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Geometrical bounds for chaotic attractors
The algebraic structure of the integrals of motion (which exist for special values of the parameters) is used to find the equations of the semi-permeable surfaces. These semi-permeable surfaces exist for wide range of values of the parameters. For chaotic values of parameters, they allow us to give algebraic bounds to the spread of the chaotic attractor in the phase space.
Examples :
Lorenz attractor :
You see here the 'positive' wing of the Lorenz chaotic attractor. The trajectory wanders around that wing until it 'decides' to cross the x=0 plane and go wandering around the other equilibrium point.
All the essence of complexity in the system comes from the fact that we do not know when the trajectory 'decides' to jump to the other side of the plane x=0. Shown also is a semi-permeable surface that split the chaotic attractor in two. The x>0 side of this surface is placed between the x=0 plane and the 'positive' wing of the attractor.
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Because the surface is semi-permeable in the x>0 half space, once the trajectory has crossed this surface, it cannot go on wandering around the other wing and it is compelled to stay winding around here. We may consider such surfaces as a separation between the two wings of the attractor. Besides, surfaces (\ref{equa lorenz S}) give a bound in phase space for the period-1 limit cycles around each equilibrium point. |
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Rössler attractor :
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This system has a non-constant divergence and there are no known integrals of motion for it. Up to order 10, there is no algebraic integral of motion. Nevertheless, using a new method, we find a family of semi-permeable surfaces, but for the first time the equation is not algebraic. |
