Eigenvalue simple

A simple eigenvalue problem can be demonstrated by the example of two coupled oscillators. The system is illustrated in fug. 2. It should be compared with the classical harmonic oscillator that was treated in Section 5.2.2. Here also, the system will be assumed to be harmonic, namely, that both springs obey Hooke s law. The potential energy can then be written in the form... 

Remark 2 For the elements of many subclasses of CBSs it can easily be proved by induction using Algorithm D that +1 are eigenvalues (in particular this is true for all linear CBSs, i.e., CBSs without bifurcation or knee cells in addition it can be shown that for all linear CBSs which have an odd number of cells the numbers + 2 are simple eigenvalues). 

The new mathematics that is introduced here is elementary bifurcation theory, in particular, bifurcation from a simple eigenvalue. Although the necessary theorems will not be proved, the material will be discussed in some detail. 

Proof. Since A has nonnegative off-diagonal entries and is irreducible, Theorem A.5 asserts that 5(A) is a simple eigenvalue of A, larger than the real parts of all other eigenvalues. The inequality hypothesis and the Gershgorin circle theorem (Theorem A.l) together imply that 5(A) < 0. If 5(A) < 0, then the final assertion of the lemma follows from Theorem A.12. If 5(A) = 0, then Theorem A.5 implies that there exists x > 0 such that Ax = 0. We can assume that Xy < 1 for all j and that x, = 1 for a nonempty subset / of indices. If J is the complementary set of indices then J is non-empty by our assumptions on the row sums of A. For iel we have... 

The hypotheses of Theorem 4.4 are stable to perturbation, in the sense that if they hold for particular values of the parameters and uptake functions then they continue to hold for all nearby values of the parameters and for nearby uptake functions. By a nearby uptake function we mean an uptake function with the properties that (a) it satisfies requirements (i) and (ii) of Section 1, and (b) it and its derivative are uniformly close to the given uptake function and its derivative on the closed interval [0,1]. The reason for this stability is that simple eigenvalues depend continuously on the entries of a matrix. 

Another important observation concerning Corollary 5.2 is that if the hypotheses are satisfied for a particular set of growth functions f and parameters, then the hypotheses continue to hold if these functions and parameters are perturbed by a small amount. Such a property is clearly important, because parameters and functions are never known precisely. The stability of the conditions to perturbations follows from the well-known continuity of simple eigenvalues of matrices to changes in their entries. 

Although stability may in principle be computed, the calculation is extremely complicated. Numerical calculations suggest the asymptotic stability of the limit cycle, but the stability has not been rigorously established. Assuming that the solution is asymptotically stable, a secondary bifurcation can be shown to occur. The argument is quite technical and requires a form of a Poincare map in the appropriate function space it is analogous to the bifurcation theorem used in Chapter 3 for bifurcation from a simple eigenvalue. The principal theorem takes the form of a bifurcation statement. 

The following simple eigenvalue problem is naturally selected for the integral transformation pair construction A(R" f ) + R" p(R) = 0 (17.a)... 

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