考虑一类微分方程$\ddot x + γ\dot x + g(x) = f(ωt)$，其中$ω\in {\bf R}^{d}$描述一维耗散系统受周期或准周期（丢番图）强迫.研究了与强迫具有相同拟周期性的轨道所描述的极限环的存在性及其性质。对于$g(x)=x^{2p+1}$，$p\in {\bf N}$，我们证明了当耗散系数足够大时，只有一个极限环，并且它是一个全局吸引子.在其他的力量，包括$g(x)=x^{2p}$（与$p=1$描述的变容方程）的情况下，我们发现的极限环的吸引区域的估计。
英文摘要：
We consider a class of differential equations, $\ddot x + γ\dot x + g(x) = f(ωt)$, with $ω\in {\bf R}^{d}$, describing one-dimensional dissipative systems subject to a periodic or quasi-periodic (Diophantine) forcing. We study existence and properties of the limit cycle described by the trajectory with the same quasi-periodicity as the forcing. For $g(x)=x^{2p+1}$, $p\in {\bf N}$, we show that, when the dissipation coefficient is large enough, there is only one limit cycle and that it is a global attractor. In the case of other forces, including $g(x)=x^{2p}$ (with $p=1$ describing the varactor equation), we find estimates for the basin of attraction of the limit cycle.
In this paper, we prove the existence of full dimensional invariant tori for a non-autonomous, almost-periodically forced nonlinear beam equation with a periodic boundary condition via KAM theory.
A periodically forced oscillator in a model for seasonality shows stability pockets and chains thereof in the parameter plane. The frequency of the oscillator and the season indicated by a value between zero and one are the two parameters. The present study is intended as a theoretical complement to the numerical study of Schmal et al. in 2015 of stability pockets or Arnol'd onions in their terminology. We construct the Poincaré map of the forced oscillator and show that the Arnol'd tongues are taken into stability pockets by a map with a number of folds. Stability pockets are already observed…
In this paper, we consider a time-periodically forced Kepler problem in any dimensions, with an external force which we only assume to be regular in a neighborhood of the attractive center. We prove that there exist infinitely many periodic orbits in this system, with possible double collisions with the center regularized, which accumulate the attractive center. The result is obtained via a localization argument combined with a result on $C^{1}$-persistence of closed orbits by a local homotopy-streching argument. Consequently, by formulating the circular and elliptic restricted three-body prob…
While periodic responses of periodically forced dissipative nonlinear mechanical systems are commonly observed in experiments and numerics, their existence can rarely be concluded in rigorous mathematical terms. This lack of a priori existence criteria for mechanical systems hinders definitive conclusions about periodic orbits from approximate numerical methods, such as harmonic balance. In this work, we establish results guaranteeing the existence of a periodic response without restricting the amplitude of the forcing or the response. Our results provide a priori justification for the use of …
We discuss the appearance of chaos in time-periodic perturbations of reversible vector fields in the plane. We use the normal forms of codimension~$1$ reversible vector fields and discuss the ways a time-dependent periodic forcing term of pulse form may be added to them to yield topological chaotic behaviour. Chaos here means that the resulting dynamics is semi-conjugate to a shift in a finite alphabet. The results rely on the classification of reversible vector fields and on the theory of topological horseshoes. This work is part of a project of studying periodic forcing of symmetric vector f…
Let $Φ$ be a quasi-periodically forced quadratic map, where the rotation constant $ω$ is a Diophantine irrational. A strange non-chaotic attractor (SNA) is an invariant (under $Φ$) attracting graph of a nowhere continuous measurable function $ψ$ from the circle $\mathbb{T}$ to $[0,1]$. This paper investigates how a smooth attractor degenerates into a strange one, as a parameter $β$ approaches a critical value $β_0$, and the asymptotics behind the bifurcation of the attractor from smooth to strange. In our model, the cause of the strange attractor is a so-called torus collision, whereby an attr…
In this work we consider a general class of $2$-dimensional hybrid systems. Assuming that the system possesses an attracting equilibrium point, we show that, when periodically driven with a square-wave pulse, the system possesses a periodic orbit which may undergo smooth and nonsmooth grazing bifurcations. We perform a semi-rigorous study of the existence of periodic orbits for a particular model consisting of a leaky integrate-and-fire model with a dynamic threshold. We use the stroboscopic map, which in this context is a $2$-dimensional piecewise-smooth discontinuous map. For some parameter …
Periodically-forced finite networks of heterogeneous coupled oscillators: a low-dimensional approach
我们研究了一个由500个耦合的改进型van der Pol振荡器组成的网络。与每个振荡器相关联的参数的值从正态分布中得出，从而给出异构网络。对于足够强的耦合，所有振荡器都具有相同的周期，并且我们考虑网络在这种状态下的周期性强迫。通过利用振荡器的状态和它的参数的值之间的快速发展的相关性，我们得到一个近似的低维描述的系统在多项式混沌展开的前几个系数。然后，可以对这个低维系统进行标准的分叉分析，并且由此获得的结果很好地预测了随机参数的任何实现集的高维系统的行为。还讨论了该方法开始失败的情况。
Consider a periodically forced nonlinear system which can be presented as a collection of smaller subsystems with pairwise interactions between them. Each subsystem is assumed to be a massive point moving with friction on a compact surface, possibly with a boundary, in an external periodic field. We present sufficient conditions for the existence of a periodic solution for the whole system. The result is illustrated by a series of examples including a chain of strongly coupled pendulums in a periodic field.
We present sufficient conditions for the existence of a periodic solution for a class of systems describing the periodically forced motion of a massive point on a compact surface with a boundary.
We study some properties of the range of the relativistic pendulum operator $\mathcal P$, that is, the set of possible continuous $T$-periodic forcing terms $p$ for which the equation $\mathcal P x=p$ admits a $T$-periodic solution over a $T$-periodic time scale $\mathbb T$. Writing $p(t)=p_0(t)+\overline p$, we prove the existence of a nonempty compact interval $\mathcal I(p_0)$, depending continuously on $p_0$, such that the problem has a solution if and only if $\overline p\in \mathcal I(p_0)$ and at least two different solutions when $\overline p$ is an interior point. Furthermore, we give…
We give a simple proof of the existence of response solutions in some quasi-periodically forced systems with a degenerate fixed points. The same questions were answered in \cite{ss18} using two versions of KAM theory. Our method is based on reformulating the existence of response solutions as a fixed point problem in appropriate spaces of smooth functions. By algebraic manipulations, the fixed point problem is transformed into a contraction. Compared to the KAM method, the present method does not incur a loss of regularity. That is, the solutions we obtain have the same regularity as the forci…
In this paper we perform a careful analysis of the forced PP04 model for climate change, in particular the behaviour of the ice-ages. This system models the transition from a glacial to an inter-glacial state through a sudden release of oceanic Carbon Dioxide into the atmosphere. This process can be cast in terms of a Filippov dynamical system, with a discontinuous change in its dynamics related to the Carbon Dioxide release. By using techniques from the theory of non-smooth dynamical systems, we give an analysis of this model in the cases of both no insolation forcing and also periodic insola…
We consider quasi-periodically systems in the presence of dissipation and study the existence of response solutions, i.e. quasi-periodic solutions with the same frequency vector as the forcing term. When the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions are known to exist without assuming any non resonance condition on the frequency vector. We analyse the case of non-simple zeroes and, in order to deal with the small divisors problem, we confine ourselves to two-dimensional frequency vectors, so as to use the properties of conti…
We prove Lr-estimates on periodic solutions of periodically-forced, linearly-damped mechanical systems with polynomially-bounded potentials. The estimates are applied to obtain a non-existence result of periodic solutions in bounded domains, depending on an upper bound on the gradient of the potential. The results are illustrated on examples.
We analyze a multiparameter periodically-forced dynamical system inspired in the SIR endemic model. We show that the condition on the \emph{basic reproduction number} $\mathcal{R}_0 < 1$ is not sufficient to guarantee the elimination of \emph{Infectious} individuals due to a \emph{backward bifurcation}. Using the theory of rank-one attractors, for an open subset in the space of parameters where $\mathcal{R}_0<1$, the flow exhibits \emph{persistent strange attractors}. These sets are not confined to a tubular neighbourhood in the phase space, are numerically observable and shadow the ghost of a…
In pattern-forming systems, localized patterns are readily found when stable patterns exist at the same parameter values as the stable unpatterned state. Oscillons are spatially localized, time-periodic structures, which have been found experimentally in systems that are driven by a time-periodic force, for example, in the Faraday wave experiment. This paper examines the existence of oscillatory localized states in a PDE model with single frequency time dependent forcing, introduced in [A. M. Rucklidge and M. Silber, SIAM J. Appl. Math., 8 (2009), pp. 298-347, arXiv:0805.0878] as a phenomenolo…
We perform a detailed study of the dynamics of a nonlinear, one-dimensional oscillator driven by a periodic force under hysteretic damping, whose linear version was originally proposed and analyzed by Bishop in [1]. We first add a small quadratic stiffness term in the constitutive equation and construct the periodic solution of the problem by a systematic perturbation method, neglecting transient terms as $t\rightarrow \infty$. We then repeat the analysis replacing the quadratic by a cubic term, which does not allow the solutions to escape to infinity. In both cases, we examine the dependence …
We consider several models of nonlinear wave equations subject to very strong damping and quasi-periodic external forcing. This is a singular perturbation, since the damping is not the highest order term. We study the existence of response solutions (i.e., quasi-periodic solutions with the same frequency as the forcing). Under very general non-resonance conditions on the frequency, we show the existence of asymptotic expansions of the response solution; moreover, we prove that the response solution indeed exists and depends analytically on $\varepsilon$ (where $\varepsilon$ is the inverse of t…
We consider the skew-product semiflow which is generated by a scalar reaction-diffusion equation \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2π\mathbb{Z}, \end{equation*} where $f$ is uniformly almost periodic in $t$. The structure of the minimal set $M$ is thoroughly investigated under the assumption that the center space $V^c(M)$ associated with $M$ is no more than $2$-dimensional. Such situation naturally occurs while, for instance, $M$ is hyperbolic or uniquely ergodic. It is shown in this paper that $M$ is a $1$-cover of the hull $H(f)$ provided that $M$ is…
We study a generic model for the interaction of negative delayed feedback and periodic forcing that was first introduced by Ghil et al. in the context of the El Niño Southern Oscillation (ENSO) climate system. This model takes the form of a delay differential equation and has been shown in previous work to be capable of producing complicated dynamics, which is organised by resonances between the external forcing and dynamics induced by feedback. For certain parameter values, we observe in simulations the sudden disappearance of (two-frequency dynamics on) tori. This can be explained by the fol…
We consider several models (including both multidimensional ordinary differential equations (ODEs) and partial differential equations (PDEs), possibly ill-posed), subject to very strong damping and quasi-periodic external forcing. We study the existence of response solutions (i.e., quasi-periodic solutions with the same frequency as the forcing). Under some regularity assumptions on the nonlinearity and forcing, without any arithmetic condition on the forcing frequency $ω$, we show that the response solutions indeed exist. Moreover, the solutions we obtained possess optimal regularity in $\var…
We study topological structure of the $ω$-limit sets of the skew-product semiflow generated by the following scalar reaction-diffusion equation \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2π\mathbb{Z}, \end{equation*} where $f(t,u,u_x)$ is $C^2$-admissible with time-recurrent structure including almost-periodicity and almost-automorphy. Contrary to the time-periodic cases (for which any $ω$-limit set can be imbedded into a periodically forced circle flow), it is shown that one cannot expect that any $ω$-limit set can be imbedded into an almost-periodically force…
In this paper, we consider the asymptotic dynamics of the skew-product semiflow generated by the following time almost-periodically forced scalar reaction-diffusion equation \begin{equation}\label{eq0} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\, 0<x<L \end{equation} with periodic boundary condition \begin{equation} \label{bdc1} u(t,0)=u(t,L),\quad u_x(t,0)=u_x(t,L), \end{equation} where $f$ is uniformly almost periodic in $t$. In particular, we study the topological structure of the limit sets of the skew-product semiflow. It is proved that any compact minimal invariant set (throughout this paper, we…
Given two distinct subsets $A,B$ in the state space of some dynamical system, Transition Path Theory (TPT) was successfully used to describe the statistical behavior of transitions from $A$ to $B$ in the ergodic limit of the stationary system. We derive generalizations of TPT that remove the requirements of stationarity and of the ergodic limit, and provide this powerful tool for the analysis of other dynamical scenarios: periodically forced dynamics and time-dependent finite-time systems. This is partially motivated by studying applications such as climate, ocean, and social dynamics. On simp…
We study the dynamics of the periodically-forced May-Leonard system. We extend previous results on the field and we identify different dynamical regimes depending on the strength of attraction $δ$ of the network and the frequency $ω$ of the periodic forcing. We focus our attention in the case $δ\gg1$ and $ω\approx 0$, where we show that, for a positive Lebesgue measure set of parameters (amplitude of the periodic forcing), the dynamics are dominated by strange attractors with fully stochastic properties, supporting a Sinai-Ruelle-Bowen (SRB) measure. The proof is performed by using the Wang an…
In Part I of this paper, we have used spectral submanifold (SSM) theory to construct reduced-order models for harmonically excited mechanical systems with internal resonances. In that setting, extracting forced response curves formed by periodic orbits of the full system was reduced to locating the solution branches of equilibria of the corresponding reduced-order model. Here we use bifurcations of the equilibria of the reduced-order model to predict bifurcations of the periodic response of the full system. Specifically, we identify Hopf bifurcations of equilibria and limit cycles in reduced m…
Seasonal variation in environmental variables, and in rates of contact among individuals, are fundamental drivers of infectious disease dynamics. Unlike most periodically-forced physical systems, for which the precise pattern of forcing is typically known, underlying patterns of seasonal variation in transmission rates can be estimated approximately at best, and only the period of forcing is accurately known. Yet solutions of epidemic models depend strongly on the forcing function, so dynamical predictions---such as changes in epidemic patterns that can be induced by demographic transitions or…
In various models and systems involving the escape of periodically forced particle from the potential well, a common pattern is observed. Namely, the minimal forcing amplitude required for the escape exhibits sharp minimum for the excitation frequency below the natural frequency of small oscillations in the well. The paper explains this regularity by exploring the transient escape dynamics in simple benchmark potential wells. In the truncated parabolic well, in absence of the damping the minimal forcing amplitude obviously tends to zero for the natural excitation frequency. Addition of weak sy…
We look at the periodic behaviour of the Earth's glacial cycles and the transitions between different periodic states when either external parameters (such as $ω$) or internal parameters (such as $d$) are varied. We model this using the PP04 model of climate change. This is a forced discontinuous Filippov (non-smooth) dynamical system. When periodically forced this has co-existing periodic orbits. We find that the transitions in this system are mainly due to grazing events, leading to grazing bifurcations. An analysis of the grazing bifurcations is given and the impact of these on the domains …
We study canard solutions of the forced van der Pol (fvdP) equation in the relaxation limit for low-, intermediate-, and high-frequency periodic forcing. A central numerical observation is that there are two branches of canards in parameter space which extend across all positive forcing frequencies. For low-frequency forcing, we demonstrate the existence of primary maximal canards induced by folded saddle-nodes of type I, and establish explicit formulas for the parameter values at which the primary maximal canards and their folds exist. We then turn to the intermediate- and high-frequency forc…
