MS15.2: Time-periodic systems

This study delves into the stability analysis of periodic solutions of dynamical systems, a topic of paramount importance across various engineering disciplines. At the core of such investigations lies the differential equation, serving as the foundational model from which periodic vibrations and their stability characteristics are derived. Although numerous methods and algorithms exist for finding the orbits and analysing their stability, they often come with limitations, including the need for equations to be problem-specific, constrained capabilities, limited convergence rates, and substantial computational demands. Our research aims to circumvent these challenges by leveraging well-established solvers/integrators, specifically utilizing the DifferentialEquations.jl package within the Julia programming environment. We propose an approach by developing an algorithm with linear time complexity that not only identifies the fixed point and its spectral properties but also achieves a convergence rate comparable to that of the integrator (e.g., fourth-order Runge-Kutta). This is accomplished through the affine mapping of the initial state (history function) for one period via direct integration. The rounding errors from the small perturbation of the fixed point are effectively eliminated by applying dual numbers. Our comprehensive testing across various scenarios, including nonlinear systems, demonstrates the algorithm's robust performance. This method is applicable to almost any time-periodic system that can be numerically simulated.

We study a conceptual model that describes the vertical mixing process in the North Atlantic Ocean between warmer surface water and cold, deep water. This mechanism, which is crucial for the transport of warm water to Northern latitudes, is subject to seasonal forcing due to variations in the amount of fresh glacial meltwater that enters the North Atlantic and seasonal atmospheric variations. The model takes the form of a periodically forced planar vector field for (appropriately rescaled) temperature and salinity, and it has two parameters: the density threshold $\eta$ for a change in mixing strength, and the virtual salinity flux $\mu$ that is subject to periodic variation of strength $c$. In the absence of forcing ($c=0$), the system has a stable periodic orbit in a bounded region of the $(\mu,\eta)$-plane. For $c \neq 0$ there is an interaction between this intrinsic oscillation and the periodic forcing. For fixed $\eta$, one finds dynamics on invariant tori in a bounded region of the $(c,\mu)$-plane. We determine the associated overall resonance structure by computing the rotation number $\rho$ over this region, as well as individual curves of saddle-node bifurcations that bound selected (lower-order) resonance tongues. This allows us to study how the resonance tongues in the $(c,\mu)$-plane change with the parameter $\eta$. More specifically, we present generic bifurcations of resonance tongues that explain the observed changes.