Normal Form Theorem Research Articles

A Nekhoroshev-type theorem for the nonlinear Schrödinger equations on the d-dimensional torus

We prove a Nekhoroshev type long time stability result for the following nonlinear Schrödinger (NLS) equations on the d-dimensional torus i u t = − △ u + V ∗ u + ∂ F ( | u | 2 ) ∂ u ¯ ( u , u ¯ ) , x ∈ T d , t ∈ R , where V is a smooth convolution potential, F ( z ) is a real-valued polynomial function in z satisfying F ( 0 ) = F ′ ( 0 ) = 0 . More precisely, it is proved that in modified Sobolev space (see (2)) if the norm of initial datum is smaller than ε ( 0 < ε ≪ 1 ) , then the corresponding solution is bounded by 2 ε over time-intervals of length e e 1 ε . The result generalizes the 1-dimensional case in L. Biasco, J.E. Massetti, M. Procesi. [An abstract Birkhoff normal form theorem and exponential type stability of the 1d NLS. Comm Math Phys. 2020;375(3):2089–2153] to d-dimensional NLS equations.

Multiteam Semantics for Interventionist Counterfactuals: Probabilities and Causation

AbstractIn (Barbero and Sandu 2020 Journal of Philosophical Logic, 50, 471-521), we showed that languages encompassing interventionist counterfactuals and causal notions based on them (as e.g. in Pearl’s and Woodward’s manipulationist approaches to causation) as well as information-theoretic notions (such as learning and dependence) can be interpreted in a semantic framework which combines the traditions of structural equation modeling and of team semantics. We now present a further extension of this framework (causal multiteams) which allows us to talk about probabilistic causal statements. We analyze the expressivity resources of two causal-probabilistic languages, one finitary and one infinitary. We show that many causal-probabilistic notions from the literature on causal inference can already be expressed in the finitary language, and we prove a normal form theorem that throws a new light on Pearl’s “ladder of causation”. In addition, we provide an exact semantic characterization of the infinitary language, which shows that this language captures precisely those causal-probabilistic statements that do not commit us to any specific interpretation of probability; and we prove that no usual, countable language is apt for this task.

Spatiotemporal Patterns in a Space–Time Discrete SIRS Epidemic Model with Self- and Cross-Diffusion

In this work, a two-dimensional Coupled Map Lattice (CML) method is applied to a continuous Reaction–Diffusion (RD) SIRS epidemic model. The model involves a complex, nonlinear incidence rate and the spatially heterogeneous self- and cross-diffusion coefficients. The CML method applied here results in a space–time discrete counterpart of the model. The basic reproduction number is derived, and the local stability of the fixed points of the discrete model is established. The two typical bifurcations of codimension 1, such as the flip and the Neimark–Sacker (NS) bifurcations, are investigated by utilizing the center manifold theory and the normal form theorem. Furthermore, the codimension-2 fold–flip bifurcation about the endemic fixed point is discussed in detail. Turing instability criterion of the cross-diffusion epidemic model is established by utilizing the CML method under periodic boundary conditions. Finally, the numerical simulations clarify both the effectiveness of all theoretical findings and the effects of time step size and cross-diffusion on the spatiotemporal patterns. The main contribution of this work is the discovery of some interesting Turing and non-Turing patterns which may be induced by flip–Turing instability, NS–Turing instability or fold–flip–Turing instability. Concretely speaking, spiral patterns are observed near the flip bifurcation threshold value and ultimately evolve into an irregular defect pattern. Stripe pattern, labyrinth pattern and circular pattern are observed near the NS bifurcation threshold value. The disorderly interwoven ribbon pattern, mosaic pattern and some irregular oscillatory patterns are observed near the fold–flip bifurcation point. Our results greatly help in predicting the long-term behavior of the space–time discrete SIRS epidemic model under study.

Investigating the Dynamic Behavior of Integer and Noninteger Order System of Predation with Holling’s Response

The paper’s primary objective is to examine the dynamic behavior of an integer and noninteger predator–prey system with a Holling type IV functional response in the Caputo sense. Our focus is on understanding how harvesting influences the stability, equilibria, bifurcations, and limit cycles within this system. We employ qualitative and quantitative analysis methods rooted in bifurcation theory, dynamical theory, and numerical simulation. We also delve into studying the boundedness of solutions and investigating the stability and existence of equilibrium points within the system. Leveraging Sotomayor’s theorem, we establish the presence of both the saddle-node and transcritical bifurcations. The analysis of the Hopf bifurcation is carried out using the normal form theorem. The model under consideration is extended to the fractional reaction–diffusion model which captures non-local and long-range effects more accurately than integer-order derivatives. This makes fractional reaction–diffusion systems suitable for modeling phenomena with anomalous diffusion or memory effects, improving the fidelity of simulations in turn. An adaptable numerical technique for solving this class of differential equations is also suggested. Through simulation results, we observe that one of the Lyapunov exponents has a negative value, indicating the potential for the emergence of a stable-limit cycle via bifurcation as well as chaotic and complex spatiotemporal distributions. We supplement our analytical investigations with numerical simulations to provide a comprehensive understanding of the system’s behavior. It was discovered that both the prey and predator populations will continue to coexist and be permanent, regardless of the choice of fractional parameter.

Stratification of the transverse momentum map

Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a refinement of the canonical stratification associated to the Lie groupoid action (the orbit type stratification, in the case of a Hamiltonian Lie group action) that seems not to have appeared before, even in the literature on Hamiltonian Lie group actions. This refinement turns out to be compatible with the Poisson geometry of the Hamiltonian action: it is a Poisson stratification of the orbit space, each stratum of which is a regular Poisson manifold that admits a natural proper symplectic groupoid integrating it. The main tools in our proofs (which we believe could be of independent interest) are a version of the Marle–Guillemin–Sternberg normal form theorem for Hamiltonian actions of proper symplectic groupoids and a notion of equivalence between Hamiltonian actions of symplectic groupoids, closely related to Morita equivalence between symplectic groupoids.

Normal forms for principal Poisson Hamiltonian spaces

We prove a normal form theorem for principal Hamiltonian actions on Poisson manifolds around the zero locus of the moment map. The local model is the generalization to Poisson geometry of the classical minimal coupling construction from symplectic geometry of Sternberg and Weinstein. Further, we show that the result implies that the quotient Poisson manifold is linearizable, and we show how to extend the normal form to other values of the moment map.

Bifurcación de Hopf en un sistema autónomo presa-predador con crecimiento logístico y respuesta funcional de Holling tipo II

The autonomous prey-predator system with logistic growth and Holling type II functional response, which describes the population dynamics of two species. In this study, the equilibrium points of the system (1.1) were identified. Two saddle-node points and one non-trivial P3 equilibrium point were found. For this latter point, the conditions were determined for the Jacobian matrix of (1.1), evaluated at P3, to have a pair of purely complex eigenvalues (necessary condition for the Hopf bifurcation). Through this analysis, values c0, δ0, k0 were found that satisfied these conditions. Subsequently, each of these values is considered as a bifurcation parameter value, and the remaining two are considered as control parameters, under the assumptions of the normal form theorem for the Hopf bifurcation, it’s concluded that by varying these values slightly, the system undergoes the Hopf bifurcation. Finally, the first Lyapunov coefficient was calculated to determine the conditions under which the system exhibits supercritical, subcritical, and degenerate Hopf bifurcation. The analysis was supported by using MAPLE and MATLAB software, which enabled graphical visualization of the obtained results.

A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF

AbstractFor any subset $Z \subseteq {\mathbb {Q}}$ , consider the set $S_Z$ of subfields $L\subseteq {\overline {\mathbb {Q}}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in L such that $C \cap {\mathbb {Q}}=Z$ . Placing a natural topology on the set ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$ of subfields of ${\overline {\mathbb {Q}}}$ , we show that if Z is not thin in ${\mathbb {Q}}$ , then $S_Z$ is meager in ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$ . Here, thin and meager both mean “small”, in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields L have the property that the ring of algebraic integers $\mathcal {O}_L$ is universally definable in L. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists $ -definable subset of an algebraic extension of ${\mathbb Q}$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.

Complex patterns in a space–time discrete mathematical model of antibiotic resistance in hospitals with self-diffusion

In this work, we extend an antibiotic resistance mathematical model in hospitals spatially on a two-dimensional coupled map lattice. The complex dynamics of the resultant space–time discrete model are investigated. An invasion reproduction number Rar and two control parameters Rsc and Rrc for sensitive bacteria and resistant bacteria, respectively, are defined. The three numbers play an important role in the existence and stability of fixed points. According to the normal form theorem and the center manifold theory, we show that the discrete model exhibits both the flip and the Turing bifurcations. We prove that the existence of the interior fixed point excludes the possibility of the occurrence of a Neimark–Sacker bifurcation. Numerical simulations, including phase diagrams, Lyapunov exponents, bifurcation diagrams, and complex patterns, are enrolled to verify the analyses. The complex patterns induced here are either a result of the pure Turing instability, or the flip instability (non-Turing), while no Turing-flip patterns are formed. The results of the current study show that the space–time discrete version of the model of antibiotic resistance in hospitals captures complex and rich nonlinear dynamics and adds to our understanding of the complicated pattern formation of the model.

Invariant curves of quasi-periodic reversible mappings and its application

We consider the existence of invariant curves of real analytic reversible mappings which are quasi-periodic in the angle variables. By the normal form theorem, we prove that under some assumptions, the original mapping is changed into its linear part via an analytic convergent transformation, so that invariants curves are obtained. In the iterative process, by solving the modified homological equations, we ensure that the transformed mapping is still reversible. As an application, we investigate the invariant curves of a class of nonlinear resonant oscillators, with the Birkhoff constants of the corresponding Poincaré mapping all zeros or not.

Dynamics of a cross-superdiffusive SIRS model with delay effects in transmission and treatment.

We investigate the dynamics of a SIRS epidemiological model taking into account cross-superdiffusion and delays in transmission, Beddington-DeAngelis incidence rate and Holling type II treatment. The superdiffusion is induced by inter-country and inter-urban exchange. The linear stability analysis for the steady-state solutions is performed, and the basic reproductive number is calculated. The sensitivity analysis of the basic reproductive number is presented, and we show that some parameters strongly influence the dynamics of the system. A bifurcation analysis to determine the direction and stability of the model is carried out using the normal form and center manifold theorem. The results reveal a proportionality between the transmission delay and the diffusion rate. The numerical results show the formation of patterns in the model, and their epidemiological implications are discussed.

Normal forms for Dirac–Jacobi bundles and splitting theorems for Jacobi structures

The aim of this paper is to prove a normal form Theorem for Dirac–Jacobi bundles using a recent techniques of Bursztyn, Lima and Meinrenken. As the most important consequence, we can prove the splitting theorems of Jacobi pairs which was proposed by Dazord, Lichnerowicz and Marle. As another application we provide an alternative proof of the splitting theorem of homogeneous Poisson structures.

Analytical bifurcation and strong resonances of a discrete Bazykin–Berezovskaya predator–prey model with Allee effect

This paper investigates multiple bifurcations analyses and strong resonances of the Bazykin–Berezovskaya predator–prey model in depth using analytical and numerical bifurcation analysis. The stability conditions of fixed points, codim-1 and codim-2 bifurcations to include multiple and generic bifurcations are studied. This model exhibits transcritical, flip, Neimark–Sacker, and [Formula: see text], [Formula: see text], [Formula: see text] strong resonances. The normal form coefficients and their scenarios for each bifurcation are examined by using the normal form theorem and bifurcation theory. For each bifurcation, various types of critical states are calculated, such as potential transformations between the one-parameter bifurcation point and different bifurcation points obtained from the two-parameter bifurcation point. To validate our analytical findings, the bifurcation curves of fixed points are determined by using MatcontM.

The dynamics of a delayed ecological model with predator refuge and cannibalism

This study has contributed to understanding a delayed prey-predator system involving cannibalism. The system is assumed to use the Holling type II functional response to describe the consuming process and incorporates the predator's refuge against the cannibalism process. The characteristics of the solution are discussed. All potential equilibrium points have been identified. All equilibrium points' local stability analyses for all time delay values are investigated. The system exhibits a Hopf bifurcation at the coexistence equilibrium, which is further demonstrated. The center manifold and normal form theorems for functional differential equations are then used to establish the direction of Hopf bifurcation and the stability of the periodic solution. To demonstrate the key findings, various numerical simulations are then run.

A local normal form for Hamiltonian actions of compact semisimple Poisson–Lie groups

The main contribution of this manuscript is a local normal form for Hamiltonian actions of Poisson-Lie groups $K$ on a symplectic manifold equipped with an $AN$-valued moment map, where $AN$ is the dual Poisson-Lie group of $K$. Our proof uses the delinearization theorem of Alekseev which relates a classical Hamiltonian action of $K$ with $\mathfrak{k}^*$-valued moment map to a Hamiltonian action with an $AN$-valued moment map, via a deformation of symplectic structures. We obtain our main result by proving a ``delinearization commutes with symplectic quotients'' theorem which is also of independent interest, and then putting this together with the local normal form theorem for classical Hamiltonian actions wtih $\mathfrak{k}^*$-valued moment maps. A key ingredient for our main result is the delinearization $\mathcal{D}(\omega_{can})$ of the canonical symplectic structure on $T^*K$, so we additionally take some steps toward explicit computations of $\mathcal{D}(\omega_{can})$. In particular, in the case $K=SU(2)$, we obtain explicit formulas for the matrix coefficients of $\mathcal{D}(\omega_{can})$ with respect to a natural choice of coordinates on $T^*SU(2)$.

Weak dual pairs in Dirac–Jacobi geometry

Adopting the omni-Lie algebroid approach to Dirac–Jacobi structures, we propose and investigate a notion of weak dual pairs in Dirac–Jacobi geometry. Their main motivating examples arise from the theory of multiplicative precontact structures on Lie groupoids. Among other properties of weak dual pairs, we prove two main results. (1) We show that the property of fitting in a weak dual pair defines an equivalence relation for Dirac–Jacobi manifolds. So, in particular, we get the existence of self-dual pairs and this immediately leads to an alternative proof of the normal form theorem around Dirac–Jacobi transversals. (2) We prove the characteristic leaf correspondence theorem for weak dual pairs paralleling and extending analogous results for symplectic and contact dual pairs. Moreover, the same ideas of this proof apply to get a presymplectic leaf correspondence for weak dual pairs in Dirac geometry (not yet present in literature).

A flow box theorem for 2d slow-fast vector fields and diffeomorphisms and the slow log-determinant integral

We prove a flow box theorem for smooth 2-dimensional slow-fast vector fields, providing a simple normal form that is obtained by smooth coordinate changes, without having to change the time. We introduce a notion of 2d slow-fast diffeomorphism, define the log-determinant integral and prove a normal form theorem similar to the flow box theorem for slow-fast vector fields. The normal form is used in proving the relevance of the log-determinant integral.

Bifurcations analysis in implicit maps through the dynamics of cumulated surface errors in milling

In this contribution, we examine the evolution of surface errors during consecutive milling operations. Its description is based on a nonlinear implicit map, which is suitable to investigate the surface quality. It describes the series of consecutive Surface Location Errors (SLE) in roughing operations. As one of the principal results of the paper, bifurcations related to the fixed point of the implicit map are analyzed via normal form theorem. We determined a formula for the criticality of the bifurcation, which allows the approximate computation of the arising period-two solution. The method is demonstrated for the surface error model of milling, and the results are verified by numerical computations. Although the amplitude of the SLE would be negligible, its derivatives has a great influence in the model, which can cause stability problems.

Retraction Note: New applications of Schr\xf6dinger type inequalities to the existence and uniqueness of Schr\xf6dingerean equilibrium

As new applications of Schrodinger type inequalities appearing in Jiang (J. Inequal. Appl. 2016:247, 2016), we first investigate the existence and uniqueness of a Schrodingerean equilibrium. Next we propose a tritrophic Hastings-Powell model with two different Schrodingerean time delays. Finally, the stability and direction of the Schrodingerean Hopf bifurcation are also investigated by using the center manifold theorem and normal form theorem.

Normal form of equivariant maps in infinite dimensions

Local normal form theorems for smooth equivariant maps between infinite-dimensional manifolds are established. These normal form results are new even in finite dimensions. The proof is inspired by the Lyapunov–Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces. It uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient by a compact group of the zero set of a smooth map. The general results are applied to the moduli space of anti-self-dual instantons, the Seiberg–Witten moduli space and the moduli space of pseudoholomorphic curves.