Inflation is always semi-classical: diffusion domination overproduces Primordial Black Holes

We use the Hamilton-Jacobi (H-J) formulation of stochastic inflation to describe the evolution of the inflaton during a period of Ultra-Slow Roll (USR), taking into account the field's velocity and its gravitational backreaction. We demonstrate how this formalism allows one to modify existing slow-roll (SR) formulae to be fully valid outside of the SR regime. We then compute the mass fraction, β, of Primordial Black Holes (PBHs) formed by a plateau in the inflationary potential. By fully accounting for the inflaton velocity as it enters the plateau, we find that PBHs are generically overproduced before the inflaton's velocity reaches zero, ruling out a period of free diffusion or even stochastic noise domination on the inflaton dynamics. We also examine a local inflection point and similarly conclude that PBHs are overproduced before entering a quantum diffusion dominated regime. We therefore surmise that the evolution of the inflaton is always predominantly classical with diffusion effects always subdominant. Both the plateau and the inflection point are characterized by a very sharp transition between the under- and over-production regimes. This can be seen either as severe fine-tunning on the inflationary production of PBHs, or as a very strong link between the fraction β and the shape of the potential and the plateau's extent.

Recent Trends of Controlling Chaotic Resonance and Future Perspectives

Stochastic resonance is a phenomenon in which the effects of additive noise strengthen the signal response against weak input signals in non-linear systems with a specific barrier or threshold. Recently, several studies on stochastic resonance have been conducted considering various engineering applications. In addition to additive stochastic noise, deterministic chaos causes a phenomenon similar to the stochastic resonance, which is known as chaotic resonance. The signal response of the chaotic resonance is maximized around the attractor-merging bifurcation for the emergence of chaos-chaos intermittency. Previous studies have shown that the sensitivity of chaotic resonance is higher than that of stochastic resonance. However, the engineering applications of chaotic resonance are limited. There are two possible reasons for this. First, the stochastic noise required to induce stochastic resonance can be easily controlled from outside of the stochastic resonance system. Conversely, in chaotic resonance, the attractor-merging bifurcation must be induced via the adjustment of internal system parameters. In many cases, achieving this adjustment from outside the system is difficult, particularly in biological systems. Second, chaotic resonance degrades owing to the influence of noise, which is generally inevitable in real-world systems. Herein, we introduce the findings of previous studies concerning chaotic resonance over the past decade and summarize the recent findings and conceivable approaches for the reduced region of orbit feedback method to address the aforementioned difficulties.

Iterative Solution of Linear Matrix Inequalities for the Combined Control and Observer Design of Systems with Polytopic Parameter Uncertainty and Stochastic Noise

Most research activities that utilize linear matrix inequality (LMI) techniques are based on the assumption that the separation principle of control and observer synthesis holds. This principle states that the combination of separately designed linear state feedback controllers and linear state observers, which are independently proven to be stable, results in overall stable system dynamics. However, even for linear systems, this property does not necessarily hold if polytopic parameter uncertainty and stochastic noise influence the system’s state and output equations. In this case, the control and observer design needs to be performed simultaneously to guarantee stabilization. However, the loss of the validity of the separation principle leads to nonlinear matrix inequalities instead of LMIs. For those nonlinear inequalities, the current paper proposes an iterative LMI solution procedure. If this algorithm produces a feasible solution, the resulting controller and observer gains ensure robust stability of the closed-loop control system for all possible parameter values. In addition, the proposed optimization criterion leads to a minimization of the sensitivity to stochastic noise so that the actual state trajectories converge as closely as possible to the desired operating point. The efficiency of the proposed solution approach is demonstrated by stabilizing the Zeeman catastrophe machine along the unstable branch of its bifurcation diagram. Additionally, an observer-based tracking control task is embedded into an iterative learning-type control framework.

Oscillations and variability in neuronal systems: interplay of autonomous transient dynamics and fast deterministic fluctuations

Neuronal systems are subject to rapidly fluctuations both intrinsically and externally. In mathematical models, these fluctuations are typically incorporated as stochastic noise (e.g., Gaussian white or colored noise). Noise can be both disruptive and constructive, for example, by creating irregularities and variability in otherwise regular patterns or by creating oscillatory patterns and increasing the signal coherence, respectively. The dynamic mechanisms underlying the interactions between rapidly fluctuating signals and the intrinsic properties of the target cells to produce variable and/or coherent responses are not fully understood. In particular, it is not clear what properties of the target cell's intrinsic dynamics control these interactions and whether the generation of this phenomena requires stochasticity of the input signal and, if yes, to what degree. In this paper we investigate these issues by using linearized and non-linear conductance-based models and piecewise constant (PWC) inputs with short duration pieces and variable amplitudes, which are arbitrarily, but not necessarily stochastically distributed. The amplitude distributions of the constant pieces consist of arbitrary permutations of a baseline PWC function with monotonically increasing amplitudes. In each trial within a given protocol we use one of these permutations and each protocol consists of a subset of all possible permutations, which is the only source of uncertainty in the protocol. We show that sustained oscillatory behavior can be generated in response to additive and multiplicative PWC inputs in both linear and nonlinear systems, independently of whether the stable equilibria of the corresponding unperturbed systems are foci (exhibiting damped oscillations) or nodes (exhibiting overshoots). The oscillatory responses are amplified by the model nonlinearities and attenuated for conductance-based PWC inputs as compared to current-based PWC inputs, consistent with previous theoretical and experimental work. In addition, the responses to PWC inputs exhibited variability across trials, which is reminiscent of the variability generated by stochastic noise (e.g., Gaussian white noise). This variability was modulated by the model parameters and the type of cellular intrinsic dynamics. Our analysis demonstrates that both oscillations and variability are the result of the interaction between the PWC input and the autonomous transient dynamics with little to no contribution from the dynamics around the steady-state. The generation of oscillations and variability does not require input stochasticity, but rather the sequential activation of the transient responses to abrupt changes in constant inputs. Each piece with the same amplitude evokes different responses across trials due to the differences in initial conditions in the corresponding regime. These initial conditions are determined by the value of the voltage at the end of the previous regime, which is different for different trials.The predictions made in this papers are amenable for experimental testing both in vitro and in vivo.

Iterative learning control for nonlinear nonaffine networked systems with stochastic noise in communication channels

This article investigates the convergence analysis of networked iterative learning control for nonlinear nonaffine systems firstly by considering stochastic noise introduced by the network channels. The convergence analysis is under a data-driven framework, which does not rely on any mechanism model information. To deal with the nonlinearity, both the state transition technique and the differential mean value principle are used to formulate the iterative dynamics of system states, tracking errors and input signals using a lifted matrix expression, respectively. In terms of the contraction mapping principle, the tracking error is shown to be iteratively convergent under the sense of mathematical expectation. Since the [Formula: see text]-norm is not used in the analysis, the convergence property of the tracking error is not affected by the operation interval and a good transient performance can be ensured in theory. Simulation studies test the theoretical results.

A Stochastic Kinetic Type Reactions Model for COVID-19

We propose two stochastic models for the Coronavirus pandemic. The statistical properties of the models, in particular the correlation functions and the probability density functions, were duly computed. Our models take into account the adoption of lockdown measures as well as the crucial role of hospitals and health care institutes. To accomplish this work we adopt a kinetic-type reaction approach where the modelling of the lockdown measures is obtained by introducing a new mathematical basis and the intensity of the stochastic noise is derived by statistical mechanics. We analysed two scenarios: the stochastic SIS-model (Susceptible ⇒ Infectious ⇒ Susceptible) and the stochastic SIS-model integrated with the action of the hospitals; both models take into account the lockdown measures. We show that, for the case of the stochastic SIS-model, once the lockdown measures are removed, the Coronavirus infection will start growing again. However, the combined contributions of lockdown measures with the action of hospitals and health institutes is able to contain and even to dampen the spread of the SARS-CoV-2 epidemic. This result may be used during a period of time when the massive distribution of vaccines in a given population is not yet feasible. We analysed data for USA and France. In the case of USA, we analysed the following situations: USA is subjected to the first wave of infection by Coronavirus and USA is in the second wave of SARS-CoV-2 infection. The agreement between theoretical predictions and real data confirms the validity of our approach.