A frequency-domain approach to the analysis of stability and bifurcations in nonlinear systems described by differential-algebraic equations

A general numerical technique is proposed for the assessment of the stability of periodic solutions and the determination of bifurcations for limit cycles in autonomous nonlinear systems represented by ordinary differential equations in the differential-algebraic form. The method is based on the harmonic balance technique, and exploits the same Jacobian matrix of the nonlinear system used in the Newton iterative numerical solution of the harmonic balance equations for the determination of the periodic steady-state. To demonstrate the approach, it is applied to the determination of the bifurcation curves in the parameters' space of Chua's circuit with cubic nonlinearity, and to study the dynamics of the limit cycle of a Colpitts oscillato

Investigation of the Stability of a Molten Salt Fast Reactor

This work focusses on analysing the stability of the MSFR – a molten salt reactor with a fast neutron spectrum. The investigations are based on a model, which was published and studied by the Politecnico di Milano using a linear approach. Since linear methods can only provide stability information to a limited extent, this work continues the conducted investigations by applying nonlinear methods. In order to examine the specified reactor model, the system equations were implemented, adjusted and verified using MATLAB code. With the help of the computational tool MatCont, a so-called fixed-point solution was tracked and its stability monitored during the variation of selected control parameters. It was found that the considered fixed point does not change its stability state and remains stable. Coexisting fixed points or periodic solutions could not be detected. Therefore, the analysed MSFR model is considered to be a stable system, in which the solutions always tend towards a steady state.:1. Introduction 2. Molten Salt Reactor Technology 2.1. Introduction 2.2. Historical Development 2.3. Working Principle of Molten Salt Reactors 2.4. Molten Salt Coolants 2.5. Advantages and Drawbacks 2.6. Classification 2.7. Molten Salt Fast Reactor Design 3. Stability Characteristics of Dynamical Systems 3.1. Introduction 3.2. Dynamical Systems 3.3. Stability Concepts 3.3.1. Introduction 3.3.2. Lagrange Stability (Bounded Stability) 3.3.3. Lyapunov Stability 3.3.4. Poincaré Stability (Orbital Stability) 3.4. Fixed-Point Solutions 3.4.1. Stability Analysis of Fixed-Point Solutions 3.4.2. Bifurcations of Fixed-Point Solutions 3.5. Periodic Solutions 3.5.1. Stability Analysis of Periodic Solutions 3.5.2. Bifurcations of Periodic Solutions 4. Analysed Reactor System 4.1. Introduction 4.2. Specified Reactor Model 4.3. Implementation and Verification of the Linearised System of Equations 4.3.1. Linearised System of Delayed Differential Equations 4.3.2. Comparison with Reference Plots 4.3.3. Adaptation of Parameter Values 4.4. Implementation and Verification of the Nonlinear System of Equations 4.4.1. Nonlinear System of Delayed Differential Equations 4.4.2. Delayed Neutron Precursor Equation Adjustments 4.4.3. Salt Temperature Equation Adjustments 4.4.4. Nonlinear System of Ordinary Differential Equations 4.4.5. Verification of the Nonlinear System of Ordinary Differential Equations 5. Conducted Stability Analyses 5.1. Introduction 5.2. Nonlinear Stability Analysis 5.2.1. Implementation 5.2.2. Results 5.2.3. Interpretation 5.3. Linear Stability Analysis 5.3.1. Comparison Between the Linearised and Nonlinearised MSFR System of Equations 5.3.2. Stability Investigations Using a Linear Criterion 5.4. MatCont Reliability Test Using an MSBR Model 6. Conclusions and Recommendations for Future StudiesIm Fokus dieser Arbeit steht die Stabilitätsanalyse des MSFR – eines Flüssigsalzreaktors mit schnellem Neutronenspektrum. Als Grundlage wurde ein Modell verwendet, das am Politecnico di Milano erstellt und dort mittels linearer Methoden untersucht wurde. Da lineare Betrachtungen nur eingeschränkte Stabilitätsaussagen treffen können, erweitert diese Arbeit die Untersuchungen um die nichtlineare Stabilitätsanalyse. Zur Untersuchung des vorgegebenen Reaktormodells wurden die Systemgleichungen in MATLAB übertragen und verifiziert. Mithilfe der Rechensoftware MatCont wurde eine sogenannten Fixpunkt-Lösung des Modells unter der Variation ausgewählter Parameter verfolgt und deren Stabilität überprüft. Es hat sich gezeigt, dass der betrachtete Fixpunkt seinen Stabilitätszustand dabei nicht verändert und stabil bleibt. Koexistierende Fixpunkte oder periodische Lösungen konnten nicht nachgewiesen werden. Daher gilt das betrachtete MSFR-Modell als ein stabiles System, dessen Lösungen immer auf einen stationären Zustand zulaufen.:1. Introduction 2. Molten Salt Reactor Technology 2.1. Introduction 2.2. Historical Development 2.3. Working Principle of Molten Salt Reactors 2.4. Molten Salt Coolants 2.5. Advantages and Drawbacks 2.6. Classification 2.7. Molten Salt Fast Reactor Design 3. Stability Characteristics of Dynamical Systems 3.1. Introduction 3.2. Dynamical Systems 3.3. Stability Concepts 3.3.1. Introduction 3.3.2. Lagrange Stability (Bounded Stability) 3.3.3. Lyapunov Stability 3.3.4. Poincaré Stability (Orbital Stability) 3.4. Fixed-Point Solutions 3.4.1. Stability Analysis of Fixed-Point Solutions 3.4.2. Bifurcations of Fixed-Point Solutions 3.5. Periodic Solutions 3.5.1. Stability Analysis of Periodic Solutions 3.5.2. Bifurcations of Periodic Solutions 4. Analysed Reactor System 4.1. Introduction 4.2. Specified Reactor Model 4.3. Implementation and Verification of the Linearised System of Equations 4.3.1. Linearised System of Delayed Differential Equations 4.3.2. Comparison with Reference Plots 4.3.3. Adaptation of Parameter Values 4.4. Implementation and Verification of the Nonlinear System of Equations 4.4.1. Nonlinear System of Delayed Differential Equations 4.4.2. Delayed Neutron Precursor Equation Adjustments 4.4.3. Salt Temperature Equation Adjustments 4.4.4. Nonlinear System of Ordinary Differential Equations 4.4.5. Verification of the Nonlinear System of Ordinary Differential Equations 5. Conducted Stability Analyses 5.1. Introduction 5.2. Nonlinear Stability Analysis 5.2.1. Implementation 5.2.2. Results 5.2.3. Interpretation 5.3. Linear Stability Analysis 5.3.1. Comparison Between the Linearised and Nonlinearised MSFR System of Equations 5.3.2. Stability Investigations Using a Linear Criterion 5.4. MatCont Reliability Test Using an MSBR Model 6. Conclusions and Recommendations for Future Studie

Instability of Turing patterns in reaction-diffusion-ODE systems

The aim of this paper is to contribute to the understanding of the pattern formation phenomenon in reaction-diffusion equations coupled with ordinary differential equations. Such systems of equations arise, for example, from modeling of interactions between cellular processes such as cell growth, differentiation or transformation and diffusing signaling factors. We focus on stability analysis of solutions of a prototype model consisting of a single reaction-diffusion equation coupled to an ordinary differential equation. We show that such systems are very different from classical reaction-diffusion models. They exhibit diffusion-driven instability (Turing instability) under a condition of autocatalysis of non-diffusing component. However, the same mechanism which destabilizes constant solutions of such models, destabilizes also all continuous spatially heterogeneous stationary solutions, and consequently, there exist no stable Turing patterns in such reaction-diffusion-ODE systems. We provide a rigorous result on the nonlinear instability, which involves the analysis of a continuous spectrum of a linear operator induced by the lack of diffusion in the destabilizing equation. These results are extended to discontinuous patterns for a class of nonlinearities.Comment: This is a new version of the paper. Presentation of results was essentially revised according to referee suggestion

Local stability of Kolmogorov forward equations for finite state nonlinear Markov processes

The focus of this work is on local stability of a class of nonlinear ordinary differential equations (ODE) that describe limits of empirical measures associated with finite-state weakly interacting N-particle systems. Local Lyapunov functions are identified for several classes of such ODE, including those associated with systems with slow adaptation and Gibbs systems. Using results from [5] and large deviations heuristics, a partial differential equation (PDE) associated with the nonlinear ODE is introduced and it is shown that positive definite subsolutions of this PDE serve as local Lyapunov functions for the ODE. This PDE characterization is used to construct explicit Lyapunov functions for a broad class of models called locally Gibbs systems. This class of models is significantly larger than the family of Gibbs systems and several examples of such systems are presented, including models with nearest neighbor jumps and models with simultaneous jumps that arise in applications.Comment: Updated to include Acknowledgement

Perturbation of the spectra for asymptotically constant differential operators and applications

We study the spectra and spectral curves for a class of differential operators with asymptotically constant coefficients. These operators widely arise as the linearizations of nonlinear partial differential equations about patterns or nonlinear waves. We present a unified framework to prove the perturbation results on the related spectra and spectral curves. The proof is based on exponential dichotomies and the Brouwer degree theory. As applications, we employ the developed theory to study the stability of quasi-periodic solutions of the Ginzburg-Landau equation, fold-Hopf bifurcating periodic solutions of reaction-diffusion systems coupled with ordinary differential equations, and periodic annulus of the hyperbolic Burgers-Fisher model.Comment: 21 page

Linearized Stability of Partial Differential Equations with Application to Stabilization of the Kuramoto--Sivashinsky Equation

This is a final draft of a work, prior to publisher editing and production, that appears in Siam J. Control Optim. Vol. 56, No 1, pp 120-147. http://dx.doi.org/10.1137/140993417.Linearization is a useful tool for analyzing the stability of nonlinear differential equations. Unfortunately, the proof of the validity of this approach for ordinary differential equations does not generalize to all nonlinear partial differential equations. General results giving conditions for when stability (or instability) of the linearized equation implies the same for the nonlinear equation are given here. These results are applied to stability and stabilization of the Kuramoto--Sivashinsky equation, a nonlinear partial differential equation that models reaction-diffusion systems. The stability of the equilibrium solutions depends on the value of a positive parameter $\nu$. It is shown that if $\nu>1$, then the set of constant equilibrium solutions is globally asymptotically stable. If $\nu<1$, then the equilibria are unstable. It is also shown that stabilizing the linearized equation implies local exponential stability of the equation. Stabilization of the Kuramoto--Sivashinsky equation using a single distributed control is considered and it is described how to use a finite-dimensional approximation to construct a stabilizing controller. The results are illustrated with simulations.Natural Sciences and Engineering Research Council of Canada (NSERC