Normal Hyperbolicity Research Articles

Extending discrete geometric singular perturbation theory to non-hyperbolic points

Abstract We extend the recently developed discrete geometric singular perturbation theory to the non-normally hyperbolic regime. Our primary tool is the Takens embedding theorem, which provides a means of approximating the dynamics of particular maps with the time-1 map of a formal vector field. First, we show that the so-called reduced map, which governs the slow dynamics near slow manifolds in the normally hyperbolic regime, can be locally approximated by the time-1 map of the reduced vector field which appears in continuous-time geometric singular perturbation theory. In the non-normally hyperbolic regime, we show that the dynamics of fast-slow maps with a unipotent linear part can be locally approximated by the time-1 map induced by a fast-slow vector field in the same dimension, which has a nilpotent singularity of the corresponding type. The latter result is used to describe (i) the local dynamics of two-dimensional fast-slow maps with non-normally singularities of regular fold, transcritical and pitchfork type, and (ii) dynamics on a (potentially high-dimensional) local center manifold in n-dimensional fast-slow maps with regular contact or fold submanifolds of the critical manifold. In general, our results show that the dynamics near a large and important class of singularities in fast-slow maps can be described via the use of formal embedding theorems which allow for their approximation by the time-1 map of a fast-slow vector field featuring a loss of normal hyperbolicity.

Existence of traveling wave solutions to reaction-diffusion-ODE systems with hysteresis

This paper establishes the existence of traveling wave solutions to a reaction-diffusion equation coupled with a singularly perturbed first order ordinary differential equation with a small parameter ϵ>0. The system is a toy model for biological pattern formation. Traveling wave solutions correspond to heteroclinic orbits of a fast-slow system. Under some conditions, the reduced problem (with ϵ=0) has a heteroclinic orbit with jump discontinuity, while the layer problem (i.e., the fast subsystem obtained as another limit of ϵ→0) has an orbit filling the gap. We thus construct a singular orbit by piecing together these two orbits. The traveling wave solution is obtained in the neighborhood of the singular orbit. However, unlike the classical FitzHugh-Nagumo equations, the singular orbit contains a fold point where the normal hyperbolicity breaks down and the standard Fenichel theory is not applicable. To circumvent this difficulty we employ the directional blowup method for geometric desingularization around the fold point.

A geometric approach of gradient descent algorithms in linear neural networks

<p style='text-indent:20px;'>In this paper, we propose a geometric framework to analyze the convergence properties of gradient descent trajectories in the context of linear neural networks. We translate a well-known empirical observation of linear neural nets into a conjecture that we call the <i>overfitting conjecture</i> which states that, for almost all training data and initial conditions, the trajectory of the corresponding gradient descent system converges to a global minimum. This would imply that the solution achieved by vanilla gradient descent algorithms is equivalent to that of the least-squares estimation, for linear neural networks of an arbitrary number of hidden layers. Built upon a key invariance property induced by the network structure, we first establish convergence of gradient descent trajectories to critical points of the square loss function in the case of linear networks of arbitrary depth. Our second result is the proof of the <i>overfitting conjecture</i> in the case of single-hidden-layer linear networks with an argument based on the notion of normal hyperbolicity and under a generic property on the training data (i.e., holding for almost all training data).</p>

Stochastic enzyme kinetics and the quasi-steady-state reductions: Application of the slow scale linear noise approximation à la Fenichel.

The linear noise approximation models the random fluctuations from the mean-field model of a chemical reaction that unfolds near the thermodynamic limit. Specifically, the fluctuations obey a linear Langevin equation up to order [Formula: see text], where [Formula: see text] is the size of the chemical system (usually the volume). In the presence of disparate timescales, the linear noise approximation admits a quasi-steady-state reduction referred to as the slow scale linear noise approximation (ssLNA). Curiously, the ssLNAs reported in the literature are slightly different. The differences in the reported ssLNAs lie at the mathematical heart of the derivation. In this work, we derive the ssLNA directly from geometric singular perturbation theory and explain the origin of the different ssLNAs in the literature. Moreover, we discuss the loss of normal hyperbolicity and we extend the ssLNA derived from geometric singular perturbation theory to a non-classical singularly perturbed problem. In so doing, we disprove a commonly-accepted qualifier for the validity of stochastic quasi-steady-state approximation of the Michaelis -Menten reaction mechanism.

Discretized Fast\u2013Slow Systems with Canards in Two Dimensions

We study the problem of preservation of maximal canards for time discretized fast–slow systems with canard fold points. In order to ensure such preservation, certain favorable structure-preserving properties of the discretization scheme are required. Conventional schemes do not possess such properties. We perform a detailed analysis for an unconventional discretization scheme due to Kahan. The analysis uses the blow-up method to deal with the loss of normal hyperbolicity at the canard point. We show that the structure-preserving properties of the Kahan discretization for quadratic vector fields imply a similar result as in continuous time, guaranteeing the occurrence of maximal canards between attracting and repelling slow manifolds upon variation of a bifurcation parameter. The proof is based on a Melnikov computation along an invariant separating curve, which organizes the dynamics of the map similarly to the ODE problem.

Sensitivity analysis for periodic orbits and quasiperiodic invariant tori using the adjoint method

<p style='text-indent:20px;'>This paper presents a rigorous framework for the continuation of solutions to nonlinear constraints and the simultaneous analysis of the sensitivities of test functions to constraint violations at each solution point using an adjoint-based approach. By the linearity of a problem Lagrangian in the associated Lagrange multipliers, the formalism is shown to be directly amenable to analysis using the COCO software package, specifically its paradigm for staged problem construction. The general theory is illustrated in the context of algebraic equations and boundary-value problems, with emphasis on periodic orbits in smooth and hybrid dynamical systems, and quasiperiodic invariant tori of flows. In the latter case, normal hyperbolicity is used to prove the existence of continuous solutions to the adjoint conditions associated with the sensitivities of the orbital periods to parameter perturbations and constraint violations, even though the linearization of the governing boundary-value problem lacks a bounded inverse, as required by the general theory. An assumption of transversal stability then implies that these solutions predict the asymptotic phases of trajectories based at initial conditions perturbed away from the torus. Example COCO code is used to illustrate the minimal additional investment in setup costs required to append sensitivity analysis to regular parameter continuation. <b>200</b> words.</p>

Analysis and profiles of travelling wave solutions to a Darcy-Forchheimer fluid formulated with a non-linear diffusion

<abstract><p>The intention along the presented analysis is to explore existence, uniqueness, regularity of solutions and travelling waves profiles to a Darcy-Forchheimer fluid flow formulated with a non-linear diffusion. Such formulation is the main novelty of the present study and requires the introduction of an appropriate mathematical treatment to deal with the introduced degenerate diffusivity. Firstly, the analysis on existence, regularity and uniqueness is shown upon definition of an appropriate test function. Afterwards, the problem is formulated within the travelling wave domain and analyzed close the critical points with the Geometric Perturbation Theory. Based on this theory, exact and asymptotic travelling wave profiles are obtained. In addition, the Geometric Perturbation Theory is used to provide evidences of the normal hyperbolicity in the involved manifolds that are used to get the associated travelling wave solutions. The main finding, which is not trivial in the non-linear diffusion case, is related with the existence of an exponential profile along the travelling frame. Eventually, a numerical exercise is introduced to validate the analytical solutions obtained.</p></abstract>

Chimera states through invariant manifold theory

We establish the existence of chimera states, simultaneously supporting synchronous and asynchronous dynamics, in a network of two symmetrically linked star subnetworks of identical oscillators with shear and Kuramoto–Sakaguchi coupling. We show that the chimera states may be metastable or asymptotically stable. If the intra-star coupling strength is of order ɛ, the chimera states persist on time scales at least of order 1/ɛ in general, and on time-scales at least of order 1/ɛ 2 if the intra-star coupling is of Kuramoto–Sakaguchi type. If the intra-star coupling configuration is sparse, the chimeras are asymptotically stable. The analysis relies on a combination of dimensional reduction using a Möbius symmetry group and techniques from averaging theory and normal hyperbolicity.

Nonparametric Bifurcation and Anti-Control of Hyperchaos in a Memristor–Memcapacitor-Based Circuit

In this paper, a new memristor model and a new memcapacitor model are proposed. Based on the two models, a simple chaotic circuit is constructed. Due to the special characteristics of the memristor and memcapacitor, the proposed circuit has two-dimensional normally hyperbolic manifolds of equilibria, and nonparametric bifurcation can occur when the conditions supporting the normal hyperbolicity of such manifolds are not satisfied. By adding a nonlinear controller to the proposed circuit, an anti-controlled system is realized, which has hyperchaotic dynamic behaviors under some suitable control parameters. The stability of equilibrium points and dynamic properties of the original system and the anti-controlled system are explored by Lyapunov exponents, bifurcation diagrams and so on. Furthermore, the anti-controlled system is applied to design a random sequence generator on digital signal processor platform.

Traveling pulses in a coupled FitzHugh–Nagumo equation

For a coupled FitzHugh–Nagumo (FHN) equation with a parameter a∈[0,1], when a∈(0,1∕2) the existence of traveling pulses of this equation was proved in 2013 by Holzer, Doelman and Kaper. Here we adopt a new approach to obtain the existence of the traveling pulse of the same equation for a∈[0,1∕2), which includes the degenerate case a=0, where the origin on the critical set loses normal hyperbolicity while it is for a∈(0,1). Also we show that the pulse does not exhibit an oscillatory tail at the homoclinic orbit when the time t→±∞, whereas the classical FHN equation could have an oscillatory tail of the traveling pulse depending on the choice of the parameters of the system. Finally we present an explanation on why traveling pulses cannot exist when a∈[1∕2,1].

Hunting $\varepsilon$: The Origin and Validity of Quasi-Steady-State Reductions in Enzyme Kinetics

The estimation of the kinetic parameters requires the careful design of experiments under a constrained set of conditions. Many estimates reported in the literature incorporate protocols that leverage simplified mathematical models known as quasi-steady-state reductions. Such reductions often - but not always - emerge as the result of a singular perturbation scenario. However, the utilization of the singular perturbation reduction method requires knowledge of a dimensionless parameter, $\varepsilon$, that is proportional to the ratio of the reaction's fast and slow timescales. Using techniques from differential equations, Fenichel theory, and center manifold theory, we derive the appropriate $\varepsilon$ whose magnitude regulates the validity of the quasi-steady-state reduction employed in the reported experimental procedures for intermolecular autocatalytic zymogen activation reaction. Although the model equations are two-dimensional, the fast/slow dynamics are rich. The phase plane exhibits a dynamic transcritical bifurcation point in a particular singular limit. The existence of such a bifurcation is relevant, because the critical manifold losses normal hyperbolicity and classical Fenichel theory is inapplicable. Furthermore, we show that in some cases chemical reversibility can be interpreted dynamically as an imperfection, since the presence of reversibility can destroy the bifurcation structure present in the singular limit. We show that the reduction method by which QSS reductions are justified can depend on the path taken in parameter space. Specifically, we show that the standard quasi-steady-state reduction for this reaction is justifiable by center manifold theory in one limit, and via Fenichel theory in a different limit.

Approximative dichotomy and persistence of nonuniformly normally hyperbolic invariant manifolds in Banach spaces

It was proved that both the normal hyperbolicity and invariant manifold for a uniformly hyperbolic compact invariant manifold and the invariant manifold for a uniformly hyperbolic noncompact invariant manifold are persistent under small perturbation. In this paper, we weaken the uniform normal hyperbolicity to the nonuniform one and prove that both the nonuniform normal hyperbolicity and invariant manifold for a nonuniformly eventually absolutely normally hyperbolic noncompact invariant manifold are persistent under small perturbation in Banach spaces. Unlike the uniform case, we need infinitely many families of norms to uniformize infinitely many exponential dichotomies correspondingly. We use the approximative exponential dichotomy and the pseudo orbit to overcome the difficulty.

Normal resonances in a double Hopf bifurcation

We introduce a framework to systematically investigate the resonant double Hopf bifurcation. We use the basic invariants of the ensuing T1-action to analyse the approximating normal form truncations in a unified manner. In this way we obtain a global description of the parameter space and thus find the organising resonance droplet, which is the present analogue of the resonant gap. The dynamics of the normal form yields a skeleton for the dynamics of the original system. In the ensuing perturbation theory both normal hyperbolicity (centre manifold theory) and kam theory are being used.

The quasi-steady-state approximations revisited: Timescales, small parameters, singularities, and normal forms in enzyme kinetics

In this work, we revisit the scaling analysis and commonly accepted conditions for the validity of the standard, reverse and total quasi-steady-state approximations through the lens of dimensional Tikhonov–Fenichel parameters and their respective critical manifolds. By combining Tikhonov–Fenichel parameters with scaling analysis and energy methods, we derive improved upper bounds on the approximation error for the standard, reverse and total quasi-steady-state approximations. Furthermore, previous analyses suggest that the reverse quasi-steady-state approximation is only valid when initial enzyme concentrations greatly exceed initial substrate concentrations. However, our results indicate that this approximation can be valid when initial enzyme and substrate concentrations are of equal magnitude. Using energy methods, we find that the condition for the validity of the reverse quasi-steady-state approximation is far less restrictive than was previously assumed, and we derive a new “small” parameter that determines the validity of this approximation. In doing so, we extend the established domain of validity for the reverse quasi-steady-state approximation. Consequently, this opens up the possibility of utilizing the reverse quasi-steady-state approximation to model enzyme catalyzed reactions and estimate kinetic parameters in enzymatic assays at much lower enzyme to substrate ratios than was previously thought. Moreover, we show for the first time that the critical manifold of the reverse quasi-steady-state approximation contains a singular point where normal hyperbolicity is lost. Associated with this singularity is a transcritical bifurcation, and the corresponding normal form of this bifurcation is recovered through scaling analysis.

Persistence of a normally hyperbolic manifold for a system of non densely defined Cauchy problems

We consider a system of non densely defined Cauchy problems and we investigate the persistence of normally hyperbolic manifolds. The notion of exponential dichotomy is used to characterize the normal hyperbolicity and a generalized Lyapunov-Perron approach is used in order to prove our main result. The result presented in this article extend the previous results on the center manifold by allowing a nonlinear dynamic in the unperturbed central part of the system. We consider two examples to illustrate our results. The first example is a parabolic equation coupled with an ODE that can be considered as an interaction between an antimicrobial and bacteria while the second one is a Ross-Macdonald epidemic model with age of infection. In both examples we were able to reduce the infinite dimensional system to an ordinary differential equation.

Relaxation oscillations and canards in the Jirsa\u2013Kelso excitator model: global flow perspective

Fenichel’s geometric singular perturbation theory and the blow-up method have been very successful in describing and explaining global non-linear phenomena in systems with multiple time-scales, such as relaxation oscillations and canards. Recently, the blow-up method has been extended to systems with flat, unbounded slow manifolds that lose normal hyperbolicity at infinity. Here, we show that transition between discrete and periodic movement captured by the Jirsa–Kelso excitator is a new example of such phenomena. We, first, derive equations of the Jirsa–Kelso excitator with explicit time scale separation and demonstrate existence of canards in the systems. Then, we combine the slow-fast analysis, blow-up method and projection onto the Poincaré sphere to understand the return mechanism of the periodic orbits in the singular case, ϵ = 0.

Quasi-normal modes for Dirac fields in the Kerr–Newman–de Sitter black holes

We provide the full asymptotic description of the quasi-normal modes (resonances) in any strip of fixed width for Dirac fields in slowly rotating Kerr–Newman–de Sitter black holes. The resonances split in a way similar to the Zeeman effect. The method is based on the extension to Dirac operators of techniques applied by Dyatlov in [Quasi-normal modes and exponential energy decay for the Kerr–de Sitter black hole, Commun. Math. Phys. 306(1) (2011) 119–163; Asymptotic distribution of quasi-normal modes for Kerr–de Sitter black holes, Ann. Henri Poincaré 13(5) (2012) 1101–1166] to the (uncharged) Kerr–de Sitter black holes. We show that the mass of the Dirac field does not have an effect on the two leading terms in the expansions of resonances. We give an expansion of the solution of the evolution equation for the Dirac fields in the outer region of the slowly rotating Kerr–Newman–de Sitter black hole which implies the exponential decay of the local energy. Moreover, using the [Formula: see text]-normal hyperbolicity of the trapped set and applying the techniques from [Asymptotics of linear waves and resonances with applications to black holes, Commun. Math. Phys. 335 (2015) 1445–1485; Resonance projectors and asymptotics for [Formula: see text]-normally hyperbolic trapped sets, J. Amer. Math. Soc. 28 (2015) 311–381], we give location of the resonance free band and the Weyl-type formula for the resonances in the band near the real axis.

Persistence Properties of Normally Hyperbolic Tori

Near-resonances between frequencies notoriously lead to small denominators when trying to prove persistence of invariant tori carrying quasi-periodic motion. In dissipative systems external parameters detuning the frequencies are needed so that Diophantine conditions can be formulated, which allow to solve the homological equation that yields a conjugacy between perturbed and unperturbed quasi-periodic tori. The parameter values for which the Diophantine conditions are not fulfilled form so-called resonance gaps. Normal hyperbolicity can guarantee invariance of the perturbed tori, if not their quasi-periodicity, for larger parameter ranges. For a 1-dimensional parameter space this allows to close almost all resonance gaps.

Transcritical Bifurcation without Parameters in Memristive Circuits

The transcritical bifurcation without parameters (TBWP) describes a stability change along a line of equilibria, resulting from the loss of normal hyperbolicity at a given point of such a line. Memristive circuits systematically yield manifolds of nonisolated equilibria, and in this paper we address a systematic characterization of the TBWP in circuits with a single memristor. To achieve this we develop two mathematical results of independent interest; the first is an extension of the TBWP theorem to explicit ordinary differential equations (ODEs) in arbitrary dimension; the second result drives the characterization of this phenomenon to semiexplicit differential-algebraic equations (DAEs), which provide the appropriate framework for the analysis of circuit dynamics. In the circuit context the analysis is performed in graph-theoretic terms: in this setting, our first working scenario is restricted to passive problems (an exception is made for the bifurcating memristor), and in a second step some results are presented for the analysis of nonpassive cases. The latter context is illustrated by means of a memristive neural network model.

Local bifurcations in differential equations with state-dependent delay

A common task when analysing dynamical systems is the determination of normal forms near local bifurcations of equilibria. As most of these normal forms have been classified and analysed, finding which particular class of normal form one encounters in a numerical bifurcation study guides follow-up computations. This paper builds on normal form algorithms for equilibria of delay differential equations with constant delay that were developed and implemented in DDE-Biftool recently. We show how one can extend these methods to delay-differential equations with state-dependent delay (sd-DDEs). Since higher degrees of regularity of local center manifolds are still open for sd-DDEs, we give an independent (still only partial) argument which phenomena from the truncated normal must persist in the full sd-DDE. In particular, we show that all invariant manifolds with a sufficient degree of normal hyperbolicity predicted by the normal form exist also in the full sd-DDE.