Rotation Intervals for Quasi-Periodically Forced Circle Maps

  • Silvia Pina Romero

Student thesis: Phd


This work investigates some aspects of the dynamics of non-invertible quasi-periodic circle maps, from the point of view of rotation numbers and their structure in parameter space.Circle maps and quasi-periodically forced circle maps have been widely used asa model for a broad range of physical phenomena. From the mathematical point of view they have also received considerable attention because of the many interesting features they exhibit.The system used is given by the maps: x_n = [ x_n-1 + a + b/(2pi) sin( 2pi x_n-1) + c sin( 2pi theta_n-1) ] mod 1, and, theta_n = theta_n-1 + omega.Where a, b and c are real constants. In addition, b and omega are restricted, respectively, to values larger than one and irrational.A fundamental part of this thesis consists of numerical approximations of rotation intervals using and adapting of the work of Boyland (1986) to the quasi-periodic case.Particular emphasis was given to the case of large coupling strength in quasi-periodicforcing.Examination of the computed rotation numbers for the large coupling case, together with previous claims suggesting that for large coupling strength the b-term could be neglected (see Ding (1989)), led to the formulation of an ergodic argument which is statistically supported. This argument indicates that, for this case, the qualitative behavior of rotation number depends linearly on a. It is also shown that the length of the rotation interval, when the transition from a trivial rotation interval (invertible case) to a non-trivial rotation interval occurs, it develops locally as a universal unfolding.A different map, piecewise monotone, and structurally similar to the maps defined to calculate the edges of rotation intervals in Boyland (1986), is studied to illustrate how the rotation number grows. The edges of rotation intervals are analytically calculated and matched with numerical observations.
Date of Award31 Dec 2012
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorPaul Glendinning (Supervisor) & James Montaldi (Supervisor)


  • Rotation Interval
  • Rotation Number
  • Quasi-Periodically Forced Circle Maps

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