Slow-fast Systems Research Articles

Increased habitat connectivity induces diversity via noise-induced symmetry breaking.

Stochasticity or noise is omnipresent in ecosystems that mediates community dynamics. The beneficial role of stochasticity in enhancing species coexistence and, hence, in promoting biodiversity is well recognized. However, incorporating stochastic birth and death processes in excitable slow-fast ecological systems to study its response to biodiversity is largely unexplored. Considering an ecological network of excitable consumer-resource systems, we study the interplay of network structure and noise on species' collective dynamics. We find that noise drives the system out of the excitable regime, and high habitat patch connectance in the ordered as well as random networks promotes species' diversity by inducing new steady states via noise-induced symmetry breaking.

The Order of Convergence in the Averaging Principle for Slow-Fast Systems of Stochastic Evolution Equations in Hilbert Spaces

In this work we are concerned with the study of the strong order of convergence in the averaging principle for slow-fast systems of stochastic evolution equations in Hilbert spaces with additive noise. In particular the stochastic perturbations are general Wiener processes, i.e their covariance operators are allowed to be not trace class. We prove that the slow component converges strongly to the averaged one with order of convergence 1/2 which is known to be optimal. Moreover we apply this result to a slow-fast stochastic reaction diffusion system where the stochastic perturbation is given by a white noise both in time and space.

Canard phenomena for a slow-fast predator-prey system with group defense of the prey

In this study, we consider that the predator reproduces much slower than the prey and propose a slow-fast predator-prey system with the group defense of the prey, which is described by the Ivlev-type functional response. We analyze the canard phenomena (relaxation oscillation and canard cycles) of our proposed model by applying the singular perturbation theory. Firstly, we apply the blow-up techniques and visualize the behavior near the special fold point, which allows us to prove the existence of the unique attractive limit cycle and imply the appearance of the canard phenomenon when crossing the fold point, and the numerical simulations verify the theoretical results. Then we further use the singular perturbation theory to investigate the existence and cyclicity of the canard cycle without head and the existence of the canard cycle with head. Our main results of the complex canard phenomenon reveal the fact that some certain species in the ecosystem suddenly burst back many years after being about extinct.

Inflation driven by non-linear electrodynamics

We investigate the inflation driven by a nonlinear electromagnetic field based on an NLED lagrangian density {mathscr {L}}_{text {nled}} = - {F} f left( {F} right) , where f left( {F}right) is a general function depending on F. We first formulate an f-NLED cosmological model with a more general function f left( {F}right) and show that all NLED models can be expressed in this framework; then, we investigate in detail two interesting examples of the function f left( {F}right) . We present our phenomenological model based on a new Lagrangian for NLED. Solutions to the field equations with the physical properties of the cosmological parameters are obtained. We show that the early Universe had no Big-Bang singularity, which accelerated in the past. We also investigate the qualitative implications of NLED by studying the inflationary parameters, like the slow-roll parameters, spectral index n_s, and tensor-to-scalar ratio r, and compare our results with observational data. Detailed phase-space analysis of our NLED cosmological model is performed with and without matter source. As a first approach, we consider the motion of a particle of unit mass in an effective potential. Our systems correspond to fast-slow systems for physical values of the electromagnetic field and the energy densities at the end of inflation. We analyze a complementary system using Hubble-normalized variables to investigate the cosmological evolution before the matter-dominated Universe.

Optimal Reaction Coordinates: Variational Characterization and Sparse Computation

Reaction coordinates (RCs) are indicators of hidden, low-dimensional mechanisms that govern the long-term behavior of high-dimensional stochastic processes. We present a novel and general variational characterization of optimal RCs and provide conditions for their existence. Optimal RCs are minimizers of a certain loss function, and reduced models based on them guarantee a good approximation of the statistical long-term properties of the original high-dimensional process. We show that for slow-fast systems, metastable systems, and other systems with known good RCs, the novel theory reproduces previous insight. Remarkably, for reversible systems, the numerical effort required to evaluate the loss function scales only with the variability of the underlying, low-dimensional mechanism, and not with that of the full system. The theory provided lays the foundation for an efficient and data-sparse computation of RCs via modern machine learning techniques.

Oscillatory and transient dynamics of a slow–fast predator–prey system with fear and its carry-over effect

In almost every ecological system, growth of various interacting species evolve in different time scales and the implementation of this time scale difference in the corresponding mathematical model exhibits some rich and complex oscillatory dynamics. In this article, we consider a predator–prey model with Beddington–DeAngelis functional response in which the prey reproduction is affected by the predation induced fear and its carry-over effect. Considering the growth of prey species occurs on a faster time scale than that of predator, the proposed system reduces to a ‘slow–fast predator–prey’ system. Using the geometric singular perturbation theory and asymptotic expansion technique, we investigate the system both analytically and numerically, and observe a wide range of rich and complex dynamics such as canard cycles (with or without head) near the singular Hopf-bifurcation threshold and relaxation oscillation cycles. The system experiences a canard explosion through which a rapid transition from small amplitude limit cycle to large amplitude limit cycle occurs in a tiny parametric interval. These types of complex oscillatory dynamics are absent in non slow–fast systems. Furthermore, it is shown that the interplay between fear and its carry-over effect, and the variation of time scale parameter may lead to a regime shift of the oscillatory dynamics. We also study the impact of fear and its carry-over effect on the properties of long transient dynamics. Thus our study provides some valuable biological insights of a slow–fast predator–prey system which will aid in understanding the interplay between fear and its carry-over effect.

Vibration reduction mechanism of Van der Pol oscillator under low-frequency forced excitation by means of nonlinear energy sink

Van der Pol (VDP) oscillator under forced excitation has complicated vibration modes, and its harm to the system cannot be ignored. The vibration reduction mechanism of VDP oscillator by means of nonlinear energy sink (NES) is explored under low-frequency forced excitation. Firstly, a mechanical model of VDP oscillator coupled NES is established by utilizing Newton’s law. Then the time history diagrams under different excitation amplitudes are compared to measure the vibration reduction effect of the coupled system. Finally, the stability and Hopf bifurcation behavior of the corresponding autonomous system are discussed, and the mechanism of cluster phenomenon and the mechanism of vibration reduction are revealed. It is interesting that, the vibration reduction effect is optimal when cluster phenomenon occurs, which is a typical vibration model in slow–fast system. The real part of eigenvalues and hysteresis time are critical factors affecting vibration reduction. These researches provide a theoretical basis for the vibration reduction of VDP oscillator coupled NES.

Averaging principle for stochastic quasi‐geostrophic flow equation with a fast oscillation

AbstractThis work is focused on the quasi‐geostrophic flow equation with a fast oscillation governed by a stochastic reaction–diffusion equation. It derives the well‐posedness of the slow–fast system, in which the fast component is ergodic and the slow component is tight. Applying the averaging principle, it is further proved that there exists a limit process, with respect to the singular perturbing parameter ε, where the fast component is averaged out. Moreover, the slow component of the slow–fast system converges to the solution of the averaged equation in some strong sense as ε tends to zero.

Numerical methods for control-based continuation of relaxation oscillations

Control-based continuation (CBC) is an experimental method that can reveal stable and unstable dynamics of physical systems. It extends the path-following principles of numerical continuation to experiments and provides systematic dynamical analyses without the need for mathematical modelling. CBC has seen considerable success in studying the bifurcation structure of mechanical systems. Nevertheless, the method is not practical for studying relaxation oscillations. Large numbers of Fourier modes are required to describe them, and the length of the experiment significantly increases when many Fourier modes are used, as the system must be run to convergence many times. Furthermore, relaxation oscillations often arise in autonomous systems, for which an appropriate phase constraint is required. To overcome these challenges, we introduce an adaptive B-spline discretisation that can produce a parsimonious description of responses that would otherwise require many Fourier modes. We couple this to a novel phase constraint that phase-locks control target and solution phase. Results are demonstrated on simulations of a slow-fast synthetic gene network and an Oregonator model. Our methods extend CBC to a much broader range of systems than have been studied so far, opening up a range of novel experimental opportunities on slow-fast systems.

Fractal codimension of nilpotent contact points in two-dimensional slow-fast systems

In this paper we introduce the notion of fractal codimension of a nilpotent contact point p, for λ=λ0, in smooth planar slow–fast systems Xϵ,λ when the contact order nλ0(p) of p is even, the singularity order sλ0(p) of p is odd and p has finite slow divergence, i.e., sλ0(p)≤2(nλ0(p)−1). The fractal codimension of p is a generalization of the “traditional” codimension of a slow-fast Hopf point of Liénard type, introduced in (Dumortier and Roussarie (2009) [7]), and it is intrinsically defined, i.e., it can be directly computed without the need to first bring the system into its normal form. The intrinsic nature of the notion of fractal codimension stems from the Minkowski dimension of fractal sequences of points, defined near p using the so-called entry-exit relation, and slow divergence integral. We apply our method to a slow-fast Hopf point and read its degeneracy (i.e., the first nonzero Lyapunov quantity) as well as the number of limit cycles near such a Hopf point directly from its fractal codimension. We demonstrate our results numerically on some interesting examples by using a simple formula for computation of the fractal codimension.

Symmetry-breaking rhythms in coupled, identical fast-slow oscillators.

Symmetry-breaking in coupled, identical, fast-slow systems produces a rich, dramatic variety of dynamical behavior-such as amplitudes and frequencies differing by an order of magnitude or more and qualitatively different rhythms between oscillators, corresponding to different functional states. We present a novel method for analyzing these systems. It identifies the key geometric structures responsible for this new symmetry-breaking, and it shows that many different types of symmetry-breaking rhythms arise robustly. We find symmetry-breaking rhythms in which one oscillator exhibits small-amplitude oscillations, while the other exhibits phase-shifted small-amplitude oscillations, large-amplitude oscillations, mixed-mode oscillations, or even undergoes an explosion of limit cycle canards. Two prototypical fast-slow systems illustrate the method: the van der Pol equation that describes electrical circuits and the Lengyel-Epstein model of chemical oscillators.

Effective filtering for slow-fast systems via Wong-Zakai approximation

Nonlinear filtering problem has been widely investigated in diverse fields. One important topic related to it is how to formulate an effective approximation scheme for a given observation. By using Wong-Zakai approximation and random invariant manifold theory, we will propose an effective approximation result for a class of slow-fast systems with respect to filtering. We will firstly establish the smooth reduced system via random invariant manifold theory, and then show exponential attractive property of it. At last, we will prove the filtering of the smooth reduced system can well approximate that of the original system.

Functional law of large numbers and central limit theorem for slow-fast McKean-Vlasov equations

In this paper, we study the asymptotic behavior of a fully-coupled slow-fast McKean-Vlasov stochastic system. Using the non-linear Poisson equation on Wasserstein space, we first establish the strong convergence in the averaging principle of the functional law of large numbers type. In particular, the diffusion coefficient of the slow process can depend on the distribution of the fast motion. Then we consider the stochastic fluctuations of the original system around its average, and prove that the normalized difference will converge weakly to a linear McKean-Vlasov Ornstein-Uhlenbeck type process, which can be viewed as a functional central limit theorem. Extra drift and diffusion coefficients involving the expectation are characterized explicitly. Furthermore, the optimal rates of the convergence are also obtained.

A unified framework for limit results in chemical reaction networks on multiple time-scales

If (XN)N=1,2,... is a sequence of Markov processes which solve the martingale problems for some operators (GN)N=1,2,..., it is a classical task to derive a limit result as N→∞, in particular a weak process limit with limiting operator G. For slow-fast systems XN=(VN,ZN) where VN is slow and ZN is fast, GN consists of two (or more) terms, and we are interested in weak convergence of VN to some Markov process V. In this case, for some f∈D(G), the domain of G, depending only on v, the limit Gf can sometimes be derived by using some gN→0 (depending on v and z), and study convergence of GN(f+gN)→Gf. We develop this method further in order to obtain functional Laws of Large Numbers (LLNs) and Central Limit Theorems (CLTs). We then apply our general result to various examples from Chemical Reaction Network theory. We show that we can rederive most limits previously obtained, but also provide new results in the case when the fast-subsystem is first order. In particular, we allow that fast species to be consumed faster than they are produced, and we derive a CLT for Hill dynamics with coefficient 2.

Clearing function in the context of the invariant manifold method

Clearing functions (CFs), which express a mathematical relationship between the expected throughput of a production facility in a planning period and its workload (or work-inprogress, WIP) in that period have shown considerable promise for modeling WIP-dependent cycle times in production planning. While steady-state queueing models are commonly used to derive analytic expressions for CFs, the finite length of planning periods calls their validity into question. We apply a different approach to propose a mechanistic model for one-resource, one-product factory shop based on the analogy between the operation of machine and enzyme molecule. The model is reduced to a singularly perturbed system of two differential equations for slow (WIP) and fast (busy machines) variables, respectively. The analysis of this slow-fast system finds that CF is nothing but a result of the asymptotic expansion of the slow invariant manifold. The validity of CF is ultimately determined by how small is the parameter multiplying the derivative of the fast variable. It is shown that sufficiently small characteristic ratio ’working machines: WIP’ guarantees the applicability of CF approximation in unsteady-state operation.

THE EFFECT OF AN ADDITIVE NOISE ON SOME SLOW-FAST EQUATION NEAR A TRANSCRITICAL POINT

We consider the effect of small additive noise with intensity $\sigma$ on trajectories of a slow-fast system with small parameter $\varepsilon$ which admits bifurcation delay at a transcritical point. We estimate the probability that the perturbed stochastic paths stay in some tubular neighborhood of the deterministic path to show that small but not exponentially small noise destroys the bifurcation delay caused by transcritical point and obtain a noise intensity threshold value $N(\varepsilon)$ of order $\varepsilon^{\frac{3}{4}}$. When <i>e</i>^{-1/$\varepsilon$}《<i>σ</i>＜$N(\varepsilon)$, the paths are likely to leave the neighborhood of the corresponding determinate path before some time of order $\sqrt{\varepsilon|log\sigma|}$. When $\sigma>N(\varepsilon) $, the paths are likely to leave before some time of order $\sigma^{\frac{2}{3}}$.

Coexistence of chaotic and non-chaotic attractors in a three-species slow–fast system

The dynamical complexity of conceptual few-species systems has long been attracting considerable attention. In particular, significant attention has been paid to the three trophic level systems to demonstrate that even a baseline prey/predator/top-predator system can exhibit complex dynamics. However, previous studies left many open questions. In this paper, we consider a generalization of the Hastings–Powell model that include intraspecific competition among the predators and top predators. To enhance the ecological relevance of the model, we incorporate different timescales for different species. Specifically, we aim to study how the dynamics of the system is affected in the presence of multiple timescales along the trophic levels. We employ the geometric singular perturbation approach to analyze the change in species abundance over three different timescales, slow, fast, and intermediate. We analytically determine the entry and exit points from the vicinity of the critical manifold; the corresponding values act as critical thresholds for a sudden, fast transition in population density. We show that the properties of this generalized form of the Hastings–Powell model are dynamically more rich than it was observed in previous studies; particularly, the system can exhibit bi-stability or tri-stability in certain parameter intervals. The existence of a homoclinic orbit in a limiting subsystem (prey–predator) indicates period-doubling cascades to chaos in the full system (prey–predator–top predator). We further investigate the impact of the intraspecific competition of predators and top predators on the system’s dynamics. Strong intraspecific competition among the predators and top predators shows periodic coexistence of all species. In contrast, weak competition can drive the system to exhibit tri-stability, which includes the coexistence of chaotic and periodic (of different periods) attractors.

Rate of homogenization for fully-coupled McKean–Vlasov SDEs

In this paper, we consider a fully-coupled slow–fast system of McKean–Vlasov stochastic differential equations with full dependence on the slow and fast component and on the law of the slow component and derive convergence rates to its homogenized limit. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. In the course of the proof we obtain related ergodic theorems and we gain results on the regularity of Poisson type of equations and of the associated Cauchy problem on the Wasserstein space that are of independent interest.

Growth and depletion in linear stochastic reaction networks

This paper is about a class of stochastic reaction networks. Of interest are the dynamics of interconversion among a finite number of substances through reactions that consume some of the substances and produce others. The models we consider are continuous-time Markov jump processes, intended as idealizations of a broad class of biological networks. Reaction rates depend linearly on "enzymes," which are among the substances produced, and a reaction can occur only in the presence of sufficient upstream material. We present rigorous results for this class of stochastic dynamical systems, the mean-field behaviors of which are described by ordinary differential equations (ODEs). Under the assumption of exponential network growth, we identify certain ODE solutions as being potentially traceable and give conditions on network trajectories which, when rescaled, can with high probability be approximated by these ODE solutions. This leads to a complete characterization of the ω-limit sets of such network solutions (as points or random tori). Dimension reduction is noted depending on the number of enzymes. The second half of this paper is focused on depletion dynamics, i.e., dynamics subsequent to the "phase transition" that occurs when one of the substances becomes unavailable. The picture can be complex, for the depleted substance can be produced intermittently through other network reactions. Treating the model as a slow-fast system, we offer a mean-field description, a first step to understanding what we believe is one of the most natural bifurcations for reaction networks.

Slow-fast torus knots

The aim of this paper is to study global dynamics of $C^\infty$-smooth slow-fast systems on the $2$-torus of class $C^\infty$ using geometric singular perturbation theory and the notion of slow divergence integral. Given any $m\in\mathbb{N}$ and two relatively prime integers $k$ and $l$, we show that there exists a slow-fast system $Y_{\epsilon}$ on the $2$-torus that has a $2m$-link of type $(k,l)$, i.e. a (disjoint finite) union of $2m$ slow-fast limit cycles each of $(k,l)$-torus knot type, for all small $\epsilon>0$. The $(k,l)$-torus knot turns around the $2$-torus $k$ times meridionally and $l$ times longitudinally. There are exactly $m$ repelling limit cycles and $m$ attracting limit cycles. Our analysis: a) proves the case of normally hyperbolic singular knots, and b) provides sufficient evidence to conjecture a similar result in some cases where the singular knots have regular nilpotent contact with the fast foliation.