Lorenz equations symmetry

There is an important symmetry in the Lorenz equations. If we replace... 

Equations (9.40) can be used to evaluate the refractive indices for the crystals of any polymer with orthorhombic symmetry, provided that the co-ordinates of all the atoms within the crystal structure are known, so that all the bond orientations can be calculated. The value of an for the crystal is substituted into the Lorentz Lorenz equation (9.9) in place of a and the value of n obtained is assumed to be equal to the refractive index for light polarised with the electric vector parallel to OXi. Similar calculations are performed to calculate the indices for light polarised parallel to OX2 and... 

In general, the bifurcation of a homoclinic butterfly is of codimension two. However, the Lorenz equation is symmetric with respect to the transformation (x y z) <-)> (—X, —y z). In such systems the existence of one homoclinic loop automatically implies the existence of another loop which is a symmetrical image of the other one. Therefore, the homoclinic butterfly is a codimension-one phenomenon for the systems with symmetry. 

In the Lorenz model, the saddle value is positive for the parameter values corresponding to the homoclinic butterfly. Therefore, upon splitting the two symmetric homoclinic loops outward, a saddle periodic orbit is born from each loop. Furthermore, the stable manifold of one of the periodic orbits intersects transversely the unstable manifold of the other one, and vice versa. The occurrence of such an intersection leads, in turn, to the existence of a hyperbolic limit set containing transverse homoclinic orbits, infinitely many saddle periodic orbits and so on [1]. In the case of a homoclinic butterfly without symmetry there is also a region in the parameter space for which such a rough limit set exists [1, 141, 149]. However, since this limit set is unstable, it cannot be directly associated with the strange attractor — a mathematical image of dynamical chaos in the Lorenz equation. 

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