Period-doubling Events Research Articles

Drops bouncing on a vibrating bath

AbstractWe present the results of a combined experimental and theoretical investigation of millimetric droplets bouncing on a vertically vibrating fluid bath. We first characterize the system experimentally, deducing the dependence of the droplet dynamics on the system parameters, specifically the drop size, driving acceleration and driving frequency. As the driving acceleration is increased, depending on drop size, we observe the transition from coalescing to vibrating or bouncing states, then period-doubling events that may culminate in either walking drops or chaotic bouncing states. The drop’s vertical dynamics depends critically on the ratio of the forcing frequency to the drop’s natural oscillation frequency. For example, when the data describing the coalescence–bouncing threshold and period-doubling thresholds are described in terms of this ratio, they collapse onto a single curve. We observe and rationalize the coexistence of two non-coalescing states, bouncing and vibrating, for identical system parameters. In the former state, the contact time is prescribed by the drop dynamics; in the latter, by the driving frequency. The bouncing states are described by theoretical models of increasing complexity whose predictions are tested against experimental data. We first model the drop–bath interaction in terms of a linear spring, then develop a logarithmic spring model that better captures the drop dynamics over a wider range of parameter space. While the linear spring model provides a faster, less accurate option, the logarithmic spring model is found to be more accurate and consistent with all existing data.

Near-grazing dynamics of base excited cantilevers with nonlinear tip interactions

In this article, nonsmooth dynamics of impacting cantilevers at different scales is explored through a combination of analytical, numerical, and experimental efforts. For off-resonance and harmonic base excitations, period-doubling events close to grazing impacts are experimentally studied in a macroscale system and a microscale system. The macroscale test apparatus consists of a base excited aluminum cantilever with attractive and repulsive tip interactions. The attractive force is generated through a combination of magnets, one located at the cantilever structure’s tip and another attached to a high-resolution translatory stage. The repulsive forces are generated through impacts of the cantilever tip with the compliant material that covers the magnet on the translatory stage. The microscale system is an atomic force microscope cantilever operated in tapping mode. In this mode, this microcantilever experiences a long-range attractive van der Waals force and a repulsive force as the cantilever tip comes close to the sample. The qualitative changes observed in the experiments are further explored through numerical studies, assuming that the system response is dominated by the fundamental cantilever vibratory mode. In both the microscale and macroscale cases, contact is modeled by using a quadratic repulsive force. A reduced-order model, which is developed on the basis of a single mode approximation, is employed to understand the period-doubling phenomenon experimentally observed close to grazing in both the macroscale and microscale systems. The associated near-grazing dynamics is examined by carrying out local analyses with Poincare map constructions to show that the observed period-doubling events are possible for the considered nonlinear tip interactions. In the corresponding experiments, the stability of the observed grazing periodic orbits has been assessed by constructing the Jacobian matrix from the experimentally obtained Poincare map. The present study also sheds light on the use of macroscale systems to understand near-grazing dynamics in microscale systems.

The effect of diffusion on the dynamics of unsteady detonations

AbstractThe dynamics of a one-dimensional detonation predicted by a one-step irreversible Arrhenius kinetic model are investigated in the presence of mass, momentum and energy diffusion. A study is performed in which the activation energy is varied and the length scales of diffusion and reaction are held constant. As the activation energy is increased, the system goes through a series of period-doubling events and eventually undergoes a transition to chaos. The rate at which these bifurcation points converge is calculated and shown to be in agreement with the Feigenbaum constant. Within the chaotic regime, there exist regions in which there are limit cycles consisting of a small number of oscillatory modes. When an appropriately fine grid is used to capture mass, momentum and energy diffusion, predictions are independent of the differencing scheme. Diffusion affects the behaviour of the system by delaying the onset of instability and strongly influencing the dynamics in the unstable regime. The use of the reactive Euler equations to predict detonation dynamics in the unstable and marginally stable regimes is called into question as the selected reactive and diffusive length scales are representative of real physical systems; reactive Navier–Stokes is a more appropriate model in such regimes.

Weak Noise Approach to the Logistic Map

Using a nonperturbative weak noise approach we investigate the interference of noise and chaos in simple 1D maps. We replace the noise-driven 1D map by an area-preserving 2D map modelling the Poincare sections of a conserved dynamical system with unbounded energy manifolds. We analyze the properties of the 2D map and draw conclusions concerning the interference of noise on the nonlinear time evolution. We apply this technique to the standard period-doubling sequence in the logistic map. From the 2D area-preserving analogue we, in addition to the usual period-doubling sequence, obtain a series of period doubled cycles which are elliptic in nature. These cycles are spinning off the real axis at parameters values corresponding to the standard period doubling events.

Simulations of pulsating one-dimensional detonations with true fifth order accuracy

A novel, highly accurate numerical scheme based on shock-fitting coupled with fifth order spatial and temporal discretizations is applied to a classical unsteady detonation problem to generate solutions with unprecedented accuracy. The one-dimensional reactive Euler equations for a calorically perfect mixture of ideal gases whose reaction is described by single-step irreversible Arrhenius kinetics are solved in a series of calculations in which the activation energy is varied. In contrast with nearly all known simulations of this problem, which converge at a rate no greater than first order as the spatial and temporal grid is refined, the present method is shown to converge at a rate consistent with the fifth order accuracy of the spatial and temporal discretization schemes. This high accuracy enables more precise verification of known results and prediction of heretofore unknown phenomena. To five significant figures, the scheme faithfully recovers the stability boundary, growth rates, and wave-numbers predicted by an independent linear stability theory in the stable and weakly unstable regime. As the activation energy is increased, a series of period-doubling events are predicted, and the system undergoes a transition to chaos. Consistent with general theories of non-linear dynamics, the bifurcation points are seen to converge at a rate for which the Feigenbaum constant is 4.66 ± 0.09, in close agreement with the true value of 4.669201…. As activation energy is increased further, domains are identified in which the system undergoes a transition from a chaotic state back to one whose limit cycles are characterized by a small number of non-linear oscillatory modes. This result is consistent with behavior of other non-linear dynamical systems, but not typically considered in detonation dynamics. The period and average detonation velocity are calculated for a variety of asymptotically stable limit cycles. The average velocity for such pulsating detonations is found to be slightly greater than the Chapman–Jouguet velocity.

Period doubling cascade in glow discharges: Local versus global differential conductivity

Short planar glow discharges coupled to a resistive layer exhibit a wealth of spontaneous spatiotemporal patterns. Due to similarities with other pattern forming systems that are described by reaction-diffusion models, several authors have tried to derive such models from discharge physics. We investigate the temporal oscillations of the discharge system and find a cascade of period doubling events. This shows that the inner structure of the discharge is more complex than can be described by a two-component reaction-diffusion-model with negative differential conductivity. We also derive an alternative reduced model.

Period-doubling events in the light curve of R Cygni: Evidence for chaotic behaviour

A detailed analysis of the century long visual light curve of the long-period Mira star R Cygni is presented and discussed. The data were collected from the publicly available databases of the AFOEV, the BAAVSS and the VSOLJ. The full light curve consists of 26655 individual points obtained between 1901 and 2001. The light curve and its periodicity were analysed with help of the O-C diagram, Fourier analysis and time-frequency analysis. The results demonstrate the limitations of these linear methods. The next step was to investigate the possible presence of low-dimensional chaos in the light curve. For this, a smoothed and noise-filtered signal was created from the averaged data and with the help of time delay embedding, we have tried to reconstruct the attractor of the system. The main result is that R Cygni shows such period-doubling events that can be interpreted as being caused by a repetitive bifurcation of the chaotic attractor between a period 2T orbit and chaos. The switch between these two states occurs in a certain compact region of the phase space, where the light curve is characterized by ∼1500-day long transients. The Lyapunov spectrum was computed for various embedding parameters confirming the chaotic attractor, although the exponents suffer from quite high uncertainty because of the applied approximation. Finally, the light curve is compared with a simple one zone model generated by a third-order differential equation which exhibits well-expressed period-doubling bifurcation. The strong resemblance is another argument for chaotic behaviour. Further studies should address the problem of global flow reconstruction, including the determination of the accurate Lyapunov exponents and dimension.

On the Spatio-Temporal Flow Phenomena in Low Prandtl Number Melts Part 1: The Case of Stationary Vertical Molten Zones

Thermal buoyancy driven flow in semiconductor melts contained in closed silica glass ampoules was studied experimentally in dependence on the height of the molten zone. The onset of temperature oscillations in the melt and further transitions of the temporal convective behaviour were detected by in-situ temperature measurements. It was shown for low PRANDTL number melts that the flow in stationary vertical molten zones strongly tends to become unsteady. For the periodic unsteady flow regime two fundamentally different periodic oscillation modes were revealed. Transitions to chaos are announced in nearly all cases by period doubling events. First return maps of selected typical time series are used to characterize the temporal dynamics of the non-linear dynamic system Based on the phase relations and amplitudes of the signals measured simultaneously in axial and azimuthal direction spatio-temporal oscillating flow structures can he carefully interpreted. Mode switching and the rotation of a vortex configuration are assumed to be the preferred oscillation mechanisms.