Discontinuity mappings in piecewise-smooth dynamical systems

In various engineering applications, mechanical systems consist of several moving parts that can frequently impact each other. Due to wear and tear, such collisions may result from limited clearance between mechanical components or loosened joints. These impacts cause undesirable chaotic vibrations, often resulting in mechanical damage like fatigue, crack propagation, fracture, etc. Nonlinear dynamical systems phenomenologically model such physical systems. The state variables, modeling such systems, evolve in the phase space and are interrupted by sudden changes almost instantaneously. The time scale over which such transitions occur is generally much smaller than the time period of oscillations observed for the respective physical system. Such transitions in the phase space occur when state variables interact with a hyper-surface, which models events like impacts and collisions in the physical system. This hyper-surface, also known as a discontinuity boundary, separates the phase space into distinct regions where the state variables evolve smoothly, i.e. their evolution is governed by functions with well-defined higher order time derivatives. Therefore, the evolution of such systems is modeled by smooth functions except at the discontinuity boundary, thus making them piecewise-smooth. Piecewise-smooth dynamical systems are often used to model a diverse range of physical systems undergoing rapid changes in biology, electrical engineering, ecology, finance, population dynamics, etc. Numerical investigation of piecewise-smooth systems shows that they exhibit a wide variety of rich dynamical behavior collectively known as discontinuity-induced bifurcations, including routes to chaotic attractors, not observed in systems governed by smooth functions of its arguments. Such unstable responses and chaotic vibrations are undesirable in most engineering applications. This necessitates to develop a unified theory for investigations of a large class of piecewise-smooth dynamical systems. Although a well-developed bifurcation theory for smooth dynamical systems exists, they are not directly applicable to piecewise-smooth systems. In the current literature on piecewise-smooth dynamical systems, the existing methodologies rely on the eigenvalue analysis of Poincar´e maps and Monodromy matrices by constructing a discontinuity mapping. Discontinuity mappings are pivotal in the analysis of non-smooth systems since their implementation accounts for the abrupt changes to the state variables evolving in phase space, after which conventional methods like stability analysis using Floquet multipliers and Lyapunov exponents can be directly applied in their investigation. However, existing methods to calculate discontinuity mappings rely on local linearization approaches. These only consider the first few terms in the Taylor expansions of vector fields that govern how the state variables evolve in time. Such approximations can lead to incorrect estimation of impact occurrences and discontinuity mappings for non-smooth systems with sensitive dependence on initial conditions. A deeper investigation of discontinuity mappings, presented in this thesis, obtained using conventional first-order approaches reveals that the linearized methods can predict incorrect mappings of the state variables during interaction or impact with the discontinuity barrier. These incorrect mappings necessitate the inclusion of higher-order correction terms to the widely accepted first-order discontinuity mappings, also known as saltation matrices. Furthermore, the current literature primarily focuses on discontinuity mappings of perturbations for certain limiting cases like grazing incidence. This thesis attempts to illustrate the limitation and problem arising during the implementation of a first-order discontinuity mapping for piecewise-smooth dynamical systems of hybrid (non-smooth systems comprising ordinary differential equation and discrete maps) and Filippov type (non-smooth systems governed by multiple vector fields). The thesis derives a higher-order discontinuity mapping for hybrid and Filippov systems which resolves the issues encountered during a first-order approximation. The proposed higher-order discontinuity mappings imply that not all state variables in the local neighborhood of an impacting state should undergo a mapping near the discontinuity barrier. This contradicts the predictions of the first-order saltation matrices, which allow discrete mappings for all values of perturbations in the local neighborhood of an impacting state. Compared to the exact numerical solution, the higher-order discontinuity mapping provides accurate estimates regarding the behavior of state variables near the discontinuity boundary, which the first-order saltation matrix fails to capture. Next, the thesis presents a numerical methodology to calculate higher-order saltation matrices from the derived higher-order transverse discontinuity mapping for hybrid and Filippov systems. The proposed approaches can be incorporated to accommodate higher-order correction terms up to any order. The thesis discusses methods to estimate Floquet multipliers and Lyapunov exponents for hybrid and Filippov systems, aided by the derived higher-order discontinuity mappings and saltation matrices. The results are verified with numerically obtained bifurcation diagrams of several representative non-smooth dynamical systems like impact oscillators undergoing instantaneous transitions in their state variables. Further, the thesis numerically investigates the dynamical responses of a phenomenological model representing the simplest case of a non-smooth hybrid fluid-structure interaction system interacting with a rigid barrier. The instantaneous changes in the system’s dynamics show a rich underlying mathematical structure in its responses. A comparison between the proposed higher-order discontinuity mappings with the first-order approaches for the representative non-smooth fluid-structure interaction system reveals that the higher-order correction terms provide better estimates of mapping as validated by the exact numerical solution. Next, numerical methods to evaluate Floquet multipliers and Lyapunov exponents are implemented, which can accurately predict when bifurcations occur as validated by the corresponding bifurcation diagrams obtained independently. The thesis concludes with an attempt to experimentally verify the occurrence of discontinuity-induced bifurcations typical to piecewise-smooth dynamical systems. An electronic circuit is implemented comprising two LCR circuits with a switching mechanism. This experimental setup is equivalent to a piecewise-smooth Filippov system. The experimental results verify topological changes in the system’s dynamics, typical of non-smooth dynamical systems. The phase portraits recorded experimentally corroborate with the analytical and numerical predictions obtained independently using the proposed higher-order transverse discontinuity mapping.