Differential Equations on the two dimensional torus

October 26, 2011 9a-1<strong>Differential</strong> <strong>Equations</strong> on the two dimensionaltorusWe will now study some simple differential equations on a two-dimensionaltorus. These will provide interesting minimal sets.The 2−torus T 2 is the product S 1 × S 1 of two circles.Writing S 1 aswe getS 1 = {z ∈ C : | z | = 1},S 1 × S 1 = {(z 1 , z 2 ) ∈ C 2 : | z 1 | = | z 2 | = 1.There is an alternative viewpoint which is useful.Consider the quotient group R/Z; i.e., the set of equivalence classes in Runder the equivalence relation x ∼ y iff x − y ∈ Z. There is a natural mapπ : R → Z assigning to each real number x its equivalence class [x].Notice that the map φ : t → e 2πit is a surjective map from R onto S 1such that each pre-image φ −1 (z) is an equivalence class in R/Z. Thus, themap ψ([x]) = φ(x) gives a well-defined bijection from R/Z to S 1 .We can use the map ψ to define a topology on R/Z, so we have notionsof continuity, etc.Note that we can also think of the circle as a closed interval [a, b] with itsendpoints identified.We can now apply these ideas to the product T 2 = S 1 × S 1 . There is abijection from R/Z × R/Z onto T 2 . This defines a topology on T 2 . There isanother natural bijection between R/Z × R/Z and R 2 /Z 2 , and we can thinkof this latter quotient set as T 2 as well. We will use all of these objects.In particular, we think of T 2 as either S 1 × S 1 or R 2 /Z 2 .Now we wish to define differential equations on T 2 . We will use therepresentation R 2 /Z 2 .For this purpose, let f(x, y), g(x, y) be two real-valued functions of tworeal variables such that they are periodic of period 1 in both variables. Thatis,f(x + 1, y + 1) = f(x, y), g(x + 1, y + 1) = g(x, y), ∀x, y.

October 26, 2011 9a-2The planar vector field X(x, y) = (f(x, y), g(x, y)) is a map from R 2 →R 2 . The periodicity conditions imply that it induces a well-defined map fromR 2 /Z 2 to R 2 . We call this a vector field on the torus T 2 . The associatedsystem of differential equationsẋ = f(x, y)ẏ = g(x, y)is called a differential equation on T 2 .The solutions (or orbits) are obtained by taking the orbits in R 2 andprojecting them down to T 2 .Let us take a simple example.Consider the constant vector field in R 2X(x, y) = ω 1 ∂ x + ω 2 ∂ y ,where ω 1 , ω 2 are positive real numbers.The associated system of differential equations isẋ = ω 1ẏ = ω 2(1)The periodicity conditions are obviously satisfied, so we get a differentialequation on T 2 . This is called the constant or linear vector field on T 2 .Let’s us consider its orbits.The solutions to (1) in R 2 are the linesx(t) = ω 1 t + c 1 , y(t) = ω 2 t + c 2 .Each such line has slope ω 2ω 1and passes through the point (c 1 , c 2 ).On the torus T 2 we simply take (x(t), y(t)) mod 1.We have two main cases:1. 2ω 1is rational. Write it as 2ω 1= p in lowest terms with p, q positiveqintegers.Consider the orbit γ 0 in R 2 through (0, 0) (i.e., c 1 = c 2 = 0).The solution in R 2 has the formx(t) = ω 1 t, y(t) =( pq)ω 1 t

October 26, 2011 9a-3Let π : R 2 → T 2 denote the canonical projection.When t = q/ω 1 , we have x(t) = q, y(t) = p, so the projection π(γ 0 ) = γinto T 2 is a closed curve (topological circle) in T 2 .Since all the solutions are vertical tranlates of γ 0 we have that allsolutions in T 2 are periodic of the same period.2. ω 2ω 1is irrational.This case is more interesting. We will show that every orbit in T 2 isdense in T 2 . Hence, all of T 2 is a minimal set.Let us show that the orbit through (0, 0) is dense in T 2 . The analogousstatement for other orbits is similar.Consider the circle S 1 = π({x = 0}).There is a first return map F : S 1 → S 1 (also called Poincaré map)obtained by taking a point (0, y) and taking the point (1, F (y)) wherethe orbit through (0, y) hits the line {x = 1}.Let us compute F .The orbit through (0, y) is {(ω 1 t, ω 2 t + y), t ∈ R}.We getHence,ω 1 t = 1ω 2 t + y = F (y).t = 1/ω 1 , F (y) = y + ω 2 /ω 1So, letting α = ω 2ω 1we get that F (y) = y + α (mod 1).To show that the orbits of X are dense, it suffices to show that theiterates {F n (y), n ≥ 0} are dense mod 1This is the number theoretic statement: for α irrational, andthenA = {nα + m : n ∈ Z, m ∈ Z},

October 26, 2011 9a-4This elementary fact is proved as follows.(a) A is clearly an additive subgroup of R.A is dense in R. (2)(b) To show that A is dense, it suffices to show that 0 is an accumulationpoint of A.(c) For this latter fact, for each n ∈ Z, let m n ∈ Z be such thatx n = nα + m n ∈ [0, 1). Since α is irrational, the numbers x n areall distinct. Since [0, 1] is compact, there is a subsequence x ni of(x n ) which converges. In particular, if ɛ > 0 is arbitrary, there arepoints x n1 , x n2 such that| x n1 − x n2 | < ɛ.But then this difference x n1 − x n2 is in A ⋂ (−ɛ, ɛ). Since, ɛ isarbitrary, we have that 0 is an accumulation point of A as required.Let us now give an application of these ideas.Consider the mass spring system whose equation ismẍ + kx = 0where m is the mass of an object suspended vertically by a spring,k > 0 is the spring constant, and x = 0 represents the equilibriumposition.√Let ω = k. The associated first order planar system can be writtenmẋ = −ω yẏ = ω x(3)Let us write this system (3) in polar coordinates.We havex = rcos(θ), y = rsin(θ)

October 26, 2011 9a-5ẋ = ṙcos(θ) − rsin(θ) ˙θẏ = ṙsin(θ) + rcos(θ) ˙θ(4)So, we see that the simple polar coordinate systemṙ = 0˙θ = ω(5)is equivalent to (4).The solutions off (0, 0) arer = c > 0, θ(t) = ω t + θ 0 .Hence, the orbits in the plane off (0, 0) consist of the circles r = c andthe motion has constant rotational speed ω with period 2π ω .Now, consider two uncoupled springs whose equations in R 4 becomex˙1 = −ω 1 y 1y˙1 = ω x 1(6)x˙2 = −ω 2 y 2y˙2 = ω 2 x 2Using polar coordinates (r 1 , θ 1 , r 2 , θ 2 ) in R 4 in the obvious way, we getthe equivalent equationsr˙1 = 0θ˙1 = ω 1(7)r˙2 = 0θ˙2 = ω 2Hence all solutions in R 4 off the union of the two subspaces R 2 ×{(0, 0} ⋃ {0, 0}×R 2 lie on two dimensional tori r 1 = c 1 , r 2 = c 2 , and the motion on each is aconstant vector field. If ω 2 /ω 1 is rational, then all these orbits are periodic,and if ω 2 /ω 1 is irrational, then all these tori are minimal sets.