(R-1))1/2, -(b(R-1))1/2 , R-1) arise. The basins of attraction of these two fixed points are separated by the stable manifold of the now unstable fixed point at the origin. At R=Rho=13.926... this stable manifold contains a homoclinic orbit and there is a dramatic change of its topology for R>Rho . Then in the upper half space, Z>0, there is an infinity of dense leaves which are interwoven in a complicated way, see figures in [Jackson85]. At the same time a countable infinity of unstable periodic orbits arises. At R=24.74... both fixed points C+ and C- become unstable by a Hopf bifurcation and the  Lorenz attractor appears.

How to use the applet: A cube in the state space with the edges  -30<X<30, -30<Y<30, 0<Z<60 is displayed and can be rotated with the left mouse-button pressed. Choose the object to display in the listbox in the upper left corner: a piece of the trajectory following the attractor, one of several unstable periodic orbits, or a piece of trajectory starting closely to the homoclinic orbit at  R=Rho and finally converging to the stable fixed point C- . Choosing the option ‘manual’ you can set the system parameters on the right side (hint: try values of  R near the prominent values mentioned above). With the option ‘slow PC’ a shorter piece of trajectory is calculated in order to speed up turning the cube.

Not all the solutions of the Lorenz equations for R > 24.74... are chaotic, but there exist ranges of R with stable periodic orbits. In these windows period doublings can be observed. These bifurcations here occur for decreasing values of  R, finally leading to chaos at the lower border of the window. The table lists values of  R in three windows [Sparrow82], the symbol P denotes one circulation around C+ , N denotes one circulation around C-:

 

R

orbit

 

PPN-doubling

99.65

PPNPPN

100.5

PPN

 

PPNN-doubling

147.5

PPNNPPNN

160.0

PPNN

 

PN-doubling

216.2

PNPNPNPN

222.0

PNPN

260.0

PN

350.0

PN

 

Note that due to the symmetry of the Lorenz equations for each asymmetric orbit an asymmetric mirror image exists. In order to watch the listed orbits in the applet, check the option ‘manual’, enter the value of R and press ‘Enter’ several times (with the cursor in the R-field). The trajectory will converge to the orbit quickly. The larger R is the larger becomes the distance between the orbit and the displayed box, since the unstable fixed points which are roughly in the centers of the orbit’s wings are always on the plane Z=R-1.

Other three-dimensional examples of autonomous differential equation systems which exhibit chaotic attractors, too, are the Chua oscillator, the Roessler system or the Rabinovich-Fabrikant equations.

 

Literature

Besides the monography [Sparrow82] and the article [Jackson85] many informations about the Lorenz system which is introduced in [Lorenz63] are provided in [Guckenheimer83] or [Schuster88]. For which Rayleigh numbers do periodic solutions exist and for which do chaotic solutions exist? This question is investigated in [Morioka78] and [Shimizu78]. For the limit of large Rayleigh numbers [Shimuzu79] introduces a method to calculate analytically a constant of motion of the (dissipative!) system and to derive a simple unstable periodic orbit. In [Froyland84] the dependence of the largest Lyapunov exponent on the Raleigh number is investigated numerically. For the fixed value   = 10 [Alfsen85] analyzes the dependence of the system’s behaviour on the parameters b and R. [Broggi90] suggests to describe the Lorenz  system in terms of its hierarchically ordered unstable periodic orbits. Following the same ansatz [Franceschini93] determines the Hausdorff  dimension and the topologic entropy of the attractor. In [Leonow87] the area of the state space containing the attractor is calculated analytically. A proof of the existance of homoclinic and heteroclinic orbits of the fixed point at (0, 0, 0) is given in [Spreuer93].

 

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