Normal Form Reduction Research Articles

Paralinearization and extended lifespan for solutions of the α-SQG sharp front equation

In this paper we paralinearize the contour dynamics equation for sharp-fronts of α-SQG, for any α∈(0,1)∪(1,2), close to a circular vortex. This turns out to be a quasi-linear Hamiltonian PDE. The key idea relies on a novel desingularization of the Hamiltonian vector field which is a convolution integral operator with nonlinear singular kernel. After deriving the asymptotic expansion of the linear frequencies of oscillations at the vortex disk and verifying the absence of three wave interactions, we prove that, in the most singular cases α∈(1,2), any initial vortex patch which is ε-close to the disk exists for a time interval of size at least ∼ε−2. This quadratic lifespan result relies on a paradifferential Birkhoff normal form reduction and exploits cancellations arising from the Hamiltonian nature of the equation. This is the first normal form long time existence result of sharp fronts.

Unconditional well-posedness for the periodic Boussinesq and Kawahara equations

<abstract><p>In this article, we obtain new results on the unconditional well-posedness for a pair of periodic nonlinear dispersive equations using an abstract framework introduced by Kishimoto. This framework is based on a normal form reductions argument coupled with a number of crucial multilinear estimates.</p></abstract>

Energy and power characteristics of nanocatalyzed Belousov-Zhabotinsky reactions via bifurcation analyses.

Active stimuli-responsive materials, intrinsically powered by chemical reactions, have immense capabilities that can be harnessed for designing synthetic systems for a variety of biomimetic applications. It goes without saying that the key aspect involved in the designing of such systems is to accurately estimate the amount of energy and power available for transduction through various mechanisms. Belousov-Zhabotinsky (BZ) reactions are dynamical systems, which exhibit self-sustained nonlinear chemical oscillations, as their catalyst undergoes periodic redox cycles in the presence of reagents. The unique feature of BZ reaction based active systems is that they can continuously perform mechanical work by transducing energy from sustained chemical oscillations. The objective of our work is to use bifurcation analyses to identify oscillatory regimes and quantify energy-power characteristics of the BZ reaction based on nanocatalyst activity and BZ reaction formulations. Our approach involves not only the computation of higher order Lyapunov and frequency coefficients but also Hamiltonian functions, through normal form reduction of the kinetic model of the BZ reaction. Ultimately, using these calculations, we determine amplitude, frequency, and energy-power densities, as a function of the nanocatalysts' activity and BZ formulations. As normal form representations are applicable to any dynamical system, we believe that our framework can be extended to other self-sustained active systems, including systems based on stimuli-responsive materials.

Semi-analytical estimates for the chaotic diffusion in the Second Fundamental Model of Resonance. Application to Earth’s navigation satellites

We discuss the applicability of the Melnikov and Landau–Teller theories in obtaining semi-analytical estimates of the speed of chaotic diffusion in systems driven by the separatrix-like stochastic layers of a resonance belonging to the ‘second fundamental model’ (SFM) (Henrard and Lemaitre, 1983). Stemming from the analytic solution for the SFM in terms of Weierstrass elliptic functions, we introduce stochastic Melnikov and Landau–Teller models allowing to locally approximate chaotic diffusion as a sequence of uncorrelated ‘jumps’ observed in the time series yielding the slow evolution of an ensemble of trajectories in the space of the adiabatic actions of the system. Such jumps occur in steps of one per homoclinic loop. We show how a semi-analytical determination of the probability distribution of the size of the jumps can be arrived at by the Melnikov and Landau–Teller approximate theories. Computing also the mean time required per homoclinic loop, we arrive at estimates of the chaotic diffusion coefficient in such systems. As a concrete example, we refer to the long-term diffusion of a small object (e.g. Earth navigation satellite or space debris) within the chaotic layers of the so-called 2g+h lunisolar resonance, which is of the SFM type. After a suitable normal form reduction of the Hamiltonian, we compute estimates of the speed of diffusion of these objects, which compare well with the results of numerical experiments.

Quasi-periodic solutions for quintic completely resonant derivative beam equations on T2

In the present paper, we consider two dimensional completely resonant, derivative, quintic nonlinear beam equations with reversible structure. Because of this reversible system without external parameters or potentials, Birkhoff normal form reduction is necessary before applying Kolmogorov–Arnold–Moser (KAM) theorem. As application of KAM theorem, the existence of partially hyperbolic, small amplitude, quasi-periodic solutions of the reversible system is proved in this paper.

Analyzing Innermost Runtime Complexity Through Tuple Interpretations

Time complexity in rewriting is naturally understood as the number of steps needed to reduce terms to normal forms. Establishing complexity bounds to this measure is a well-known problem in the rewriting community. A vast majority of techniques to find such bounds consist of modifying termination proofs in order to recover complexity information. This has been done for instance with semantic interpretations, recursive path orders, and dependency pairs. In this paper, we follow the same program by tailoring tuple interpretations to deal with innermost complexity analysis. A tuple interpretation interprets terms as tuples holding upper bounds to the cost of reduction and size of normal forms. In contrast with the full rewriting setting, the strongly monotonic requirement for cost components is dropped when reductions are innermost. This weakened requirement on cost tuples allows us to prove the innermost version of the compatibility result: if all rules in a term rewriting system can be strictly oriented, then the innermost rewrite relation is well-founded. We establish the necessary conditions for which tuple interpretations guarantee polynomial bounds to the runtime of compatible systems and describe a search procedure for such interpretations.

Sharp Resolvent Estimate for the Damped-Wave Baouendi–Grushin Operator and Applications

In this article we study the semiclassical resolvent estimate for the non-selfadjoint Baouendi-Grushin operator on the two-dimensional torus $\mathbb{T}^2=\mathbb{R}^2/(2\pi\mathbb{Z})^2$ with H\"older dampings. The operator is subelliptic degenerating along the vertical direction at $x=0$. We exhibit three different situations: (i) the damping region verifies the geometric control condition with respect to both the non-degenerate Hamiltonian flow and the vertical subelliptic flow; (ii) the undamped region contains a horizontal strip; (iii) the undamped part is a line. In all of these situations, we obtain sharp resolvent estimates. Consequently, we prove the optimal energy decay rate for the associated damped waved equations. For (i) and (iii), our results are in sharp contrast to the Laplace resolvent since the optimal bound is governed by the quasimodes in the subelliptic regime. While for (ii), the optimality is governed by the quasimodes in the elliptic regime, and the optimal energy decay rate is the same as for the classical damped wave equation on $\mathbb{T}^2$. Our analysis contains the study of adapted two-microlocal semiclassical measures, construction of quasimodes and refined Birkhoff normal-form reductions in different regions of the phase-space. Of independent interest, we also obtain the propagation theorem for semiclassical measures of quasimodes microlocalized in the subelliptic regime.

Bifurcation based quadratic feedback control for Moore‐Greitzer model with surge and multiple stall modes

AbstractWe consider the ‐mode truncation of the Moore‐Greitzer partial differential equation model for axial compressors with bleed valve as actuator. It is well known that all the rotating stall modes are linearly unstabilizable past the peak of the compressor characteristic. By using the center manifold and normal form reduction, we obtain a model describing the nonlinear interactions of rotating stall and surge modes when the compressor operates in a neighborhood of the peak pressure rise. The normal form is a special class of the the famous Lotka–Volterra system. Typically, there are equilibria as the throttle coefficient varies. We analyze the stability of these equilibria for the normal form, and interpret their relevance to the robustness of the nominal equilibrium near the peak pressure rise. We also derive the necessary and sufficient conditions for stabilizing the peak of the characteristic using a quadratic feedback control law.

Non-constant steady states and Hopf bifurcation of a species interaction model

In this paper, we are concerned with the species interaction model with the ratio-dependent Holling III functional response and strong Allee effect. To explore nonhomogeneous solutions of the model, we consider the existence and non-existence of non-constant steady states and temporal bifurcation. Then we show the boundedness of the global positive solutions for the parabolic system and present the upper and lower bounds of positive solutions for the associated elliptic system. The non-existence and existence of the non-constant steady states of the elliptic system with the homogeneous Neumann boundary conditions are obtained by using the priori estimates, maximum principle and index theory. Furthermore, the existence and the direction of Hopf bifurcation are investigated via the stability analysis, center manifold theory and normal form reduction. Numerical simulations are carried out to verify our theoretical analysis and to illustrate that the ratio-dependent Holling III functional response and strong Allee effect have strong impact on dynamical behaviors of the species interaction systems.

Sharp Decay Rate for the Damped Wave Equation With Convex-Shaped Damping

Abstract We revisit the damped wave equation on two-dimensional torus where the damped region does not satisfy the geometric control condition. It was shown in [1] that, for sufficiently regular damping, the damped wave equation is stale at a rate sufficiently close to $t^{-1}$. We show that if the damping vanishes like a Hölder function $|x|^{\beta }$, and in addition, the boundary of the damped region is locally strictly convex with positive curvature, the wave is stable at rate $t^{-1+\frac {2}{2\beta +7}}$, which is better than the known optimal decay rate $t^{-1+\frac {1}{\beta +3}}$ for strip-shaped dampings of the same Hölder regularity. Moreover, we show by example that the decay rate is optimal. This illustrates the fact that the sharp energy decay rate depends not only on the order of vanishing of the damping but also on the shape of the damped region. The main ingredient of the proof is the averaging method (normal form reduction) developed by Hitrik and Sjöstrand ([20], [35]).

Codimension two bifurcation in a coupled FitzHugh–Nagumo system with multiple delays

In this paper, a coupled FitzHugh–Nagumo system with multiple delays is considered. In a first step, the critical point at which a zero root of multiplicity two occurs in the characteristic equation is constructed. In a second step, in order to ensure that all the roots of the characteristic equation except for the double zero root have negative real parts, the zeros of a third and a fourth degree exponential polynomial are studied. Moreover, the critical values where the Bogdanov–Takens bifurcation occurs are derived. By using the normal form theory and reduction on the centre manifold, the truncated normal form is obtained, and throughout the bifurcation diagram, its dynamical behaviours are studied. Finally, a numerical example is given to demonstrate our results.

Delayed carrying capacity induced subcritical and supercritical Hopf bifurcations in a predator–prey system

The effects of past activities of prey population on the dynamics of a predator–prey system are explored by introducing a delayed carrying capacity of the prey. The delay-induced stability changes, bifurcation patterns, and long-term dynamics are studied. The delay instigates three types of stability scenarios at the coexisting steady state: no stability change, stability change (stable to unstable only), and stability switching, depending on the other parameters. This is one of the novelties of the present study in obtaining all these results in a single model with a single time delay. The relationships between delay and other system parameters for the existence of the stability scenarios are established analytically. Nature of the bifurcations is determined using normal-form reduction. Another important contribution is to unveil the occurrence of delay-induced subcritical and supercritical, non-degenerate and degenerate Hopf bifurcations around the coexisting steady state. Finite-time oscillation death resulting in mass extinction of populations is observed in the unsteady regime.

Solitary waves in a Whitham equation with small surface tension

AbstractUsing a nonlocal version of the center‐manifold theorem and a normal form reduction, we prove the existence of small‐amplitude generalized solitary‐wave solutions and modulated solitary‐wave solutions to the steady gravity‐capillary Whitham equation with weak surface tension. Through the application of the center‐manifold theorem, the nonlocal equation for the solitary wave profiles is reduced to a four‐dimensional system of ODEs inheriting reversibility. Along particular parameter curves, relating directly to the classical gravity‐capillary water wave problem, the associated linear operator is seen to undergo either a reversible bifurcation or a reversible bifurcation. Through a normal form transformation, the reduced system of ODEs along each relevant parameter curve is seen to be well approximated by a truncated system retaining only second‐order or third‐order terms. These truncated systems relate directly to systems obtained in the study of the full gravity‐capillary water wave equation and, as such, the existence of generalized and modulated solitary waves for the truncated systems is guaranteed by classical works, and they are readily seen to persist as solutions of the gravity‐capillary Whitham equation due to reversibility. Consequently, this work illuminates further connections between the gravity‐capillary Whitham equation and the full two‐dimensional gravity‐capillary water wave problem.

Normal form approach to the one-dimensional periodic cubic nonlinear Schr\xf6dinger equation in almost critical Fourier-Lebesgue spaces

In this paper, we study the one-dimensional cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, we develop a normal form approach to study NLS in almost critical Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions introduced by the first author with Z. Guo and S. Kwon (2013), we derive a normal form equation which is equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. We then prove that this normal form equation is unconditionally globally well-posed in the Fourier-Lebesgue spaces ℱLp({cal F}{L^p}(mathbb{T})), 1 ≤ p < ∞. By inverting the transformation, we conclude global well-posedness of the renormalized cubic NLS in almost critical Fourier-Lebesgue spaces in a suitable sense. This approach also allows us to prove unconditional uniqueness of the (renormalized) cubic NLS in ℱLp({cal F}{L^p}(mathbb{T})) for 1 leq p leq {3 over 2}.

Unconditional well-posedness for the Kawahara equation

This article is concerned with the unconditional well-posedness for the Kawahara equation on the real line and shows that this holds true for initial data in L2(R). This is achieved by applying an infinite iteration scheme of normal form reductions.

English

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schr\"odinger equation with initial data $u_{0}\in X$, where $X\in\{M_{2,q}^{s}(\mathbb R), H^{\sigma}(\mathbb T), H^{s_{1}}(\mathbb R)+H^{s_{2}}(\mathbb T)\}$ and $q\in[1,2]$, $s\geq0$, or $\sigma\geq0$, or $s_{2}\geq s_{1}\geq0$. Moreover, if $M_{2,q}^{s}(\mathbb R)\hookrightarrow L^{3}(\mathbb R)$, or if $\sigma\geq\frac16$ or if $s_{1}\geq\frac16$ and $s_{2}>\frac12$ we show that the Cauchy problem is unconditionally wellposed in $X$. Similar results hold true for all higher order nonlinear Schr\"odinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ the normal form reduction via the differentiation by parts technique and build upon our previous work

Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line

In this paper, we revisit the infinite iteration scheme of normal form reductions, introduced by the first and second authors (with Z. Guo), in constructing solutions to nonlinear dispersive PDEs. Our main goal is to present a simplified approach to this method. More precisely, we study normal form reductions in an abstract form and reduce multilinear estimates of arbitrarily high degrees to successive applications of basic trilinear estimates. As an application, we prove unconditional well-posedness of canonical nonlinear dispersive equations on the real line. In particular, we implement this simplified approach to an infinite iteration of normal form reductions in the context of the cubic nonlinear Schr\odinger equation (NLS) and the modified KdV equation (mKdV) on the real line and prove unconditional well-posedness in $H^s(\mathbb R)$ with (i) $s\geq \frac 16$ for the cubic NLS and (ii) $s > \frac 14$ for the mKdV. Our normal form approach also allows us to construct weak solutions to the cubic NLS in $H^s(\mathbb R)$, $0 \leq s < \frac 16$, and distributional solutions to the mKdV in $H^\frac{1}{4}(\mathbb R)$ (with some uniqueness statements).

Unconditional local well-posedness for periodic NLS

Nonlinear Schrödinger equations with nonlinearities |u|2ku on the d-dimensional torus are considered for arbitrary positive integers k and d. The solution of the Cauchy problem is shown to be unique in the class CtHxs for a certain range of scale-subcritical regularities s, which is almost optimal in the case d≥4 or k≥2. The proof is based on various multilinear estimates and the infinite normal form reduction argument.

Control of Neimark–Sacker bifurcation in a type of weak impulse excited centrifugal governor system

Centrifugal governors play an important role in rotating machinery such as diesel engines and steam engines. This paper considers two impulse excitations of the freewheel. The feedback control issue of the Neimark–Sacker bifurcation design of the centrifugal governor system is studied. A feedback control method is addressed to realize the control objectives of the existence, stability and the mean radius of cross section of the torus solution. An explicit criterion, without the use of eigenvalue calculations, including eigenvalue assignments and transversality conditions, is used to push the linear gain responsible for controlling the existence of bifurcation. Using the central manifold theory and the normal form reduction method, the nonlinear gain of the control torus stability is obtained. The expression of mean radius of cross section of the limit torus is developed to derive the linear gains, which are responsible for controls of torus. Numerical simulations of the centrifugal governor system show that the Neimark–Sacker​ torus with the desired characteristics can be produced at any of the pre-set parameter points.

Eigenvalues and eigenfunctions for the two dimensional Schrödinger operator with strong magnetic field

We study the eigenvalues of the two-dimensional Schrödinger operator with a large constant magnetic field perturbed by a decaying scalar potential. For each Landau level, we give the precise asymptotic distribution of eigenvalues created by the minimum, maximum and the closed energy curve of the potential. Normal form reduction, WKB construction and pseudodifferential calculus are applied to the effective Hamiltonian.