On the Existence of Linearly Oscillating Galaxies

We consider two classes of steady states of the three-dimensional, gravitational Vlasov-Poisson system: the spherically symmetric Antonov-stable steady states (including the polytropes and the King model) and their plane symmetric analogues. We completely describe the essential spectrum of the self-adjoint operator governing the linearized dynamics in the neighborhood of these steady states. We also show that for the steady states under consideration, there exists a gap in the spectrum. We then use a version of the Birman-Schwinger principle first used by Mathur to derive a general criterion for the existence of an eigenvalue inside the first gap of the essential spectrum, which corresponds to linear oscillations about the steady state. It follows in particular that no linear Landau damping can occur in the neighborhood of steady states satisfying our criterion. Verification of this criterion requires a good understanding of the so-called period function associated with each steady state. In the plane symmetric case we verify the criterion rigorously, while in the spherically symmetric case we do so under a natural monotonicity assumption for the associated period function. Our results explain the pulsating behavior triggered by perturbing such steady states, which has been observed numerically.

Simultaneous Bifurcation of Limit Cycles and Critical Periods

The present work introduces the problem of simultaneous bifurcation of limit cycles and critical periods for a system of polynomial differential equations in the plane. The simultaneity concept is defined, as well as the idea of bi-weakness in the return map and the period function. Together with the classical methods, we present an approach which uses the Lie bracket to address the simultaneity in some cases. This approach is used to find the bi-weakness of cubic and quartic Liénard systems, the general quadratic family, and the linear plus cubic homogeneous family. We finish with an illustrative example by solving the problem of simultaneous bifurcation of limit cycles and critical periods for the cubic Liénard family.

Monotonicity of a Quadratic Lienard Equation

We study the period function of the quadratic Lienard equation of a certain type in order to give necessary and sufficient conditions for monotonicity and isochronicty of the period function. We apply this result to identify the region of monotonicity of the period function of particular cases.

Estimation of Synaptic Activity during Neuronal Oscillations

In the study of brain connectivity, an accessible and convenient way to unveil local functional structures is to infer the time trace of synaptic conductances received by a neuron by using exclusively information about its membrane potential (or voltage). Mathematically speaking, it constitutes a challenging inverse problem: it consists in inferring time-dependent parameters (synaptic conductances) departing from the solutions of a dynamical system that models the neuron’s membrane voltage. Several solutions have been proposed to perform these estimations when the neuron fluctuates mildly within the subthreshold regime, but very few methods exist for the spiking regime as large amplitude oscillations (revealing the activation of complex nonlinear dynamics) hinder the adaptability of subthreshold-based computational strategies (mostly linear). In a previous work, we presented a mathematical proof-of-concept that exploits the analytical knowledge of the period function of the model. Inspired by the relevance of the period function, in this paper we generalize it by providing a computational strategy that can potentially adapt to a variety of models as well as to experimental data. We base our proposal on the frequency versus synaptic conductance curve (f−gsyn), derived from an analytical study of a base model, to infer the actual synaptic conductance from the interspike intervals of the recorded voltage trace. Our results show that, when the conductances do not change abruptly on a time-scale smaller than the mean interspike interval, the time course of the synaptic conductances is well estimated. When no base model can be cast to the data, our strategy can be applied provided that a suitable f−gsyn table can be experimentally constructed. Altogether, this work opens new avenues to unveil local brain connectivity in spiking (nonlinear) regimes.

Real-Time Traffic Flow Forecasting via a Novel Method Combining Periodic-Trend Decomposition

Accurate and timely traffic flow forecasting is a critical task of the intelligent transportation system (ITS). The predicted results offer the necessary information to support the decisions of administrators and travelers. To investigate trend and periodic characteristics of traffic flow and develop a more accurate prediction, a novel method combining periodic-trend decomposition (PTD) is proposed in this paper. This hybrid method is based on the principle of “decomposition first and forecasting last”. The well-designed PTD approach can decompose the original traffic flow into three components, including trend, periodicity, and remainder. The periodicity is a strict period function and predicted by cycling, while the trend and remainder are predicted by modelling. To demonstrate the universal applicability of the hybrid method, four prevalent models are separately combined with PTD to establish hybrid models. Traffic volume data are collected from the Minnesota Department of Transportation (Mn/DOT) and used to conduct experiments. Empirical results show that the mean absolute error (MAE), mean absolute percentage error (MAPE), and mean square error (MSE) of hybrid models are averagely reduced by 17%, 17%, and 29% more than individual models, respectively. In addition, the hybrid method is robust for a multi-step prediction. These findings indicate that the proposed method combining PTD is promising for traffic flow forecasting.