Families of piecewise linear maps with constant Lyapunov exponent

We consider families of piecewise linear maps in which the moduli of the two slopes take different values. In some parameter regions, despite the variations in the dynamics, the Lyapunov exponent and the topological entropy remain constant. We provide numerical evidence of this fact and we prove it analytically for some special cases. The mechanism is very different from that of the logistic map and we conjecture that the Lyapunov plateaus reflect arithmetic relations between the slopes.Comment: 26 pages, 13 figure

Emergence of hierarchical networks and polysynchronous behaviour in simple adaptive systems

We describe the dynamics of a simple adaptive network. The network architecture evolves to a number of disconnected components on which the dynamics is characterized by the possibility of differently synchronized nodes within the same network (polysynchronous states). These systems may have implications for the evolutionary emergence of polysynchrony and hierarchical networks in physical or biological systems modeled by adaptive networks.Comment: 4 pages, 4 figure

Imperfect Homoclinic Bifurcations

Experimental observations of an almost symmetric electronic circuit show complicated sequences of bifurcations. These results are discussed in the light of a theory of imperfect global bifurcations. It is shown that much of the dynamics observed in the circuit can be understood by reference to imperfect homoclinic bifurcations without constructing an explicit mathematical model of the system.Comment: 8 pages, 11 figures, submitted to PR

Combinatorics of linear iterated function systems with overlaps

Let $\bm p_0,...,\bm p_{m-1}$ be points in ${\mathbb R}^d$, and let $\{f_j\}_{j=0}^{m-1}$ be a one-parameter family of similitudes of ${\mathbb R}^d$: $$f_j(\bm x) = \lambda\bm x + (1-\lambda)\bm p_j, j=0,...,m-1,$$ where $\lambda\in(0,1)$ is our parameter. Then, as is well known, there exists a unique self-similar attractor $S_\lambda$ satisfying $S_\lambda=\bigcup_{j=0}^{m-1} f_j(S_\lambda)$. Each $\bm x\in S_\lambda$ has at least one address $(i_1,i_2,...)\in\prod_1^\infty\{0,1,...,m-1\}$, i.e., $\lim_n f_{i_1}f_{i_2}... f_{i_n}({\bf 0})=\bm x$. We show that for $\lambda$ sufficiently close to 1, each $\bm x\in S_\lambda\setminus\{\bm p_0,...,\bm p_{m-1}\}$ has $2^{\aleph_0}$ different addresses. If $\lambda$ is not too close to 1, then we can still have an overlap, but there exist $\bm x$'s which have a unique address. However, we prove that almost every $\bm x\in S_\lambda$ has $2^{\aleph_0}$ addresses, provided $S_\lambda$ contains no holes and at least one proper overlap. We apply these results to the case of expansions with deleted digits. Furthermore, we give sharp sufficient conditions for the Open Set Condition to fail and for the attractor to have no holes. These results are generalisations of the corresponding one-dimensional results, however most proofs are different.Comment: Accepted for publication in Nonlinearit

Unstable dimension variability, heterodimensional cycles, and blenders in the border-collision normal form

Chaotic attractors commonly contain periodic solutions with unstable manifolds of different dimensions. This allows for a zoo of dynamical phenomena not possible for hyperbolic attractors. The purpose of this Letter is to demonstrate these phenomena in the border-collision normal form. This is a continuous, piecewise-linear family of maps that is physically relevant as it captures the dynamics created in border-collision bifurcations in diverse applications. Since the maps are piecewise-linear they are relatively amenable to an exact analysis and we are able to explicitly identify parameter values for heterodimensional cycles and blenders. For a one-parameter subfamily we identify bifurcations involved in a transition through unstable dimension variability. This is facilitated by being able to compute periodic solutions quickly and accurately, and the piecewise-linear form should provide a useful test-bed for further study

Oscillatory bubbles induced by geometrical constraint

International audienc

Synchronization of many nano-mechanical resonators coupled via a common cavity field

Using amplitude equations, we show that groups of identical nano-mechanical resonators, interacting with a common mode of a cavity microwave field, synchronize to form a single mechanical mode which couples to the cavity with a strength dependent on the square sum of the individual mechanical-microwave couplings. Classically this system is dominated by periodic behaviour which, when analyzed using amplitude equations, can be shown to exhibit multi-stability. In contrast groups of sufficiently dissimilar nano-mechanical oscillators may lose synchronization and oscillate out of phase at significantly higher amplitudes. Further the method by which synchronization is lost resembles that for large amplitude forcing which is not of the Kuramoto form.Comment: 23 pages, 11 figure

Analysis of the shearing instability in nonlinear convection and magnetoconvection

Numerical experiments on two-dimensional convection with or without a vertical magnetic field reveal a bewildering variety of periodic and aperiodic oscillations. Steady rolls can develop a shearing instability, in which rolls turning over in one direction grow at the expense of rolls turning over in the other, resulting in a net shear across the layer. As the temperature difference across the fluid is increased, two-dimensional pulsating waves occur, in which the direction of shear alternates. We analyse the nonlinear dynamics of this behaviour by first constructing appropriate low-order sets of ordinary differential equations, which show the same behaviour, and then analysing the global bifurcations that lead to these oscillations by constructing one-dimensional return maps. We compare the behaviour of the partial differential equations, the models and the maps in systematic two-parameter studies of both the magnetic and the non-magnetic cases, emphasising how the symmetries of periodic solutions change as a result of global bifurcations. Much of the interesting behaviour is associated with a discontinuous change in the leading direction of a fixed point at a global bifurcation; this change occurs when the magnetic field is introduced

Assessing the influence of one astronomy camp over 50 years

The International Astronomical Youth Camp has benefited thousands of lives during its 50-year history. We explore the pedagogy behind this success, review a survey taken by more than 300 previous participants, and discuss some of the challenges the camp faces in the future.Comment: 10 pages, 4 figure

In good company: risk, security and choice in young people's drug decisions

This article draws on original empirical research with young people to question the degree to which 'individualisation of risk', as developed in the work of Beck and Giddens, adequately explains the risks young people bear and take. It draws on alternative understandings and critiques of 'risk' not to refute the notion of the reflexive individual upon which 'individualisation of risk' is based but to re-read that reflexivity in a more hermeneutic way. It explores specific risk-laden moments – young people's drug use decisions – in their natural social and cultural context of the friendship group. Studying these decisions in context, it suggests, reveals the meaning of 'risk' to be not given, but constructed through group discussion, disagreement and consensus and decisions taken to be rooted in emotional relations of trust, mutual accountability and common security. The article concludes that 'the individualisation of risk' fails to take adequate account of the significance of intersubjectivity in risk-decisions. It argues also that addressing the theoretical overemphasis on the individual bearer of risk requires not only further empirical testing of the theory but appropriate methodological reflection