STABILITY Math 21b, O. KnillLINEAR DYNAMICAL SYSTEM. A linear map x 7→ Ax defines a dynamical system. Iterating the mapproduces an orbit x0, x1= Ax, x2= A2= AAx, .... The vector xn= Anx0describes the situation of thesystem at time n.Where does xngo when time evolves? Can one describe what happens asymptotically when time n goes toinfinity?In the case of the Fibonacci sequence xnwhich gives the number of rabbits in a rabbit population at time n, thepopulation grows essentially exponentially. Such a behavior would be called unstable. On the other hand, if Ais a rotation, then An~v stays bounded which is a type of stability. If A is a dilation with a dilation factor < 1,then An~v → 0 for all ~v, a thing which we will call asymptotic stability. The next pictures show experimentswith some orbits An~v with different matrices.0.99 −11 0stable (notasymptotic)0.54 −10.95 0asymptoticstable0.99 −10.99 0asymptoticstable0.54 −11.01 0unstable2.5 −11 0unstable1 0.10 1unstableASYMPTOTIC STABILITY. The origin~0 is invariant under a linear map T (~x) = A~x. It is called asymptot-ically stable if An(~x) →~0 for all ~x ∈ IRn.EXAMPLE. Let A =p −qq pbe a dilation rotation matrix. Because multiplication wich such a matrix isanalogue to the multiplication with a complex number z = p+iq, the matrix Ancorresponds to a multiplicationwith (p+iq)n. Since |(p + iq)|n= |p+iq|n, the origin is asymptotically stable if and only if |p +iq| < 1. Becausedet(A) = |p+iq|2= |z|2, rotation-dilation matrices A have an asymptotic stable origin if and only if |det(A)| < 1.Dilation-rotation matricesp −qq phave eigenvalues p ± iq and can be diagonalized in the complex.EXAMPLE. If a matrix A has an eigenvalue |λ| ≥ 1 to an eigenvector ~v, then An~v = λn~v, whose length is |λn|times the length of ~v. So, we have no asymptotic stability if an eigenvalue satisfies |λ| ≥ 1.STABILITY. The book also writes ”stable” for ”asymptotically stable”. This is ok to abbreviate. Note howeverthat the commonly used term ”stable” also includes linear maps like rotations, reflections or the identity. It istherefore preferable to leave the attribute ”asymptotic” in front of ”stable”.ROTATIONS. Rotationscos(φ) −sin(φ)sin(φ) cos(φ)have the eigenvalue exp(±iφ) = cos(φ) + i sin(φ) and are notasymptotically stable.DILATIONS. Dilationsr 00 rhave the eigenvalue r with algebraic and geometric multiplicity 2. Dilationsare asymptotically stable if |r| < 1.CRITERION.A linear dynamical system x 7→ Ax has an asymptotically stable origin if andonly if all its eigenvalues have an absolute value < 1.PROOF. We have already seen in Example 3, that if one eigenvalue satisfies |λ| > 1, then the origin is notasymptotically stable. If |λi| < 1 for all i and all eigenvalues are different, there is an eigenbasis v1, . . . , vn.Every x can be written as x =Pnj=1xjvj. Then, Anx = An(Pnj=1xjvj) =Pnj=1xjλnjvjand because |λj|n→ 0,there is stability. The proof of the general (nondiagonalizable) case will be accessible later.THE 2-DIMENSIONAL CASE. The characteristic polynomial of a 2 × 2 matrix A =a bc disfA(λ) = λ2− tr(A)λ + det(A). If c 6= 0, the eigenvalues are λ±= tr(A)/2 ±p(tr(A)/2)2− det(A). Ifthe discriminant (tr(A)/2)2− det(A) is nonnegative, then the eigenvalues are real. This happens below theparabola, where the discriminant is zero.CRITERION. In two dimensions we have asymptoticstability if and only if (tr(A), det(A)) is containedin the stability triangle bounded by the linesdet(A) = 1, det(A) = tr(A)−1 and det(A) = −tr(A)−1.PROOF. Write T = tr(A)/2, D = det(A). If |D| ≥ 1,there is no asymptotic stability. If λ = T +√T2− D =±1, then T2− D = (±1 − T )2and D = 1 ± 2T . ForD ≤ −1 + |2T | we have a real eigenvalue ≥ 1. Theconditions for stability is therefore D > |2T | − 1. Itimplies automatically D > −1 so that the triangle canbe described shortly as|tr(A)|− 1 < det(A) < 1 .EXAMPLES.1) The matrix A =1 1/2−1/2 1has determinant 5/4 and trace 2 and the origin is unstable. It is a dilation-rotation matrix which corresponds to the complex number 1 + i/2 which has an absolute value > 1.2) A rotation A is never asymptotically stable: det(A)1 and tr(A) = 2 cos(φ). Rotations are the upper side ofthe stability triangle.3) A dilation is asymptotically stable if and only if the scaling factor has norm < 1.4) If det(A) = 1 and tr(A) < 2 then the eigenvalues are on the unit circle and there is no asymptotic stability.5) If det(A) = −1 (like for example Fibonacci) there is no asymptotic stability. For tr(A) = 0, we are a cornerof the stability triangle and the map is a reflection, which is not asymptotically stable neither.SOME PROBLEMS.1) If A is a matrix with asymptotically stable origin, what is the stability of 0 with respect to AT?2) If A is a matrix which has an asymptotically stable origin, what is the stability with respect to to A−1?3) If A is a matrix which has an asymptotically stable origin, what is the stability with respect to to A100?ON THE STABILITY QUESTION.For general dynamical systems, the question of stability can be very difficult. We deal here only with lineardynamical systems, where the eigenvalues determine everything. For nonlinear systems, the story is not sosimple even for simple maps like the Henon map. The questions go deeper: it is for example not known,whether our solar system is stable. We don’t know whether in some future, one of the planets could getexpelled from the solar system (this is a mathematical question because the escape time would be larger thanthe life time of the sun). For other dynamical systems like the atmosphere of the earth or the stock market,we would really like to know what happens in the near future ...A pioneer in stability theory was Alek-sandr Lyapunov (1857-1918). For nonlin-ear systems like xn+1= gxn− x3n− xn−1the stability of the origin is nontrivial.As with Fibonacci, this can be writtenas (xn+1, xn) = (gxn− x2n− xn−1, xn) =A(xn, xn−1) called cubic Henon map inthe plane. To the right are orbits in thecases g = 1.5, g = 2.5.The first case is stable (but proving this requires a fancy theory called KAM theory), the second case isunstable (in this case actually the linearization at~0 determines the
View Full Document
Unlocking...