Sensitivity analysis of oscillator models in the space of phase-response curves: Oscillators as open systems

Oscillator models are central to the study of system properties such as entrainment or synchronization. Due to their nonlinear nature, few system-theoretic tools exist to analyze those models. The paper develops a sensitivity analysis for phase-response curves, a fundamental one-dimensional phase reduction of oscillator models. The proposed theoretical and numerical analysis tools are illustrated on several system-theoretic questions and models arising in the biology of cellular rhythms

Critical Transitions In a Model of a Genetic Regulatory System

We consider a model for substrate-depletion oscillations in genetic systems, based on a stochastic differential equation with a slowly evolving external signal. We show the existence of critical transitions in the system. We apply two methods to numerically test the synthetic time series generated by the system for early indicators of critical transitions: a detrended fluctuation analysis method, and a novel method based on topological data analysis (persistence diagrams).Comment: 19 pages, 8 figure

Parameter-sweeping techniques for temporal dynamics of neuronal systems: case study of Hindmarsh-Rose model

Background: Development of effective and plausible numerical tools is an imperative task for thorough studies of nonlinear dynamics in life science applications. Results: We have developed a complementary suite of computational tools for twoparameter screening of dynamics in neuronal models. We test a ‘brute-force’ effectiveness of neuroscience plausible techniques specifically tailored for the examination of temporal characteristics, such duty cycle of bursting, interspike interval, spike number deviation in the phenomenological Hindmarsh-Rose model of a bursting neuron and compare the results obtained by calculus-based tools for evaluations of an entire spectrum of Lyapunov exponents broadly employed in studies of nonlinear systems. Conclusions: We have found that the results obtained either way agree exceptionally well, and can identify and differentiate between various fine structures of complex dynamics and underlying global bifurcations in this exemplary model. Our future planes are to enhance the applicability of this computational suite for understanding of polyrhythmic bursting patterns and their functional transformations in small networks

Exact control of genetic networks in a qualitative framework: the bistable switch example

International audienceA qualitative method to control piecewise affine differential systems is proposed and explored for application to genetic regulatory networks. This study considers systems whose outputs and inputs are of a qualitative form, well suited to experimental devices: the measurements indicate whether the variables are "strongly" or "weakly" expressed, that is, only the region of the state space where trajectories evolve at each instant can be known. The control laws are piecewise constant functions in each region and in time, and are only allowed to take three qualitative values corresponding to no control (u=1u=1), high synthesis rates (View the MathML sourceu=umax) or low synthesis rates (View the MathML sourceu=umin). The problems of controlling the bistable switch to each of its steady states is considered. Exact solutions are given to asymptotically control the system to either of its two stable steady states. Two approximate solutions are suggested to the problem of controlling the system to the unstable steady state: either control to a neighborhood of the state, or in the form of a periodic cycle that passes through the state

Dynamic analysis of self-oscillating H-bridge inverters with state feedback

This paper presents a comprehensive approach to analyze the dynamics of a generalized model of resonant inverters using nonsmooth dynamical system theory. The model simultaneously covers both parallel and series resonant inverters under state feedback control. The multi-parametric physical space is reduced to a plane, which is divided in several regions with different dynamical behavior. The boundaries separating these regions are located by solving their corresponding equations and it is found that they all emerge from a singular point in the parameter plane. Suitability for applications of these regions is emphasized, thus providing useful criteria for parameter selection.Postprint (author's final draft

Exact Coherent Structures and Data-Driven Modeling in a Two-Dimensional Fluid

Turbulence is one of the most ubiquitous features of the world around us. Its signatures can be found at every scale, in the small eddies of streams to the vortex wakes of airliners, from ocean currents to the cosmic swirls of interstellar gases. Fluids like these have been studied for centuries, but while many advances have been made toward understanding the captivating patterns that they create, predicting the evolution of turbulent fluids remains one of the most notoriously difficult unsolved problems in classical physics. The Navier-Stokes equation is a deterministic, high-dimensional partial differential equation which allows us to make limited predictions about fluids under particular conditions. However, a characteristic feature of turbulence is that it is chaotic, meaning the evolution of a turbulent fluid is highly sensitive to its initial conditions. In practice these initial conditions can never be measured precisely enough to make long-term predictions tractable, and often measurements cannot be made of every physical quantity that goes into the Navier-Stokes equation. Physicists have thus turned recently to developing data-driven computational fluid models to gather statistics about recurring patterns that guide the flow's evolution, which can then be used to characterize an experimental flow. In this dissertation, two data-driven approaches are explored in the context of a shallow, driven fluid flow, an experimental approximation of two-dimensional (2D) Kolmogorov flow. The first approach provides experimental evidence for the statistical role of exact, unstable solutions of the Navier-Stokes equation known as exact coherent structures. In particular, periodic orbits are shown to play an important role in guiding the dynamics of a turbulent flow, as the fluid flow spends a large amount of time shadowing the most relevant orbits as predicted by periodic orbit theory. In the second approach, a weak formulation of the symbolic regression algorithm is used to develop a model of the 3D fluid using only a 2D approximation of the velocity field. The model can then be used to recreate the pressure and forcing fields, yielding a modified, quasi-2D Navier-Stokes equation that governs the flow and agrees with the first-principles model derived in previous studies. Finally, as fluid properties change, the variation in the coefficients of this quasi-2D model are also in agreement with predictions from previous work and provide a useful diagnostic tool for common experimental errors. The substantial progress provided by this dissertation suggests that physics-informed data-driven analysis of turbulent flows provides an important validation of existing models and theories.Ph.D

Robust and Adaptive Attitude Control Of Spacecraft Using Solar Radiation Pressure

Satellites in orbit are expected to maintain a preset attitude either pointing towards Earth (in case of satellites for weather) or pointing towards space for the purpose of research and exploration. The satellite as a system though is extremely nonlinear and the system parameters are not easily available. The goal of this thesis is to develop robust and adaptive control laws that can be used to control the attitude of satellites in elliptic orbits. The attitude of the satellite is controlled by the use of Solar Radiation Pressure (SRP) on the solar panels of the satellite. The SRP is basically a mechanical pressure caused by the photons impinging on the solar panels. By deflecting the solar panels the area that is impinged by photons is varied and therefore the torque on the satellite is also varied. This torque is used to control the attitude of the satellite which is expected to be maintained at a preset orientation. In this work, different methods will be used to control the satellite under different conditions involving state and output feedback. For state feedback all the states of the system are assumed to be available. In this case the states would involve the pitch angle, angular velocity and acceleration of the satellite. However the information on these states may not be easily available. Output feedback is when the output alone of the system is used and only an estimation of other states is used. Simulation is used to project the results of the different types of controllers