Chaos, Fractals and Inflation

In order to draw out the essential behavior of the universe, investigations of early universe cosmology often reduce the complex system to a simple integrable system. Inflationary models are of this kind as they focus on simple scalar field scenarios with correspondingly simple dynamics. However, we can be assured that the universe is crowded with many interacting fields of which the inflaton is but one. As we describe, the nonlinear nature of these interactions can result in a complex, chaotic evolution of the universe. Here we illustrate how chaotic effects can arise even in basic models such as homogeneous, isotropic universes with two scalar fields. We find inflating universes which act as attractors in the space of initial conditions. These universes display chaotic transients in their early evolution. The chaotic character is reflected by the fractal border to the basin of attraction. The broader implications are likely to be felt in the process of reheating as well as in the nature of the cosmic background radiation.Comment: 16 pages, RevTeX. See published version for fig

Effects of additive noise on the stability of glacial cycles

It is well acknowledged that the sequence of glacial-interglacial cycles is paced by the astronomical forcing. However, how much is the sequence robust against natural fluctuations associated, for example, with the chaotic motions of atmosphere and oceans? In this article, the stability of the glacial-interglacial cycles is investigated on the basis of simple conceptual models. Specifically, we study the influence of additive white Gaussian noise on the sequence of the glacial cycles generated by stochastic versions of several low-order dynamical system models proposed in the literature. In the original deterministic case, the models exhibit different types of attractors: a quasiperiodic attractor, a piecewise continuous attractor, strange nonchaotic attractors, and a chaotic attractor. We show that the combination of the quasiperiodic astronomical forcing and additive fluctuations induce a form of temporarily quantised instability. More precisely, climate trajectories corresponding to different noise realizations generally cluster around a small number of stable or transiently stable trajectories present in the deterministic system. Furthermore, these stochastic trajectories may show sensitive dependence on very small amounts of perturbations at key times. Consistently with the complexity of each attractor, the number of trajectories leaking from the clusters may range from almost zero (the model with a quasiperiodic attractor) to a significant fraction of the total (the model with a chaotic attractor), the models with strange nonchaotic attractors being intermediate. Finally, we discuss the implications of this investigation for research programmes based on numerical simulators. }Comment: Parlty based on a lecture given by M. Crucifix at workshop held in Rome in 2013 as a part of Mathematics of Planet Earth 201

Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks

We introduce a data-driven forecasting method for high-dimensional chaotic systems using long short-term memory (LSTM) recurrent neural networks. The proposed LSTM neural networks perform inference of high-dimensional dynamical systems in their reduced order space and are shown to be an effective set of nonlinear approximators of their attractor. We demonstrate the forecasting performance of the LSTM and compare it with Gaussian processes (GPs) in time series obtained from the Lorenz 96 system, the Kuramoto-Sivashinsky equation and a prototype climate model. The LSTM networks outperform the GPs in short-term forecasting accuracy in all applications considered. A hybrid architecture, extending the LSTM with a mean stochastic model (MSM-LSTM), is proposed to ensure convergence to the invariant measure. This novel hybrid method is fully data-driven and extends the forecasting capabilities of LSTM networks.Comment: 31 page

Anisotropic Inflation from Charged Scalar Fields

We consider models of inflation with U(1) gauge fields and charged scalar fields including symmetry breaking potential, chaotic inflation and hybrid inflation. We show that there exist attractor solutions where the anisotropies produced during inflation becomes comparable to the slow-roll parameters. In the models where the inflaton field is a charged scalar field the gauge field becomes highly oscillatory at the end of inflation ending inflation quickly. Furthermore, in charged hybrid inflation the onset of waterfall phase transition at the end of inflation is affected significantly by the evolution of the background gauge field. Rapid oscillations of the gauge field and its coupling to inflaton can have interesting effects on preheating and non-Gaussianities.Comment: minor changes, references added, figures are modified, conforms JCAP published versio

Chaos-based communication scheme using proportional and proportional-integral observers

In this paper, we propose a new chaos-based communication scheme using the observers. The novelty lies in the masking procedure that is employed to hide the confidential information using the chaotic oscillator. We use a combination of the addition and inclusion methods to mask the information. The performance of two observers, the proportional observer (P-observer) and the proportional integral observer (PI-observer) is compared that are employed as receivers for the proposed communication scheme. We show that the P-observer is not suitable scheme since it imposes unpractical constraints on the messages to be transmitted. On the other hand, we show that the PI-observer is the better solution because it allows greater flexibility in choosing the gains of the observer and does not impose any unpractical restrictions on the message