Continuous periodic solution of a nonlinear pseudo-oscillator equation in which the restoring force is inversely proportional to the dependent variable

In this paper, we consider a method for the simple exact analytical solution of autonomous nonlinear oscillator equations. While the approach can be used to solve nonlinear oscillator equations with smooth solutions (and we demonstrate this with an application of the approach to an autonomous Duffing equation), our primary interest will be on solving equations with non-smooth yet continuous solutions. To this end, we consider the second-order pseudo-oscillator equation yy″ + 1 = 0 used as a simple model of the path taken by an electron in an electron beam injected into a plasma tube. In recent results of Gadella and Lara, the authors claim the non-existence of periodic solutions to this equation, but actually show that there are no smooth periodic solutions. We show that although there are no smooth solutions to this equation, there does exist a type of continuous periodic solution on the whole problem domain, hence periodic solutions do indeed exist. These periodic solutions can be constructed to have any arbitrary positive period, and the amplitude of these solutions increases as the period is increased. The approach allows one to construct periodic solutions to a variety of nonlinear oscillator equations, even if the solutions are not smooth, and hence could be a useful tool for those interested in physical applications in which nonlinear oscillator models arise

Continuous periodic solution of a nonlinear pseudo-oscillator equation in which the restoring force is inversely proportional to the dependent variable

In this paper, we consider a method for the simple exact analytical solution of autonomous nonlinear oscillator equations. While the approach can be used to solve nonlinear oscillator equations with smooth solutions (and we demonstrate this with an application of the approach to an autonomous Duffing equation), our primary interest will be on solving equations with non-smooth yet continuous solutions. To this end, we consider the second-order pseudo-oscillator equation yy″ + 1 = 0 used as a simple model of the path taken by an electron in an electron beam injected into a plasma tube. In recent results of Gadella and Lara, the authors claim the non-existence of periodic solutions to this equation, but actually show that there are no smooth periodic solutions. We show that although there are no smooth solutions to this equation, there does exist a type of continuous periodic solution on the whole problem domain, hence periodic solutions do indeed exist. These periodic solutions can be constructed to have any arbitrary positive period, and the amplitude of these solutions increases as the period is increased. The approach allows one to construct periodic solutions to a variety of nonlinear oscillator equations, even if the solutions are not smooth, and hence could be a useful tool for those interested in physical applications in which nonlinear oscillator models arise

Motion of isolated open vortex filaments evolving under the truncated local induction approximation

The study of nonlinear waves along open vortex filaments continues to be an area of active research. While the local induction approximation (LIA) is attractive due to locality compared with the non-local Biot-Savart formulation, it has been argued that LIA appears too simple to model some relevant features of Kelvin wave dynamics, such as Kelvin wave energy transfer. Such transfer of energy is not feasible under the LIA due to integrability, so in order to obtain a non-integrable model a truncated LIA, which breaks the integrability of the classical LIA, has been proposed as a candidate model with which to study such dynamics. Recently Laurie et al. [“Interaction of Kelvin waves and nonlocality of energy transfer in superfluids,” Physical Review B 81, 104526 (2010)] derived such as truncated LIA systematically from Biot-Savart dynamics. The focus of the present paper is to study the dynamics of a section of common open vortex filaments under the truncated LIA dynamics. We obtain the analogue of helical, planar, and more general filaments which rotate without change in form in the classical LIA, demonstrating that while quantitative differences do exist, qualitatively such solutions still exist under the truncated LIA. Conversely, solitons and breather solutions found under the LIA should not be expected under the truncated LIA, as the existence of such solutions relies on the existence of an infinite number of conservation laws which is violated due to loss of integrability. On the other hand, similarity solutions under the truncated LIA can be quite different to their counterparts found for the classical LIA, as they must obey a t1/3 type scaling rather than the t1/2 type scaling commonly found in the LIA and Biot-Savart dynamics. This change in similarity scaling mean that Kelvin waves are radiated at a slower rate from vortex kinks formed after reconnection events. The loss of soliton solutions and the difference in similarity scaling indicate that dynamics emergent under the truncated LIA can indeed differ a great deal from those previously studied under the classical LIA

Motion of isolated open vortex filaments evolving under the truncated local induction approximation

The study of nonlinear waves along open vortex filaments continues to be an area of active research. While the local induction approximation (LIA) is attractive due to locality compared with the non-local Biot-Savart formulation, it has been argued that LIA appears too simple to model some relevant features of Kelvin wave dynamics, such as Kelvin wave energy transfer. Such transfer of energy is not feasible under the LIA due to integrability, so in order to obtain a non-integrable model a truncated LIA, which breaks the integrability of the classical LIA, has been proposed as a candidate model with which to study such dynamics. Recently Laurie et al. [“Interaction of Kelvin waves and nonlocality of energy transfer in superfluids,” Physical Review B 81, 104526 (2010)] derived such as truncated LIA systematically from Biot-Savart dynamics. The focus of the present paper is to study the dynamics of a section of common open vortex filaments under the truncated LIA dynamics. We obtain the analogue of helical, planar, and more general filaments which rotate without change in form in the classical LIA, demonstrating that while quantitative differences do exist, qualitatively such solutions still exist under the truncated LIA. Conversely, solitons and breather solutions found under the LIA should not be expected under the truncated LIA, as the existence of such solutions relies on the existence of an infinite number of conservation laws which is violated due to loss of integrability. On the other hand, similarity solutions under the truncated LIA can be quite different to their counterparts found for the classical LIA, as they must obey a t1/3 type scaling rather than the t1/2 type scaling commonly found in the LIA and Biot-Savart dynamics. This change in similarity scaling mean that Kelvin waves are radiated at a slower rate from vortex kinks formed after reconnection events. The loss of soliton solutions and the difference in similarity scaling indicate that dynamics emergent under the truncated LIA can indeed differ a great deal from those previously studied under the classical LIA

Dynamics from a predator-prey-quarry-resource-scavenger model

Allochthonous resources can be found in many foodwebs, and can influence both the structure and stability of an ecosystem. In order to better understand the role of how allochthonous resources are transferred as quarry from one predator-prey system to another, we propose a predator-prey-quarry-resource-scavenger (PPQRS) model, which is an extension of an existing model for quarry-resource-scavenger (a predator-prey-subsidy (PPS) model). Instead of taking the allochthonous resource input rate as a constant, as has been done in previous theoretical work, we explicitly incorporated the underlying predator-prey relation responsible for the input of quarry. The most profound diggerences between PPS and PPQRS system are found when the predator-prey system has limit cycles, resulting in a periodic rather than constant influx of quarry (the allochthonous resource) into the scavenger-resource interactions. This suggest that the way in which allochthonous resources are input into a predator-prey system can have a strong influence over the population dynamics. In order to understand the role of seasonality, we incorporated non-autonomous terms, and show that these terms can either stabilize or destabilize the dynamics, depending on the parameter regime. We also considered the influence of spatial motion (via diffusion) by constructing a continuum PDE model over space. We determine when such spatial dynamics essentially give the same information as the ODE system, versus other cases where there are strong spatial differences (such as spatial pattern formation) in the populations. In situations where increasing the carrying capacity in the ODE model drives the amplitude of the oscillations up, we found that a large carrying capacity in the PDE model results in a very small variation in average population size, showing that spatial diffusion is stabilizing for the PPQRS model

Nonlinear dynamics from discrete time two-player status-seeking games

We study the dynamics of two-player status-seeking games where moves are made simultaneously in discrete time. For such games, each player's utility function will depend on both non-positional goods and positional goods (the latter entering into "status"). In order to understand the dynamics of such games over time, we sample a variety of different general utility functions, such as CES, composite log-Cobb-Douglas, and King-Plosser-Rebelo utility functions (and their various simplifications). For the various cases considered, we determine asymptotic dynamics of the two-player game, demonstrating the existence of stable equilibria, periodic orbits, or chaos, and we show that the emergent dynamics will depend strongly on the utility functions employed. For periodic orbits, we provide bifurcation diagrams to show the existence or non-existence of period doubling or chaos resulting from bifurcations due to parameter shifts. In cases where multiple feasible solution branches exist at each iteration, we consider both cases where deterministic or random selection criteria are employed to select the branch used, the latter resulting in a type of stochastic game

Motion of open vortex-current filaments under the Biot-Savart model

Vortex-current filaments have been used to study phenomenon such as coronal loops and solar flares as well as Tokamak, and recent experimental work has demonstrated dynamics akin to vortex-current filaments on a table-top plasma focus device. While MHD vortex dynamics and related applications to turbulence have attracted consideration in the literature due to a wide variety of applications, not much analytical progress has been made in this area, and the analysis of such vortex-current filament solutions under various geometries may motivate further experimental efforts. To this end, we consider the motion of open, isolated vortex-current filaments in the presence of magnetohydrodynamic (MHD) as well as the standard hydrodynamic effects. We begin with the vortex-current model of Yatsuyanagi et al. giving the self-induced motion of a vortex-current filament. We give the "cut-off" formulation of the Biot-Savart integrals used in this model, to avoid the singularity at the vortex core. We then study the motion of a variety of vortex-current filaments, including helical, planar, and self-similar filament structures. In the case where MHD effects are weak relative to hydrodynamic effects, the filaments behave as expected from the pure hydrodynamic theory. However, when MHD effects are strong enough to dominate, then we observe structural changes to the filaments in all cases considered. The most common finding is reversal of vortex-current filament orientation for strong enough MHD effects. Kelvin waves along a vortex filament (as seen for helical and selfsimilar structures) will reverse their translational and rotational motion under strong MHD effects. Our findings support the view that vortex-current filaments can be studied in a manner similar to classical hydrodynamic vortex filaments, with the primary role of MHD effects being to change the filament motion, while preserving the overall geometric structure of such filaments

Nonlinear dynamics from discrete time two-player status-seeking games

We study the dynamics of two-player status-seeking games where moves are made simultaneously in discrete time. For such games, each player's utility function will depend on both non-positional goods and positional goods (the latter entering into "status"). In order to understand the dynamics of such games over time, we sample a variety of different general utility functions, such as CES, composite log-Cobb-Douglas, and King-Plosser-Rebelo utility functions (and their various simplifications). For the various cases considered, we determine asymptotic dynamics of the two-player game, demonstrating the existence of stable equilibria, periodic orbits, or chaos, and we show that the emergent dynamics will depend strongly on the utility functions employed. For periodic orbits, we provide bifurcation diagrams to show the existence or non-existence of period doubling or chaos resulting from bifurcations due to parameter shifts. In cases where multiple feasible solution branches exist at each iteration, we consider both cases where deterministic or random selection criteria are employed to select the branch used, the latter resulting in a type of stochastic game