A Bendixson-Dulac theorem for some piecewise systems

The Bendixson-Dulac Theorem provides a criterion to find upper bounds for the number of limit cycles in analytic differential systems. We extend this classical result to some classes of piecewise differential systems. We apply it to three different Liénard piecewise differential systems ¨ x+f±(x)˙ x+x = 0. The first is linear, the second is rational and the last corresponds to a particular extension of the cubic van der Pol oscillator. In all cases, the systems present regions in the parameter space with no limit cycles and others having at most one

Self-excited oscillations of flexible-channel flow with fixed upstream flux

Self-excited oscillations in a collapsible-tube flow driven by fixed upstream flux have been observed by numerical and laboratory experiments. In this thesis we attempt to understand the mechanism of onset of these oscillations by focusing on a reduced physical model. We consider flow in a finite-length planar channel, where a segment of one wall is replaced by a membrane under longitudinal tension. The upstream flux and downstream pressure are prescribed and an external linear pressure distribution is applied to the membrane such that the system admits uniform Poiseuille flow as a steady solution. We describe the system using a one-dimensional model that accounts for viscous and fluid inertial effects. We perform linear stability analysis and weakly nonlinear analysis on the one-dimensional model, the resulting predictions are tested against two-dimensional Navier–Stokes numerical simulation. When the membrane has similar length to the rigid segment of channel downstream of the membrane, we find that in a narrow parameter regime we consider “mode-2” oscillations (i.e. membrane displacements with two extrema) are largely independent of the downstream segment but are driven by divergent instabilities of two non-uniform steady configurations of the membrane. When the downstream segment is much longer than the membrane, our analysis reveals how instability is promoted by a 1:1 resonant interaction between two modes, with the resulting oscillations described by a fourth-order amplitude equation. This predicts the existence of saturated sawtooth oscillations, which we reproduce in full Navier–Stokes simulations of the same system. In this case, our analysis shows some agreements with experimental observations, namely that increasing the length of the downstream tube reduces the frequency of oscillations but has little effect on the conditions for onset. We also use linear stability analysis to show that steady highly-collapsed solutions, constructed by utilizing matched asymptotic expansions, are very unstable, which allows the possibility that they are a precursor to slamming motion whereby the membrane becomes transiently constricted very close to the opposite rigid wall before rapidly recovering

Self-excited oscillations of flexible-channel flow with fixed upstream flux

Self-excited oscillations in a collapsible-tube flow driven by fixed upstream flux have been observed by numerical and laboratory experiments. In this thesis we attempt to understand the mechanism of onset of these oscillations by focusing on a reduced physical model. We consider flow in a finite-length planar channel, where a segment of one wall is replaced by a membrane under longitudinal tension. The upstream flux and downstream pressure are prescribed and an external linear pressure distribution is applied to the membrane such that the system admits uniform Poiseuille flow as a steady solution. We describe the system using a one-dimensional model that accounts for viscous and fluid inertial effects. We perform linear stability analysis and weakly nonlinear analysis on the one-dimensional model, the resulting predictions are tested against two-dimensional Navier–Stokes numerical simulation. When the membrane has similar length to the rigid segment of channel downstream of the membrane, we find that in a narrow parameter regime we consider “mode-2” oscillations (i.e. membrane displacements with two extrema) are largely independent of the downstream segment but are driven by divergent instabilities of two non-uniform steady configurations of the membrane. When the downstream segment is much longer than the membrane, our analysis reveals how instability is promoted by a 1:1 resonant interaction between two modes, with the resulting oscillations described by a fourth-order amplitude equation. This predicts the existence of saturated sawtooth oscillations, which we reproduce in full Navier–Stokes simulations of the same system. In this case, our analysis shows some agreements with experimental observations, namely that increasing the length of the downstream tube reduces the frequency of oscillations but has little effect on the conditions for onset. We also use linear stability analysis to show that steady highly-collapsed solutions, constructed by utilizing matched asymptotic expansions, are very unstable, which allows the possibility that they are a precursor to slamming motion whereby the membrane becomes transiently constricted very close to the opposite rigid wall before rapidly recovering

Rate-induced critical transitions

This thesis focuses on rate-induced critical transitions or tipping points (R-tipping points), where the system undergoes a critical transition if the time-varying external conditions vary faster than some critical rate. Such a critical transition is usually a sudden and unexpected change of the system state. The change can be either irreversible: a permanent tipping point with no return to the original state, or reversible: a temporary tipping point with self-recovery back to the original state, both of which may cause significant consequences in applications. Indeed, R-tipping is an ubiquitous nonlinear phenomenon in nature that remains largely unexplored by the scientists. From a mathematical viewpoint, it is a genuine nonautonomous instability that cannot be explained by the classical (autonomous) bifurcation theory and requires an alternative approach. The first part of the thesis focuses on a mathematical framework for R-tipping in systems of nonautonomous differential equations, where the nonautonomous terms representing time-varying external conditions decay asymptotically. In particular, special compactification techniques for asymptotically autonomous systems are developed to simplify analysis of R-tipping. In the second part of the thesis, the main concepts of edge states and thresholds are introduced to define the R-tipping phenomenon. Then, simple testable criteria for the occurrence of reversible and irreversible R-tipping in arbitrary dimension are given. This part extends the previous results on irreversible R-tipping in one dimension. The third part of the thesis identifies canonical examples of R-tipping based on the system dimension, timescales and the threshold type. These examples are relatively simple low-dimensional nonlinear systems that capture different R-tipping mechanisms. R-tipping analysis of canonical examples, which is underpinned by the compactification framework developed in the second part, reveals intricate R-tipping diagrams with multiple critical rates and transitions between different types of R-tipping

Nonlinear resonance and excitability in interconnected systems

Engineering design amounts to develop components and interconnect them to obtain a desired behaviour. While in the context of equilibrium dynamics there is a well-developed theory that can account for robustness and optimality in this process, we still lack a corresponding methodology for nonequilibrium dynamics and in particular oscillatory behaviours. With the aim of fostering such a theory, this thesis studies two basic interconnections in the contexts of nonlinear resonance and excitability, two phenomena with the potential of encompassing a large number of applications. The first interconnection is considered in the context of vibration absorption. It corresponds to coupling two Duffing oscillators, the prototypical example of nonlinear resonator. Of primary interest is the frequency response of the system, which quantifies the behaviour in presence of harmonic forces. The analysis focuses on how isolated families of solutions appear and merge with a main one. Using singularity theory it is possible to organise these solutions in the space of parameters and delimit their presence through numerical methods. The second interconnection studied in this dissertation appears in the context of excitable circuits. Combining a fast excitable system and a slower oscillatory system that share a similar structure naturally leads to bursting. The resulting system has a slow-fast structure that can be leveraged in the analysis. The first step of this analysis is a novel slow-fast model of bistability between a rest state and a spiking attractor. Following this, the analysis moves to the complete interconnection, and in particular on how it can generate different patterns of bursting activity