We study a network of 500 coupled modified van der Pol oscillators. The value of a parameter associated with each oscillator is drawn from a normal distribution, giving a heterogeneous network. For strong enough coupling the oscillators all have the same period, and we consider periodic forcing of the network when it is in this state. By exploiting the correlations that quickly develop between the state of an oscillator and the value of its parameter we obtain an approximate low-dimensional description of the system in terms of the first few coefficients in a polynomial chaos expansion. Standard bifurcation analysis can then be performed on this low-dimensional system, and the results obtained from this predict very well the behaviour of the high-dimensional system for any set of realisations of the random parameter. Situations in which the method begins to fails are also discussed.
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.
In a previous work by the authors the one dimensional (doubling) renormalization operator was extended to the case of quasi-periodically forced one dimensional maps. The theory was used to explain different self-similarity and universality observed numerically in the parameter space of the Forced Logistic Maps. The extension proposed was not complete in the sense that we assumed a total of four conjectures to be true. In this paper we present numerical support for these conjectures. We also discuss the applicability of this theory to the Forced Logistic Map.
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 …
In this paper, we study the global attractivity for a class of periodic difference equation with delay which has a generalized form of Pielou's difference equation. The global dynamics of the equation is characterized by using a relation between the upper and lower limit of the solution. There are two possible global dynamics: zero solution is globally attractive or there exists a periodic solution which is globally attractive. Recent results in [E. Camouzis, G. Ladas, Periodically forced Pielou's equation, J. Math. Anal. Appl. 333 (1) (2007) 117-127] is generalized. Two examples are given to …
Consider a difference equation which takes the k-th largest output of m functions of the previous m terms of the sequence. If the functions are also allowed to change periodically as the difference equation evolves this is analogous to a differential equation with periodic forcing. A large class of such non-autonomous difference equations are shown to converge to a periodic limit which is independent of the initial condition. The period of the limit does not depend on how far back each term is allowed to look back in the sequence, and is in fact equal to the period of the forcing.
We propose an extension of the one dimensional (doubling) renormalization operator to the case of maps on the cylinder. The kind of maps considered are commonly referred as quasi-periodic forced one dimensional maps. We prove that the fixed point of the one dimensional renormalization operator extends to a fixed point of the quasi-periodic forced renormalization operator. We also prove that the operator is differentiable around the fixed point and we study its derivative. Then we consider a two parametric family of quasi-periodically forced maps which is a unimodal one dimensional map with a f…
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…
In this paper we are concerned with quasi-periodic forced one dimensional maps. We consider a two parametric family of quasi-periodically forced maps such that the one dimensional map (before forcing) is unimodal and it has a full cascade of period doubling bifurcations. Between one period doubling and the next one it is known that there exist a parameter value where the $2^n$-periodic orbit is superatracting. In a previous work we proposed an extension of the one-dimensional (doubling) renormalization operator to the quasi-periodic case. We proved that, if the family satisfies suitable hypoth…
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 …
We study the dynamics of the Forced Logistic Map in the cylinder. We compute a bifurcation diagram in terms of the dynamics of the attracting set. Different properties of the attracting set are considered, as the Lyapunov exponent and, in the case of having a periodic invariant curve, its period and its reducibility. This reveals that the parameter values for which the invariant curve doubles its period are contained in regions of the parameter space where the invariant curve is reducible. Then we present two additional studies to explain this fact. In first place we consider the images and th…
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…
We propose a method to identify and to locate "repellers'' in quasi-periodically forced logistic map (QPLM), using a kind of Morse decomposition of nested attracting invariant sets. In order to obtain the invariant sets, we use an auxiliary 1+2-dimensional skew-product map system describing the evolution of a line segment in the phase space of QPLM. With this method, detailed structure of repellers can be visualized, and the emergence of a repeller in QPLM can be detected as an easily observable bifurcation in the auxiliary system. In addition to the method to detect the repellers, we propose …
We prove that the seasonally-forced SIR model with a T-periodic forcing has a periodic solution with period T whenever the basic reproductive number R0>1. The proof uses the Leray-Schauder degree theory. We also describe some numerical results in which we compute the T-periodic solution, where in order to obtain the T-periodic solution when the behavior of the system is subharmonic or chaotic, we use a Galerkin scheme.
We consider a system of rotators subject to a small quasi-periodic forcing. We require the forcing to be analytic and satisfy a time-reversibility property and we assume its frequency vector to be Bryuno. Then we prove that, without imposing any non-degeneracy condition on the forcing, there exists at least one quasi-periodic solution with the same frequency vector as the forcing. The result can be interpreted as a theorem of persistence of lower-dimensional tori of arbitrary codimension in degenerate cases.
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 recent papers by the authors (S.~Motonaga and K.~Yagasaki, Obstructions to integrability of nearly integrable dynamical systems near regular level sets, submitted for publication, and K.~Yagasaki, Nonintegrability of nearly integrable dynamical systems near resonant periodic orbits, submitted for publication), two different techniques which allow us to prove the real-analytic or complex-meromorphic nonintegrability of forced nonlinear oscillators having the form of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems were provided. Here the concept of nonintegrability…
Our proof is based on a generalization of action-angle variables, a convergent Lie transformation, and Moser's invariant curve theorem. As an overall outline we give a quick proof of Morris' original theorem. Then the full theorem is established in two steps. In the first step we define a special case and then follow the proof given for Morris' Theorem to prove boundedness for the special case. In the second step we use the method of Lie transforms to reduce the general case to the special case.
Spatially localized, time-periodic structures are common in pattern-forming systems, appearing in fluid mechanics, chemical reactions, and granular media. We examine the existence of oscillatory localized states in a PDE model with single frequency time dependent forcing, introduced in [20] as phenomenological model of the Faraday wave experiment. In this study, we reduce the PDE model to the forced complex Ginzburg-Landau equation in the limit of weak forcing and weak damping. This allows us to use the known localized solutions found in [7]. We reduce the forced complex Ginzburg-Landau equati…
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 varacto…
We analyze the dynamics of the forced singularly perturbed differential equation of Duffing's type. We explain the appearance of the large frequency nonlinear oscillations of the solutions. It is shown that the frequency can be controlled by a small parameter at the highest derivative. We give some generalizations of results obtained recently by B.S. Wu, W.P. Sun and C.W. Lim, Analytical approximations to the double-well Duffing oscillator in large amplitude oscillations, Journal of Sound and Vibration, Volume 307, Issues 3-5, (2007), pp. 953-960. The new method for an analysis of the nonlinea…
We review some recent results surrounding a general mechanism for producing chaotic behavior in periodically-kicked oscillators. The key geometric ideas are illustrated via a simple linear shear model.
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 show how spectral submanifold (SSM) theory can be used to extract forced-response curves, including isolas, without any numerical simulation in high-degree-of-freedom, periodically forced mechanical systems. We use multivariate recurrence relations to construct the SSMs, achieving a major speed-up relative to earlier autonomous SSM algorithms. The increase in computational efficiency promises to close the current gap between studying lower-dimensional academic examples and analyzing larger systems obtained from finite-element modeling, as we illustrate on a discretization of a damped-forced…
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 study the ordinary differential equation ${\varepsilon}\ddot x+\dot x + {\varepsilon} g(x) = {\varepsilon} f(ωt)$, with $f$ and $g$ analytic and $f$ quasi-periodic in $t$ with frequency vector $ω\in R^{d}$. We show that if there exists $c_0\in R$ such that $g(c_0)$ equals the average of $f$ and the first non-zero derivative of $g$ at $c_0$ is of odd order $n$, then, for ${\varepsilon}$ small enough and under very mild Diophantine conditions on $ω$, there exists a quasi-periodic solution close to $c_0$, with the same frequency vector as $f$. In particular if $f$ is a trigonometric polynomial…
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…
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 explore different families of quasi-periodically Forced Logistic Maps for the existence of universality and self-similarity properties. In the bifurcation diagram of the Logistic Map it is well known that there exist parameter values $s_n$ where the $2^n$-periodic orbit is superattracting. Moreover these parameter values lay between one period doubling and the next. Under quasi-periodic forcing, the superattracting periodic orbits give birth to two reducibility-loss bifurcations in the two dimensional parameter space of the Forced Logistic Map, both around the points $s_n$. In the present w…
The phase space for a periodically driven nonlinear oscillator consists of many resonance zones. Let the strength of periodic excitation and the strength of the damping be indexed by a small parameter $\varepsilon$. It is well known that, as $\varepsilon \to 0$, the measure of the set of initial conditions which lead to 'capture in a resonance zone' goes to zero. In this paper we study the effect of weak noise on the escape from a resonance zone.
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.
The paper revisits a well-known model of forced vibro-impact oscillator with Amonton-Coulomb friction. In vast majority of the existing studies, this model included also viscous friction, and its global dynamics in the state space is governed by periodic, quasiperiodic or chaotic attractors. We demonstrate that removal of the viscous friction leads to qualitative modification of the global dynamics. Namely, the state space is divided into the regions with "regular" attraction to the aforementioned special solutions, and the regions with profoundly Hamiltonian dynamics. The latter regions conta…
We study one-parameter families of quasi-periodically forced monotone interval maps and provide sufficient conditions for the existence of a parameter at which the respective system possesses a non-uniformly hyperbolic attractor. This is equivalent to the existence of a sink-source orbit, that is, an orbit with positive Lyapunov exponent both forwards and backwards in time. The attractor itself is a non-continuous invariant graph with negative Lyapunov exponent, often referred to as "SNA". In contrast to former results in this direction, our conditions are C^2-open in the fibre maps. By applyi…
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 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…
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.
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…
We consider dissipative periodically forced systems and investigate cases in which having information as to how the system behaves for constant dissipation may be used when dissipation varies in time before settling at a constant final value. First, we consider situations where one is interested in the basins of attraction for damping coefficients varying linearly between two given values over many different time intervals: we outline a method to reduce the computation time required to estimate numerically the relative areas of the basins and discuss its range of applicability. Second, we obse…
The dynamics of mechanical systems such as turbomachinery with multiple blades are often modeled by arrays of periodically driven coupled nonlinear oscillators. It is known that such systems may have multiple stable vibrational modes, and transitions between them may occur under the influence of random factors. A methodology for finding most probable escape paths and estimating the transition rates in the small noise limit is developed and applied to a collection of arrays of coupled monostable oscillators with cubic nonlinearity, small damping, and harmonic external forcing. The methodology i…
One-shot learning for the complex dynamical behaviors of weakly nonlinear forced oscillators
State Key Lab of Disaster Reduction in Civil Engineering（灾害防治ivil工程国家重点实验室）；Department of Civil and Environmental Engineering（土木与环境工程系）；Guangxi Laboratory of Whole Life Safety for Land-Sea Corridor Engineering（广西陆海通道工程全生命周期安全实验室）
The ability of a circadian system to entrain to the 24-hour light-dark cycle is one of its most important properties. A new tool, called the entrainment map, was recently introduced to study this process for a single oscillator. Here we generalize the map to study the effects of light-dark forcing in a hierarchical system consisting of a central circadian oscillator that drives a peripheral circadian oscillator. We develop techniques to reduced the higher dimensional phase space of the coupled system to derive a generalized 2-D entrainment map. Determining the nature of various fixed points, t…
We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, giving a simple construction of unstable KAM tori and their stable and unstable manifolds for analytic perturbations. We extend analytically the solutions of the equations of motion, order by order in the perturbation parameter, to a uniform neighbourhood of the time axis.
