Averaging theory and catastrophes

TL;DR

This work addresses how time-periodic perturbations affect bifurcations in near-autonomous systems using averaging. By focusing on $ ext{K}$-universal unfoldings of the guiding system, the authors prove that catastrophes persist in the non-autonomous setting under strong fibred diffeomorphisms, while non-versal bifurcations like transcritical and pitchfork may not persist in their simplest form. They provide precise theorems linking the catastrophe surface of the Poincaré map to the catastrophe surface of the guiding system, and they illustrate both persistence (fold) and non-persistence (transcritical, pitchfork) through concrete time-periodic examples. Additionally, the paper discusses topological equivalence and demonstrates that catastrophe geometry can persist even when topological conjugacy fails, using a saddle-focus counterexample. Overall, the results offer a cohesive framework for predicting how averaged, time-varying perturbations shape the qualitative dynamics of nonlinear systems, with implications for models with seasonal or periodic parameter variation.

Abstract

When a dynamical system is subject to a periodic perturbation, the averaging method can be applied to obtain an autonomous leading order "guiding system", placing the time dependence at higher orders. Recent research focused on investigating invariant structures in non-autonomous differential systems arising from hyperbolic structures in the guiding system, such as periodic orbits and invariant tori. Complementarily, the effect that bifurcations in the guiding system have on the original non-autonomous one has also been recently explored, albeit less frequently. This paper extends this study by providing a broader description of the dynamics that can emerge from non-hyperbolic structures of the guiding system. Specifically, we prove here that $\mathcal{K}$-universal bifurcations in the guiding system `persist' in the original non-autonomous one, while non-versal bifurcations, such as the transcritical and pitchfork, do not. We illustrate the results on examples of a fold, a transcritical, a pitchfork, and a saddle-focus.

Figures (8)

• Figure 1: The catastrophe surface $Z_{g_\ell}$ of the guiding system (left, suspended through $\varepsilon\in\mathbb R$), and the catastrophe surface $M_\Pi$ of the time-dependent system (right). $M_\Pi$ is the image of $Z_{g_\ell}$ under the diffeomorphism $\Phi$, and $Z_{g_\ell}\times\{0\}$ is invariant under $\Phi$.
• Figure 2: Solutions of the system \ref{['foldeg1']} exhibiting a fold, with $\mu=\pm0.5$ and $\varepsilon=0.4$. The upper picture shows the Poincaré map of the original system (red/orange points), converging in forward/backward time to the fixed points of the guiding system (black curves). For two values of $\mu$ we plot the solutions below. For $\mu<\mu^*(0.5)$, the exact solutions (red dotted curves) and averaged solutions (blue curves) oscillate around the guiding solutions (black dotted curves), all converging in forward/backward time onto the stable/unstable fixed points (black lines). For $\mu>\mu^*(0.5)$ there are no fixed points and the solutions diverge. (For the exact solution we plot the variable $x=X+\varepsilon\cos t$).
• Figure 3: The surface of fixed points $x^2+\mu x+\varepsilon c=0$ plotted for $c=1$ in $(x,\mu,\varepsilon)$ space (with $x_*$ denoting the fixed-point value of $x$). The transcritical bifurcation at $\varepsilon=0$ degenerates into a pair of fold bifurcations for $\varepsilon>0$ and two stable families of fixed points for $\varepsilon<0$. Sections of the surface at different $\varepsilon$ give the bifurcation diagrams with varying $\mu$ (stable/unstable branches indicated by full/dotted curves).
• Figure 4: Solutions of the perturbed transcritical system. The Poincaré map of $x(t)$$\mod(t,2\pi)$, showing exact solutions converging in forward time (red) and backward time (orange) onto the stable and unstable fixed points (black curves), respectively, from initial conditions close to the fixed points if they exist, or close to the origin otherwise. The parameters used are: left $\varepsilon=-0.02,c=1$, middle $\varepsilon=0.02,c=0$, right $\varepsilon=0.02,c=1$. For $\varepsilon<0$ there are always two fixed points. For $\varepsilon>0$ and $c\neq0$ there are two fixed points only for $|\mu|>\mu_{\rm fold}$, so between the folds the solutions diverge.
• Figure 5: Solutions of the perturbed transcritical system with $c=0.1$, for $\varepsilon=0.3$ (top row) and $\mu=-0.3$ (bottom row), with values $\mu=-0.5,0,0.5$, (from left to right). The original solutions (red curves) and averaged solutions (blue curves) oscillate around the guiding solutions (black dotted curves), converging in forward/backward time onto the stable/unstable fixed points (blue curves) if they exist. For $\varepsilon>0$ there are two fixed points only for $|\mu|>\mu_{\rm fold}$, so in the middle picture the solutions diverge. For $\varepsilon<0$ there are always two fixed points.

Theorems & Definitions (50)

• Definition 1
• Definition 2
• Proposition 1
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 1
• proof