Nonlinear Dynamics of Systems with Discontinuities

My research focuses on dynamical systems that exhibit switching conditions, such as frictional contact, threshold-based decisions, or mode-dependent control. Such nonsmooth behaviour arises naturally across mechanical, biological, ecological, and engineered systems, as well as in control, robotics, and hybrid dynamical systems, where it can lead to sudden transitions, unexpected vibrations, multistability, instability, loss of predictability, and chaos that cannot be explained using classical smooth models. My work aims to understand and model these nonsmooth dynamics through a combination of analytical tools and numerical simulations, with particular emphasis on capturing physically meaningful behaviour at switching interfaces and translating it into predictive, system-level insight.

Self-excited stick–slip vibrations arise when contacting surfaces alternate between sticking due to static friction and slipping due to kinetic friction. This repeated transition leads to sudden changes in frictional forces, producing oscillations commonly observed in both engineering systems and everyday life. Familiar examples include the squeaking chalk on a blackboard, the noise of a creaking door, sound from violin string vibration, disc brake squeal, vibration in machining and drilling processes, seismic motion along fault lines, and so on. Despite their ubiquity, stick–slip vibrations remain difficult to predict and control because they originate from abrupt switching at the friction interface. Understanding the mechanisms behind these vibrations is therefore essential for reducing noise, limiting wear, and improving the reliability and performance of engineering systems.

A major challenge in modelling such systems is defining well-posed and physically meaningful dynamics at the discontinuity. Classical approaches from piecewise-smooth dynamics, most notably Filippov’s convex method, define motion on the switching surface through a linear combination of vector fields on either side. While mathematically elegant, this approach implicitly assumes that the coefficient of static friction is equal to the coefficient of kinetic friction. In reality, static friction is typically larger, and this discrepancy can have a profound influence on the system’s behaviour, particularly near the sticking–slipping transition.

My research addresses this limitation by analysing friction-induced vibrations in a self-excited Smooth–Discontinuous (SD) oscillator with geometric nonlinearity, using the concept of hidden dynamics, which introduces nonlinear switching behaviour that is active only on the discontinuity surface. This oscillator serves as a representative mechanical model for a wide range of real systems, including earthquake fault motion, brake disc–pad interactions, and the sliding behaviour at the base of large ice streams. By focusing on the dynamics at the friction interface rather than treating it as an idealised boundary, the model captures physically realistic features of dry friction. A blow-up transformation is used to replace the discontinuity with a switching layer, allowing the dynamics to be analysed using ideas from geometric singular perturbation theory (multiple time scale dynamics). This reveals that the friction boundary can support a rich variety of behaviours.

Population dynamics in ecological systems are often governed by threshold-based decisions rather than smooth responses. In integrated pest management (IPM), intervention strategies change abruptly when pest or predator populations cross critical levels, for example switching between no action, biological control, or chemical intervention. These decisions naturally introduce switching behaviour into predator–prey models, making their dynamics nonsmooth. Understanding how such switching affects long-term population behaviour is essential for designing management strategies that are both effective and reliable, especially in the presence of environmental variability. Similar threshold-based frameworks also arise in plant disease dynamics, avian influenza transmission, and other decision-driven biological systems. 

When both prey and predator thresholds are present, the resulting model contains two interacting switching surfaces that divide the system into three regimes, corresponding to different control actions: no intervention, combined biological and chemical control, or reduced chemical control. The intersection of these surfaces creates a codimension-2 discontinuity, where multiple vector fields interact simultaneously, and standard modelling approaches provide limited guidance. In addition, environmental effects are often time-dependent, and in this work they are modelled through periodic forcing whose frequency itself switches across regions, capturing realistic ecological mechanisms. Together, these features generate rich but challenging dynamics that are representative of many real-world systems governed by interacting thresholds.

(a) Threshold-based intervention strategy in IPM, (b) Blow-up of the switching surfaces & cylindrical blow-up of codimension-2 discontinuity, (c) Dynamical behaviour of the system.

Currently under active investigation.