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Winding Numbers

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Encyclopedia of Nonlinear Science

WINDING NUMBERS

If you wrap a rubber ring (rubber band) around a pencil, the intuitive idea of an integer invariant for the wrapping process arises. The number of oriented turns around the pencil is an integer, and it is independent of how tight or creased the ring is. The only way to change this integer is to wrap/unwrap one loop of the ring at the endpoints of the pencil. Using a rubber string, the number of turns could be some real number instead.

How does this fact connect with dynamics? Consider a three-dimensional (3-d) dynamical system having a torus as phase space. Let the hole in the torus represent the pencil and the rubber ring represent a closed trajectory within the torus. Small modifications to the trajectory will not alter the wrapping property.

The underlying feature in the above examples is the forbidden region given by the pencil or the hole of the torus. To further analyze its structure, decompose 3-d space as the product of the pencil direction times a perpendicular plane (equivalent to ). The role of the pencil is to identify a special point on that plane given by the projection along the pencil direction.

Similarly, consider a periodic orbit in as the forbidden region. Since the points in the periodic orbit are regular points in a small tube around the periodic orbit, the flow can be decomposed in a component parallel to the periodic orbit and a projected flow onto a perpendicular section. Hence, any other closed trajectory sufficiently near to the orbit will wind around it. A similar situation arises around a period-one orbit of a periodically forced flow in

The winding number characterizes the topological properties of “the plane minus one point.” Moreover, the topology of the plane minus n points gives a deeper characterization of the periodic orbit structure of 3-d dynamical systems admitting a Poincaré section.

Definition

Consider a simple continuous closed curve γ in the complex plane and a point The winding number n is defined as

(1)

We may regard our wrapped rubber ring as a suitable complex function γ of the unit circle, thus connecting our motivating idea with the formal definition (similarly for the second example if we project the torus along the direction perpendicular to the hole when seen as a disc). Where appropriate in the sequel, we will let z₀=0 for simplicity and recast γ as a map of the unit circle It follows from Equation (1) that n(γ, 0) is a real integer given by

(2)

which is called the degree of γ (Rotman, 1988, p. 50).

Applications

Homotopy classes of the circle. A loop is a continuous map g of the circle to itself such that g(0)=g(2π)=2π. Two loops α and β are homotopic if there exists a continuous map such that H(·, 0)=α, H(·, 1)=β and for each t, H(·, t) is a loop. In other words, two loops are homotopic if one can continuously deform one of them into the other, keeping it as a loop all the way throughout the deformation. The winding number classifies the homotopy classes of loops, namely, if a is homotopic to γ, then n(σ, 0)=n(γ, 0) (Rotman, 1988, p. 52).

Braids and periodic orbits. While the plane minus one point produces the winding number as a class invariant under homotopies, the homotopy classes of the plane with n special points requires a more elaborated structure which connects nicely with the dynamical properties of 3-d flows admitting a Poincaré section. In fact, periodic orbits of such flows can be regarded as imbeddings of the unit disk in phase space parametrized with time in units of 2π/T where T is the minimal period. On the Poincaré section, these special trajectories appear as an invariant set of n periodic points. The homotopy classes of loops on the plane with an invariant set of n points are classified by elements of the Braid group on n strands (Thurston, 1988; Hall, 1994; Natiello & Solari; 1994).

Linking number. In the same lines, given a pair of periodic orbits in phase space, we may think of the number of turns that one orbit does around the other when completing one excursion along itself. Such number is a link invariant which has a natural interpretation in terms of winding number, and it is called linking number (Uezu & Aizawa, 1982; Solari et al., 1996).

For 3-d flows admitting a Poincaré section, the periods of the orbits are commensurate and one may compute the average rotation per period of one orbit around another. This is called the relative rotation rate (Solari et al., 1996) and helps in understanding the orbit organization of such flows.

Poincaré index (PI). Consider a planar dynamical system and a simple closed counterclockwise curve C not passing through any equilibrium points. The PI k computed along C is defined as

(3)

(See below for a discussion of the PI in terms of complex analysis and winding number.)

In the context of planar dynamics, the PI of a node or a center is +1, of a hyperbolic saddle point is −1, and of a closed orbit is +1. Also the PI of a closed curve not containing fixed points is zero, and the PI of a closed curve equals the sum of the indices of the fixed points within (Guckenheimer & Holmes, 1983, p. 51).

Fixed point theorems. The degree of a map can be generalized to higher dimensions. In fact, this property (or the winding number when adequate) is a basic ingredient in the proof of Brouwer’s fixed point theorem. An interesting discussion of this fact along with some philosophical considerations can be found in www.mathpages.com.

Complex analysis. The computation of the winding number is a standard tool in the proof of the Fundamental Theorem of Algebra.

Also, let C be a closed contour on the complex plane not passing through any singularities or zeroes of the complex function f, which is analytic inside C except at most at a finite number of poles. Then

(4)

where N is the number of zeroes of f and P the number of poles inside C. This is called the Principle of the Argument in standard textbooks (Wunsch, 1994, p. 458).

This result is related to Equation 2 and to the PI. Concerning Equation (2), taking the special point of the plane to be the origin (or any point inside the unit circle), n counts how many turns γ performs around this point when running along the unit circle. Assume now that f has only one zero inside C with multiplicity n and no poles. Then f restricted to C is exactly the same as γ with a suitable choice of parametrization for C and Concerning the PI, let z=x+iy and F(z)=f(x, y)+ig(x, y), regarding the xy-plane as the complex plane. If the vector field (f, g) is continuous, F will not have poles within C and the Poincaré index reduces to the Principle of the Argument calculation for F.