For conservative systems of hyperbolic chaos is the Anosov dynamics, when the uniformly hyperbolic invariant set occupies entire compact phase space (for iterative maps), the surface of constant energy (for systems with continuous time).

The figure schematically shows a hinge mechanism [1,2] composed of three disks in a common plane with the centers at the vertices of an equilateral triangle. Each disk is able to rotate about its axis, and contain a hinges on the edge. These hinges P_1,2,3 three identical rods are attached to these hinges, and their opposite ends are joined together with one more movable hinge P₀. The instantaneous configuration of the system is determined by the angles of rotation of the disks, but only two of them are independent variables due to the imposed mechanical constraint.

Hunt and MacKay showed [2] that at appropriate selection of masses and sizes of the elements motion of such a mechanism by inertia with conservation of the kinetic energy corresponds to the geodesic flow on a manifold of negative curvature, which is the dynamics of Anosov. The simplest case for the analysis occurs if we set the radius r small, and the length of the rod R equal to the distance from the origin to the centers of the disks (taken a unit), and, in addition, suppose that the only massive elements are the disks. Conditions imposed by the mechanical constraint are expressed then in a very simple form:

Although in this case the curvature is negative not everywhere (there are a finite number of points where it is zero), but this is enough to ensure the dynamics of Anosov.

Taking unit moments of inertia of the disks, we write the equations of motion in the form

where the Lagrange multiplier A has to be determined taking into account the conditions of the algebraic holonomic mechanical constraint. Differentiating the relation twice over time and substituting the resulting the second derivatives from the equations of motion, we find explicitly the Lagrange multiplier, and the set of equations takes a closed form

The equation of constraint and the equality obtained as a result of its differentiation give us two integrals of motion of the system; the initial conditions must be consistent with these integrals.

The figure shows a typical trajectory in the configuration space obtained by numerical solution of the equations. The trajectory is continuous: at the intersection of an edge of the cubic cell in the figure it emanates from the opposite face of the cube because of the periodicity of the configuration space in three variables. It can be seen that the trajectory covers the surface topologically equivalent to a "pretzel with three holes." (This is known as the Schwartz surface.) The next figure shows the time dependences of the generalized coordinates and velocities on time for the energy W = 0.1. Doe to the cyclic nature of the generalized coordinates the continuous curves on the top diagram looks in such manner that at the intersection of the upper border the curve continues, starting from the lower border and vice versa. It is evident that they are of chaotic nature.

To characterize the observed chaos quantitatively, it is natural to use Lyapunov exponents. In our case, there are four Lyapunov exponent, consistent with the imposed mechanical constraints, and two of them are zero. Since the total sum of the exponents also vanishes, in the numerical calculations it is sufficient to evaluate only one the largest exponent; then other three are uniquely determined. For W = 0.1, which corresponds to the Figure, the Lyapunov exponents obtained numerically are L₁=0.157, L_2,3=0, L₄=-0.157. For other energies the Lyapunov exponents can also be easily determined from these data, since they behave in proportion to the square root of the energy.

Fundamental mathematical fact is that the uniformly hyperbolic invariant sets are rough, or structurally stable. In this regard, the hyperbolic nature of chaotic dynamics is preserved if we modify the system with introduction of dissipation and feedback, making it a self-oscillator. It has been demonstrated by numerical calculations in variants, when attractor corresponds to an invariant energy surface, as well as in models where the energy for the trajectories on the attractor fluctuates over time around some average value [3, 4]. Moreover, the hyperbolic attractor persists, when the mechanical constraint of three rotators constituting the system, is replaced by their interaction through appropriately chosen potential field, with the potential minimum corresponding to the initially assumed equation of the constraint.

The page is elaborated under support of RSF grant No 15-12-20035 in the Udmurt State University (Izhevsk).

REFERENCES

[1] Thurston W.P. and Weeks J.R. The mathematics of three-dimensional manifolds. Sci. Am., 1984, vol. 251, pp. 94–106.

[2] Hunt T.J., MacKay R.S. Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor. Nonlinearity. 2003. V. 16. P. 1499-1510.

[3] Kuznetsov S.P. Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories. Regular and Chaotic Dynamics. 2015. V. 20. No 6. P. 649–666.

[4] Kuznetsov S.P. Chaos in the system of three coupled rotators: from Anosov dynamics to hyperbolic attractor. Izv. Saratov. Univ. (N. S.), Ser. Fiz., 2015, vol. 15, no. 2, pp. 5–17