Escaping Orbits Are also Rare in the Almost Periodic Fermi–Ulam Ping-Pong

We study the one-dimensional Fermi–Ulam ping-pong problem with a Bohr almost periodic forcing function and show that the set of initial condition leading to escaping orbits typically has Lebesgue measure zero.

Bifurcation Analysis of an Ecological System with State-Dependent Feedback Control and Periodic Forcing

By assuming a periodic variation in the intrinsic growth rate of the prey, a nonlinear ecological system with periodic forcing and state-dependent feedback control is proposed. The main purpose of the present paper is to study the dynamical behavior generated by periodic forcing and nonlinear impulse perturbations and their effects on pest control. To do this, we first investigate the existence and stability of the boundary periodic solution, and then we employ the numerical bifurcation techniques, mainly including one-dimensional and two-dimensional parameter bifurcation analyses, to reveal that the system exhibits rich and complex dynamic behaviors. Especially, period-adding bifurcation with chaos is found in the two-parameter bifurcation plane. Moreover, we find the periodic structure similar to Arnold tongues, and they are arranged according to the sequence of a Farey tree. In addition, one-dimensional bifurcation diagrams reveal the existence of order-[Formula: see text] periodic, and chaotic solutions, multiple coexisting attractors, period-doubling bifurcations, period-halving bifurcations, and so on. Finally, the effects of the initial population density of pests and natural enemies on the pulse frequency and the biological significance related to the numerical results are studied and discussed.

Long Time Behaviour of a Local Perturbation in the Isotropic XY Chain Under Periodic Forcing

We study the isotropic XY quantum spin chain with a time-periodic transverse magnetic field acting on a single site. The asymptotic dynamics is described by a highly resonant Floquet–Schrödinger equation, for which we show the existence of a periodic solution if the forcing frequency is away from a discrete set of resonances. This in turn implies the state of the quantum spin chain to be asymptotically a periodic function synchronised with the forcing, also at arbitrarily low non-resonant frequencies. The behaviour at the resonances remains a challenging open problem.

Analysis of a mass-spring-relay system with periodic forcing

The dynamics of a hysteretic relay oscillator with harmonic forcing is investigated. Periodic excitation of the system results in periodic, quasi-periodic, chaotic and unbounded behavior. An explicit Poincaré map is constructed with an implicit constraint on the switching time. The stability of the fixed points of the Poincaré map corresponding to period-one solutions is investigated. By varying the forcing parameters, we observed a saddle-center and a pitchfork bifurcation of two centers and a saddle-type fixed point. The global dynamics of the system exhibits discontinuity induced bifurcations of the fixed points.

The Role of Acoustics in the Conservation of the Ivory-Billed Woodpecker (Campephilus principalis)

The Ivory-billed Woodpecker (Campephilus principalis) is an iconic species that has survived in barely detectable numbers for the past 100 years, during which it has been feared extinct only to be rediscovered several times. The most recent rediscovery was announced in an article that was featured on the cover of Science in 2005. The persistence of the Ivory-billed Woodpecker became controversial when ornithologists were unable to obtain a clear photo for documenting this ultra-elusive bird during multi-year searches at sites in Arkansas and Florida, where they had several sightings and were convinced these birds were present. Audio recordings of ‘kent’ calls and double knocks were obtained at both sites, but such recordings are not regarded as conclusive evidence of persistence. A debate on this issue that took place in Science and Nature focused on relatively weak video evidence obtained in Arkansas but excluded three videos obtained in Louisiana and Florida that show flights, field marks, and other behaviors and characteristics that are consistent with the Ivory-billed Woodpecker but no other species of the region. Kent calls were recorded in the 1930s, when other types of vocalizations were observed but not recorded, including a high-pitched alarm call. On two occasions in Louisiana, high-pitched calls were observed coming from the direction of an alarmed Ivory-billed Woodpecker, and several of them were recorded. The spectrograms of the high-pitched calls and all other known and putative vocalizations of the Ivory-billed Woodpecker consist of simultaneously excited harmonics. A harmonic oscillator model has been used to draw a connection between the drumming that is typical of most woodpeckers and the double knocks of the Ivory-billed Woodpecker and other Campephilus woodpeckers. Drumming corresponds to periodic forcing; double knocks correspond to impulsive forcing, and a single thrust of the body is sufficient to produce two impacts of the bill in rapid succession. The audio recordings from Arkansas and Florida were obtained with single microphones. A horizontal array of microphones would make it possible to detect weaker sounds and determine the directions of sources. This approach has the potential to lead to the discovery of a nest, and it might be more effective if the array is placed above the treetops, where sounds might propagate to longer ranges.

Entrainment Dynamics Organised by Global Manifolds in a Circadian Pacemaker Model

Circadian rhythms are established by the entrainment of our intrinsic body clock to periodic forcing signals provided by the external environment, primarily variation in light intensity across the day/night cycle. Loss of entrainment can cause a multitude of physiological difficulties associated with misalignment of circadian rhythms, including insomnia, excessive daytime sleepiness, gastrointestinal disturbances, and general malaise. This can occur after travel to different time zones, known as jet lag; when changing shift work patterns; or if the period of an individual’s body clock is too far from the 24 h period of environmental cycles. We consider the loss of entrainment and the dynamics of re-entrainment in a two-dimensional variant of the Forger-Jewett-Kronauer model of the human circadian pacemaker forced by a 24 h light/dark cycle. We explore the loss of entrainment by continuing bifurcations of one-to-one entrained orbits under variation of forcing parameters and the intrinsic clock period. We show that the severity of the loss of entrainment is dependent on the type of bifurcation inducing the change of stability of the entrained orbit, which is in turn dependent on the environmental light intensity. We further show that for certain perturbations, the model predicts counter-intuitive rapid re-entrainment if the light intensity is sufficiently high. We explain this phenomenon via computation of invariant manifolds of fixed points of a 24 h stroboscopic map and show how the manifolds organise re-entrainment times following transitions between day and night shift work.