Bounds on the unstable eigenvalue for the asymmetric renormalization operator for period doubling

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Asymptotics of scaling parameters for period-doubling in unimodal maps with asymmetric critical points

The universal period-doubling scaling of a unimodal map with an asymmetric critical point is governed by a period-2 point of a renormalisation operator. The period-2 point is parametrised by the degree of the critical point and the asymmetry modulus. In this paper we study the asymptotics of period-2 points and their associated scaling parameters in the singular limit of degree tending to 1

Renormalization analysis of correlation properties in a quasiperiodically forced two-level system

We give a rigorous renormalization analysis of the self-similarity of correlation functions in a quasiperiodically forced two-level system. More precisely, the system considered is a quantum two-level system in a time-dependent field consisting of periodic kicks with amplitude given by a discontinuous modulation function driven in a quasiperiodic manner at golden mean frequency. Mathematically, our analysis consists of a description of all piecewise-constant periodic orbits of an additive functional recurrence. We further establish a criterion for such orbits to be globally bounded functions. In a particular example, previously only treated numerically, we further calculate explicitly the asymptotic height of the main peaks in the correlation function

Continued fractions and solutions of the Feigenbaum-Cvitanovic equation

In this paper, we develop a new approach to the construction of solutions of the Feigenbaum-Cvitanovic equation whose existence was shown by H.Epstein. Our main tool is the analytic theory of continued fractions

The effect of seasonal host birth rates on disease persistence

In this paper, we add seasonality to the birth rate of an SIR model with density dependence in the death rate. We find that disease persistence can be explained by considering the average value of the seasonal term. If the basic reproductive ratio with this average value then the disease will persist and if with this average value then the disease will die out. However, if the underlying non-seasonal model displays oscillations towards the equilibrium then the dynamics of the seasonal model can become more complex. In this case the seasonality can interact with the underlying oscillations, resonate and the population can display a range of complex behaviours including chaos. We discuss these results in terms of two examples, Cowpox in bank voles and Rabbit Haemorrhagic Disease in rabbits

A computer assisted proof of universality for cubic critical maps of the circle with golden mean rotation number

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Golden mean renormalization for a generalized Harper equation: The strong coupling fixed point

We construct a renormalisation fixed point corresponding to the strong coupling limit of the golden mean Harper equation. We give an analytic expression for this fixed point, establish its existence and uniqueness, and verify properties previously seen only in numerical calculations. The spectrum of the linearisation of the renormalisation operator at this fixed point is also explicitly determined. This strong coupling fixed point also helps describe the onset of a strange nonchaotic attractor in quasiperiodically forced systems