Slow-fast Systems Research Articles

Bifurcation on 4-dimensional Canards with Hyper Catastrophe

In 4-dimensional slow-fast system, under the condition of ”symmetry”, there exists ”structural stability”. It is, however, of one parameter for the slow vector. On other parameters for all slow/fast vectors, it is not yet discussed still now as it is very complicated geometrical structure. In the ”Hyper catastrophe on 4-dimensional canards”, it is confirmed due to the existence of ”bifurcation”, because ”catastrophe” is a bifurcation problem itself. In the beginning of catastrophe theory, the word ”structural stability” is used for the original differential equations and not be used for the multi variable functions. In the slow-fast system having canards, what kinds of structure are there? It is used for the parameter, which depends on the existence of canards. Through this paper, it will become clear.

Regimes of stratified turbulence at low Prandtl number

Quantifying transport by strongly stratified turbulence in low Prandtl number ( $Pr$ ) fluids is critically important for the development of better models for the structure and evolution of stellar and planetary interiors. Motivated by recent numerical simulations showing strongly anisotropic flows suggestive of a scale-separated dynamics, we perform a multiscale asymptotic analysis of the governing equations. We find that, in all cases, the resulting slow–fast systems take a quasilinear form. Our analysis also reveals the existence of several distinct dynamical regimes depending on the emergent buoyancy Reynolds and Péclet numbers, $Re_b = \alpha ^2 Re$ and $Pe_b = Pr Re_b$ , respectively, where $\alpha$ is the aspect ratio of the large-scale turbulent flow structures, and $Re$ is the outer-scale Reynolds number. Scaling relationships relating the aspect ratio, the characteristic vertical velocity and the strength of the stratification (measured by the Froude number $Fr$ ) naturally emerge from the analysis. When $Pe_b \ll \alpha$ , the dynamics at all scales is dominated by buoyancy diffusion, and our results recover the scaling laws empirically obtained from direct numerical simulations by Cope et al. (J. Fluid Mech., vol. 903, 2020, A1). For $Pe_b \ge O(1)$ , diffusion is negligible (or at least subdominant) at all scales and our results are consistent with those of Chini et al. (J. Fluid Mech., vol. 933, 2022) for strongly stratified geophysical turbulence at $Pr =O(1)$ . Finally, we have identified a new regime for $\alpha \ll Pe_b \ll 1$ , in which slow, large scales are diffusive while fast, small scales are not. We conclude by presenting a map of parameter space that clearly indicates the transitions between isotropic turbulence, non-diffusive stratified turbulence, diffusive stratified turbulence and viscously dominated flows, and by proposing parameterisations of the buoyancy flux, mixing efficiency and turbulent diffusion coefficient for each regime.

Traveling front in a diffusive predator–prey model with Beddington–DeAngelis functional response

In this paper, we consider a singular diffusive predator–prey model with Beddington–DeAngelis functional response, employing geometric singular perturbation theory and Bendixson's criteria. Our investigation revolves around transforming the reaction–diffusion equation into a multi‐scale four‐dimensional slow–fast system with two different orders of small parameters. Through once singular perturbation analysis, our focus shifts towards exploring the existence of heteroclinic orbits in a three‐dimensional system. We analyze these dynamics through the perspective of the Fisher–KPP equation in two limit cases. In the first case, only the normal to the two‐dimensional slow manifold is unstable. This allows for the deduction of existence of heteroclinic orbits in the three‐dimensional system through investigating the dynamics on the two‐dimensional slow manifold. Consequently, we obtain both monotonic traveling fronts and non‐monotonic fronts with oscillatory tails. In the second case, the normal to the one‐dimensional slow manifold exhibits both stable and unstable directions, then it is impossible to restrict the dynamics of the three‐dimensional system entirely to the slow manifold. Instead, we integrate the slow orbits of the reduced system with the fast orbits of the layer system to construct a singular heteroclinic orbit. According to Fenichel's theorem, we discover the existence of exact heteroclinic orbits of three‐dimensional system and derive the monotonic traveling fronts under weaker parameter conditions. Additionally, we also discuss the nonexistence of traveling fronts. Finally, we demonstrate our theoretical results with numerical simulations.

AI-Lorenz: A physics-data-driven framework for Black-Box and Gray-Box identification of chaotic systems with symbolic regression

Discovering mathematical models that characterize the observed behavior of dynamical systems remains a major challenge, especially for systems in a chaotic regime, due to their sensitive dependence on initial conditions and the complex non-linear interactions within the system. The challenge is even greater when the physics underlying such systems is not yet understood, and scientific inquiry must solely rely on empirical data. Despite advancements such as the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm, which has shown success in identifying chaotic systems with reasonable accuracy, challenges remain in dealing with noise, sparse data, and the need for models that generalize well across different chaotic systems. Driven by the need to fill this gap, we develop a framework named AI-Lorenz that learns mathematical expressions modeling complex dynamical behaviors by identifying differential equations from noisy and sparse observable data. We train a physics-informed neural network (PINN) with the eXtreme Theory of Functional Connections (X-TFC) algorithm by using data and known physics (when available) to learn the dynamics of a system, its rate of change in time, and missing model terms, which are used as input for a symbolic regression algorithm (PySR) to autonomously distill the explicit mathematical terms. This, in turn, enables us to predict the future evolution of the dynamical behavior. The performance of this framework is validated by recovering the right-hand sides and unknown terms of certain complex, chaotic systems, such as the well-known Lorenz system, a six-dimensional hyperchaotic system, the non-autonomous Sprott chaotic system, and the slow-fast Duffing system, and comparing them with their known analytical expressions and state-of-the-art regression and system identification methods.

Canard cycle, relaxation oscillation and cross-diffusion induced pattern formation in a slow–fast ecological system with weak Allee effect

In this paper, we consider a Holling type IV functional response that describes a situation in which the predator’s per capita rate of predation decreases at sufficiently high prey density. Meanwhile it can be transformed into a Holling type II functional response in the limiting sense where predator’s attack rate increases at a decreasing rate with prey density until it becomes satiated. To explore the different dynamics of these two functional responses, we consider the slow–fast dynamic behavior of predator–prey system with Holling type IV and II functional responses, respectively, and make some simple comparisons of the dynamics of the system with these two functional responses. Specifically speaking, in the nonspatial case, firstly, the system with Holling type II functional response does not undergo a higher-codimension Hopf bifurcation. Then, the system with Holling type IV is extensively proved the existence of canard cycles and relaxation oscillations by using a range of analytical methods such as geometric singular perturbation technique, normal form of the slow–fast system, and the way in–way out function. In the spatial case, the temporal systems are extended to reaction–diffusion predator–prey systems. For the reaction–diffusion system with Holling IV, the different types of traveling wave are observed. Moreover, for the reaction–diffusion predator–prey system with Holling II, it is demonstrated that Turing instability occurs, which induces spatial heterogeneity patterns. Finally, the comparisons of the above dynamics of Holling Type IV and II functional responses in the non-spatial and spatial cases, respectively, are presented. From these comparisons, different Holling functional responses may be adopted for species at different stages or states, which is more conducive to maintaining species diversity and coexistence.

A complete analysis of a slow–fast modified Leslie–Gower predator–prey system

In this paper, we transform a modified Leslie–Gower predator–prey system into the corresponding fast–slow version by assuming that prey reproduces much faster than predator, and then perform a complete analysis about its dynamics. More specifically, we find the necessary and sufficient conditions of the exact number (zero, one or two) of positive equilibria of the slow–fast system and its (or their) location, and then we further fully determine its (or their) dynamics under explicit conditions. Besides, by converting the slow–fast system into its slow–fast normal form, we are able to characterize its rich dynamics completely, including relaxation oscillation, singular Hopf bifurcation, canard explosion, homoclinic orbits, heteroclinic orbits and global attraction of equilibrium. Moreover, the sufficient conditions to guarantee these various rich dynamics are explicitly given, including the explicit conditions to determine whether singular Hopf bifurcation is supercritical or subcritical, which generally cannot be explicitly derived in the existing literatures. Additionally, the cyclicity of diverse canard cycles is found under explicit conditions. Of particular interest is that we show the existence and uniqueness of one canard cycle without head whose cyclicity is at most two under explicit parameters conditions. Our results complement and enrich the previous work (relaxation oscillation, and global attraction of a boundary equilibrium) about this fast–slow system. We also employ numerical simulations to illustrate our theoretical analysis.

The approximation and portray of stochastic Poincaré maps for stochastic slow-fast systems in Hilbert spaces

The approximation and portray of stochastic Poincaré maps for stochastic slow-fast systems in Hilbert spaces

The hyperbolic umbilic singularity in fast-slow systems

Fast-slow systems with three slow variables and gradient structure in the fast variables have, generically, hyperbolic umbilic, elliptic umbilic or swallowtail singularities. In this article we provide a detailed local analysis of a fast-slow system near a hyperbolic umbilic singularity. In particular, we show that under some appropriate non-degeneracy conditions on the slow flow, the attracting slow manifolds jump onto the fast regime and fan out as they cross the hyperbolic umbilic singularity. The analysis is based on the blow-up technique, in which the hyperbolic umbilic point is blown up to a 5-dimensional sphere. Moreover, the reduced slow flow is also blown up and embedded into the blown-up fast formulation. Further, we describe how our analysis is related to classical theories such as catastrophe theory and constrained differential equations.

Rate and Bifurcation Induced Transitions in Asymptotically Slow-Fast Systems

Rate and Bifurcation Induced Transitions in Asymptotically Slow-Fast Systems

Polynomial slow-fast systems on the Poincaré–Lyapunov sphere

Polynomial slow-fast systems on the Poincaré–Lyapunov sphere

A dynamical systems approach to WKB-methods: The simple turning point

In this paper, we revisit the classical linear turning point problem for the second order differential equation ϵ2x″+μ(t)x=0 with μ(0)=0,μ′(0)≠0 for 0<ϵ≪1. Written as a first order system, t=0 therefore corresponds to a turning point connecting hyperbolic and elliptic regimes. Our main result is that we provide an alternative approach to WBK that is based upon dynamical systems theory, including GSPT and blowup, and we bridge – perhaps for the first time – hyperbolic and elliptic theories of slow-fast systems. As an advantage, we only require finite smoothness of μ. The approach we develop will be useful in other singular perturbation problems with hyperbolic–to–elliptic turning points.

A two-timescale model of plankton–oxygen dynamics predicts formation of oxygen minimum zones and global anoxia

Decline of the dissolved oxygen in the ocean is a growing concern, as it may eventually lead to global anoxia, an elevated mortality of marine fauna and even a mass extinction. Deoxygenation of the ocean often results in the formation of oxygen minimum zones (OMZ): large domains where the abundance of oxygen is much lower than that in the surrounding ocean environment. Factors and processes resulting in the OMZ formation remain controversial. We consider a conceptual model of coupled plankton–oxygen dynamics that, apart from the plankton growth and the oxygen production by phytoplankton, also accounts for the difference in the timescales for phyto- and zooplankton (making it a “slow-fast system”) and for the implicit effect of upper trophic levels resulting in density dependent (nonlinear) zooplankton mortality. The model is investigated using a combination of analytical techniques and numerical simulations. The slow-fast system is decomposed into its slow and fast subsystems. The critical manifold of the slow-fast system and its stability is then studied by analyzing the bifurcation structure of the fast subsystem. We obtain the canard cycles of the slow-fast system for a range of parameter values. However, the system does not allow for persistent relaxation oscillations; instead, the blowup of the canard cycle results in plankton extinction and oxygen depletion. For the spatially explicit model, the earlier works in this direction did not take into account the density dependent mortality rate of the zooplankton, and thus could exhibit Turing pattern. However, the inclusion of the density dependent mortality into the system can lead to stationary Turing patterns. The dynamics of the system is then studied near the Turing bifurcation threshold. We further consider the effect of the self-movement of the zooplankton along with the turbulent mixing. We show that an initial non-uniform perturbation can lead to the formation of an OMZ, which then grows in size and spreads over space. For a sufficiently large timescale separation, the spread of the OMZ can result in global anoxia.

Canard Cycles and Their Cyclicity of a Fast–Slow Leslie–Gower Predator–Prey Model with Allee Effect

We study canard cycles and their cyclicity of a fast–slow Leslie–Gower predator–prey system with Allee effect. More specifically, we find necessary and sufficient conditions of the exact number (zero, one or two) of positive equilibria of the slow–fast system and its location (or their locations), and then we further completely determine its (or their) dynamics under explicit conditions. Besides, by geometric singular perturbation theory and the slow–fast normal form, we find explicit sufficient conditions to characterize singular Hopf bifurcation and canard explosion of the system. Additionally, the cyclicity of canard cycles is completely solved, and of particular interest is that we show the existence and uniqueness of a canard cycle, whose cyclicity is at most two, under corresponding precise explicit conditions.

Exploring the geometry of the bifurcation sets in parameter space

By studying a nonlinear model by inspecting a p-dimensional parameter space through (p-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p-1)$$\\end{document}-dimensional cuts, one can detect changes that are only determined by the geometry of the manifolds that make up the bifurcation set. We refer to these changes as geometric bifurcations. They can be understood within the framework of the theory of singularities for differentiable mappings and, in particular, of the Morse Theory. Working with a three-dimensional parameter space, geometric bifurcations are illustrated in two models of neuron activity: the Hindmarsh–Rose and the FitzHugh–Nagumo systems. Both are fast-slow systems with a small parameter that controls the time scale of a slow variable. Geometric bifurcations are observed on slices corresponding to fixed values of this distinguished small parameter, but they should be of interest to anyone studying bifurcation diagrams in the context of nonlinear phenomena.

Cyclicity of slow-fast cycles with two canard mechanisms.

In this paper, we study the cyclicity of some degenerate slow-fast cycles with two canard mechanisms in planar slow-fast systems. One canard mechanism originates from a slow-fast Hopf point and the other from a point of self-intersection where the so-called entry-exit relation can be used. By studying the difference map, we show that the cyclicity of such slow-fast cycles is at most two (the associated slow divergence integral is nonzero or vanishes). As an example, we apply this result to the modified Holling-Tanner model.

Modelling and analysis of cAMP-induced mixed-mode oscillations in cortical neurons: Critical roles of HCN and M-type potassium channels.

Cyclic AMP controls neuronal ion channel activity. For example hyperpolarization-activated cyclic nucleotide-gated (HCN) and M-type K+ channels are activated by cAMP. These effects have been suggested to be involved in astrocyte control of neuronal activity, for example, by controlling the action potential firing frequency. In cortical neurons, cAMP can induce mixed-mode oscillations (MMOs) consisting of small-amplitude, subthreshold oscillations separating complete action potentials, which lowers the firing frequency greatly. We extend a model of neuronal activity by including HCN and M channels, and show that it can reproduce a series of experimental results under various conditions involving and inferring with cAMP-induced activation of HCN and M channels. In particular, we find that the model can exhibit MMOs as found experimentally, and argue that both HCN and M channels are crucial for reproducing these patterns. To understand how M and HCN channels contribute to produce MMOs, we exploit the fact that the model is a three-time scale dynamical system with one fast, two slow, and two super-slow variables. We show that the MMO mechanism does not rely on the super-slow dynamics of HCN and M channel gating variables, since the model is able to produce MMOs even when HCN and M channel activity is kept constant. In other words, the cAMP-induced increase in the average activity of HCN and M channels allows MMOs to be produced by the slow-fast subsystem alone. We show that the slow-fast subsystem MMOs are due to a folded node singularity, a geometrical structure well known to be involved in the generation of MMOs in slow-fast systems. Besides raising new mathematical questions for multiple-timescale systems, our work is a starting point for future research on how cAMP signalling, for example resulting from interactions between neurons and glial cells, affects neuronal activity via HCN and M channels.

Pseudo-Abelian integrals and divergence integrals on slow-fast model system

We study the model of slow-fast Darboux integrable system and its polynomial perturbations. We obtain explicit formulas for the pseudo-Abelian integral associated with polynomial deformations of slow-fast as well as for slow-fast (pseudo-)divergence integral. The comparison of both tool detecting limit cycles is the result of the paper.

Slow divergence integral in regularized piecewise smooth systems

In this paper we define the notion of slow divergence integral along sliding segments in regularized planar piecewise smooth systems. The boundary of such segments may contain diverse tangency points. We show that the slow divergence integral is invariant under smooth equivalences. This is a natural generalization of the notion of slow divergence integral along normally hyperbolic portions of curve of singularities in smooth planar slow-fast systems. We give an interesting application of the integral in a model with visible-invisible two-fold of type V I 3 . It is related to a connection between so-called Minkowski dimension of bounded and monotone "entry-exit" sequences and the number of sliding limit cycles produced by so-called canard cycles.

Relaxation Oscillations in an SIS Epidemic Model with a Nonsmooth Incidence

In this study, we extend an SIS epidemic model by introducing a piecewise smooth incidence rate. By assuming that the demographic parameters are much smaller than the disease-related ones, the proposed model is converted to a slow–fast system. Utilizing the geometrical singular perturbation theory and entry-exit function, we prove the coexistence of two relaxation oscillations surrounding the unique positive equilibrium of the model. Numerical simulations are performed to verify our theoretical results. The phenomenon presented in this study can be a potential explanation for that several infectious diseases can re-emerge many years after being almost extinct.

Slow-Fast Systems with an Equilibrium Near the Folded Slow Manifold

Slow-Fast Systems with an Equilibrium Near the Folded Slow Manifold