Numerical studies of complex and circle maps
Gerald Charles Stewart · ERA · 1992
This thesis deals with problems in mappings of the complex plane and of the circle.The principal motivation comes from circle mapping problems, discussed in [Da,2] and we summarise the results, of interest to us, in Chapter 0.In Chapters 1, 2 and 3, we look at mappings, which can represented by the form, 4>, where $(0) = 0 and ' (0) = A, where A is a root of unity.Each of these chapters is designated by the study of a particular mapping, starting with z H ze z , in Chapter 1, then moving onto : M z+l+-z~l , in Chapter 2 and then z M -ze 7-, in i Chapter 3. Our approach is very similar in each case, though we pay more attention to detail in Chapters 1 and 2.In short, we are interested in the behaviour of invariant curves of , both locally, near 0 (or for Re(z) large in Chapter 2) and globally.We define limit functions of , mapping into the w-plane and parametrise the invariant curves in terms of Im(w).We then evaluate numerical approximations for intervals of Im(w) , from which we can specify invariant curves with homoclinic intersections.In Chapter 4, we look at the family of circle dif f eomorphisms , fn( x ) = x+^l+ E SRe(z k a k e 2n J kx ^ and our interest k= 1 centres round the occurence of rational values of the rotation number, namely p(fn).Specifically, we are interested in forms of / n , for which e is small and the coefficients, a k , k = 1,2,..., are Fourier coefficients of periodic functions, defined in terms of in Chapters 1, 2 and 3. Hence, we derive three forms of / n , using estimates for n k , from each of the complex analysis problems.We define the interval, 7 a = (ft: p(fn) = a}, where o = *-is a rational, expressed in it lowest terms and we go on to show that |/ a | = |$a fa t ,... ,a q )\£ q +0(e q * 1 j, where ^o, is a polynomial with complex coefficients.Finally, we estimate | V&fai,... ,a^)\ , for several values of a and we make numerical calculations which show, or n t 1 cas I i ncl i en.Lc , t,ha i, | / a | > 0 .I am indebted to my supervisor, Dr.A.M.Davie, for his constant technical assistance and guidance, without which, the composition of this thesis would not have been possible.I would also like to thank the staff of the Department of Mathematics and Statistics at the University of Edinburgh for their encouragement and practical assistance, with my gratitude reserved, in particular, for the departmental secretaries and the computing officers.On a personal note, I would like to pay tribute to a few friends, relatives and colleagues; for both their moral support and the enduring patience they have shown, in spite of my eccentricities and my several (appalling) attempts to accurately forecast a submission date.My thoughts are always with