Perturbations Of Vector Fields Research Articles

Black hole perturbations in Maxwell-Horndeski theories

We study the linear stability of black holes in Maxwell-Horndeski theories where a $U(1)$ gauge-invariant vector field is coupled to a scalar field with the Lagrangian of full Horndeski theories. The perturbations on a static and spherically symmetric background can be decomposed into odd- and even-parity modes under the expansion of spherical harmonics with multipoles $l$. For $l\ensuremath{\ge}2$, the odd-parity sector contains two propagating degrees of freedom associated with the gravitational and vector field perturbations. In the even-parity sector, there are three dynamical perturbations arising from the scalar field besides the gravitational and vector field perturbations. For these five propagating degrees of freedom, we derive conditions for the absence of ghost/Laplacian stabilities along the radial and angular directions. We also discuss the stability of black holes for $l=0$ and $l=1$, in which case no additional conditions are imposed to those obtained for $l\ensuremath{\ge}2$. We apply our general results to Einstein-Maxwell-dilaton-Gauss-Bonnet theory and Einstein-Born-Infeld-dilaton gravity and show that hairy black hole solutions present in these theories can be consistent with all the linear stability conditions. In regularized four-dimensional Einstein-Gauss-Bonnet gravity with a Maxwell field, however, exact charged black hole solutions known in the literature are prone to instabilities of even-parity perturbations besides a strong coupling problem with a vanishing kinetic term of the radion mode.

Rigorous justification for the space–split sensitivity algorithm to compute linear response in Anosov systems

Ruelle (1997 Commun. Math. Phys. 187 227–41; 2003 Commun. Math. Phys. 234 185–90) (see also Jiang 2012 Ergod. Theor. Dynam. Syst. 32 1350–69) gave a formula for linear response of transitive Anosov diffeomorphisms. Recently, practically computable realizations of Ruelle’s formula have emerged that potentially enable sensitivity analysis of certain high-dimensional chaotic numerical simulations encountered in the applied sciences. In this paper, we provide full mathematical justification for the convergence of one such efficient computation, the space–split sensitivity, or S3, algorithm (Chandramoorthy and Wang 2022 SIAM J. Appl. Dyn. Syst. 21 735–81). In S3, Ruelle’s formula is computed as a sum of two terms obtained by decomposing the perturbation vector field into a coboundary and a remainder that is parallel to the unstable direction. Such a decomposition results in a splitting of Ruelle’s formula that is amenable to efficient computation. We prove the existence of the S3 decomposition and the convergence of the computations of both resulting components of Ruelle’s formula.

Schwarzschild quasi-normal modes of non-minimally coupled vector fields

We study perturbations of massive and massless vector fields on a Schwarzschild black-hole background, including a non-minimal coupling between the vector field and the curvature. The coupling is given by the Horndeski vector-tensor operator, which we show to be unique, also when the field is massive, provided that the vector has a vanishing background value.We determine the quasi-normal mode spectrum of the vector field, focusing on the fundamental mode of monopolar and dipolar perturbations of both even and odd parity, as a function of the mass of the field and the coupling constant controlling the non-minimal interaction. In the massless case, we also provide results for the first two overtones, showing in particular that the isospectrality between even and odd modes is broken by the non-minimal gravitational coupling.We also consider solutions to the mode equations corresponding to quasi-bound states and static configurations. Our results for quasi-bound states provide strong evidence for the stability of the spectrum, indicating the impossibility of a vectorization mechanism within our set-up. For static solutions, we analytically and numerically derive results for the electromagnetic susceptibilities (the spin-1 analogs of the tidal Love numbers), which we show to be non-zero in the presence of the non-minimal coupling.

Intensity---A Metric Approach to Quantifying Attractor Robustness in ODEs

Although mathematical models do not fully match reality, robustness of dynamical objects to perturbation helps bridge from theoretical to real-world dynamical systems. Classical theories of structural stability and isolated invariant sets treat robustness of qualitative dynamics to sufficiently small errors. But they do not indicate just how large a perturbation can become before the qualitative behavior of our system changes fundamentally. Here we introduce a quantity, intensity of attraction, that measures the robustness of attractors in metric terms. Working in the setting of ordinary differential equations on $\mathbb{R}^n$, we consider robustness to vector field perturbations that are time dependent or independent. We define intensity in a control-theoretic framework, based on the magnitude of control needed to steer trajectories out of a domain of attraction. Our main result is that intensity also quantifies the robustness of an attractor to time-independent vector field perturbations; we prove this by connecting the reachable sets of control theory to isolating blocks of Conley theory. In addition to treating classical questions of robustness in a new metric framework, intensity of attraction offers a novel tool for resilience quantification in ecological applications. Unlike many measurements of resilience, intensity detects the strength of transient dynamics in a domain of attraction.

A mathematical model for treatment using chemo-immunotherapy

In this study, we investigated a mathematical model for chemoimmunotherapy (a combination of chemotherapy and immunotherapy) for brain cancer. In most cases, the standard protocol for cancer treatment is fixed in terms of treatment time intervals and dosages. We offer a wide range of non-fixed protocols, which essentially vary in terms of time intervals and dosages (i.e., personalised medicine). The functions that describe this treatment are explicit and analytical. Hence, the parameters of the function can be easily changed and a new protocol can be obtained. We compared different protocols and obtained an optimal solution. In addition, we applied the singular perturbed vector field (SPVF) method to determine the hierarchy of the system of equations, which enabled us to identify the equilibrium points of the mathematical model and investigate their stability.

Applying Laplace Adomian decomposition method (LADM) for solving a model of Covid-19

In this study, we apply the Laplace Adomian decomposition method (LADM) for the mathematical model of Covid-19. The mathematical model includes a system of nonlinear ordinary differential equations. Therefore, the model cannot be solved analytically but only by approximation. The application of LADM approximates the solution profiles of the dynamical variables of the Covid-19 model by an analytical power series. The conventional way to calculate the expressions of the approximation solutions is complicated both in terms of mathematical calculations and in terms of computer run time. In this paper, we propose a new algorithm for implementing the LADM method combined with the singularly perturbed vector field (SPVF) method. The new algorithm we offer is significantly reducing the running time of both the computer and the mathematical calculations. We compared the results obtained from the LADM to the numerical simulations. Some plots are presented to show the reliability and simplicity of the new algorithm.

Normal Form Near Orbit Segments of Convex Hamiltonian Systems

Abstract In the study of Hamiltonian systems on cotangent bundles, it is natural to perturb Hamiltonians by adding potentials (functions depending only on the base point). This led to the definition of Mañé genericity [8]: a property is generic if, given a Hamiltonian $H$, the set of potentials $g$ such that $H+g$ satisfies the property is generic. This notion is mostly used in the context of Hamiltonians that are convex in $p$, in the sense that $\partial ^2_{pp} H$ is positive definite at each point. We will also restrict our study to this situation. There is a close relation between perturbations of Hamiltonians by a small additive potential and perturbations by a positive factor close to one. Indeed, the Hamiltonians $H+g$ and $H/(1-g)$ have the same level one energy surface, hence their dynamics on this energy surface are reparametrisation of each other, this is the Maupertuis principle. This remark is particularly relevant when $H$ is homogeneous in the fibers (which corresponds to Finsler metrics) or even fiberwise quadratic (which corresponds to Riemannian metrics). In these cases, perturbations by potentials of the Hamiltonian correspond, up to parametrisation, to conformal perturbations of the metric. One of the widely studied aspects is to understand to what extent the return map associated to a periodic orbit can be modified by a small perturbation. This kind of question depends strongly on the context in which they are posed. Some of the most studied contexts are, in increasing order of difficulty, perturbations of general vector fields, perturbations of Hamiltonian systems inside the class of Hamiltonian systems, perturbations of Riemannian metrics inside the class of Riemannian metrics, and Mañé perturbations of convex Hamiltonians. It is for example well known that each vector field can be perturbed to a vector field with only hyperbolic periodic orbits, this is part of the Kupka–Smale Theorem, see [5, 13] (the other part of the Kupka–Smale Theorem states that the stable and unstable manifolds intersect transversally; it has also been studied in the various settings mentioned above but will not be discussed here). In the context of Hamiltonian vector fields, the statement has to be weakened, but it remains true that each Hamiltonian can be perturbed to a Hamiltonian with only non-degenerate periodic orbits (including the iterated ones), see [11, 12]. The same result is true in the context of Riemannian metrics: every Riemannian metric can be perturbed to a Riemannian metric with only non-degenerate closed geodesics, this is the bumpy metric theorem, see [1, 2, 4]. The question was investigated only much more recently in the context of Mañé perturbations of convex Hamiltonians, see [9, 10]. It is proved in [10] that the same result holds: if $H$ is a convex Hamiltonian and $a$ is a regular value of $H$, then there exist arbitrarily small potentials $g$ such that all periodic orbits (including iterated ones) of $H+g$ at energy $a$ are non-degenerate. The proof given in [10] is actually rather similar to the ones given in papers on the perturbations of Riemannian metrics. In all these proofs, it is very useful to work in appropriate coordinates around an orbit segment. In the Riemannian case, one can use the so-called Fermi coordinates. In the Hamiltonian case, appropriate coordinates are considered in [10,Lemma 3.1] itself taken from [3, Lemma C.1]. However, as we shall detail below, the proof of this Lemma in [3], Appendix C, is incomplete, and the statement itself is actually wrong. Our goal in the present paper is to state and prove a corrected version of this normal form Lemma. Our proof is different from the one outlined in [3], Appendix C. In particular, it is purely Hamiltonian and does not rest on the results of [7] on Finsler metrics, as [3] did. Although our normal form is weaker than the one claimed in [10], it is actually sufficient to prove the main results of [6, 10], as we shall explain after the statement of Theorem 1, and probably also of the other works using [3, Lemma C.1].

Second order directional shape derivatives of integrals on submanifolds

<p style='text-indent:20px;'>We compute first and second order shape sensitivities of integrals on smooth submanifolds using a variant of shape differentiation. The result is a quadratic form in terms of one perturbation vector field that yields a second order quadratic model of the perturbed functional. We discuss the structure of this derivative, derive domain expressions and Hadamard forms in a general geometric framework, and give a detailed geometric interpretation of the arising terms.</p>

High-Order Approximation of Heteroclinic Bifurcations in Truncated 2D-Normal Forms for the Generic Cases of Hopf-Zero and Nonresonant Double Hopf Singularities

Based on the nonlinear time transformation method, in this paper we propose a recursive algorithm for arbitrary order approximation of heteroclinic orbits. This approach works fine for a wide class of systems that are perturbations of non-Hamiltonian integrable planar vector fields. Specifically, our method can provide an approximation up to any desired order, both for the locus where the heteroclinic bifurcation occurs in the parameter space and for the orbit in the phase space. We also give proofs that guarantee the existence and uniqueness of the solution. As illustrations, we apply it to the generic Hopf-zero and nonresonant double Hopf singularities which give rise to an intricate bifurcation scenario in the cases where a heteroclinic cycle appears in the corresponding unfolding of normal forms. The obtained high-order approximations are expressed in terms of symbolic nonzero normal form coefficients which improve the existing first-order approximations in the literature. They are also in excellent agreement with numerical continuations. Note that the derived formulas can be practically important in many control engineering applications in which certain constants of the problem may not be known in advance.

Generalized loxodromes with application to time-optimal navigation in arbitrary wind

This article aims at generalizing loxodromic (rhumb line) navigation to conformally flat Riemannian manifolds. We admit space-time dependence of both perturbing vector field and ship’s self-speed. Thereafter, the findings are applied to time-efficient navigation by a variational approach, referring to the local solutions of Zermelo’s navigation problem under arbitrary wind. This yields the corresponding conditions for loxodromic time-minimal and time-maximal navigation in relation to the navigation data. Our research is also illustrated by a two-dimensional example (an oblate ellipsoid), which distinguishes perturbations of different force: weak, critical and strong. It includes numerical simulations and discussion, emphasizing and comparing loxodromic solutions among the minimizing, maximizing and anomalous time extremals.

Inflation with multiple vector fields and non-Gaussianities

We consider a model of inflation consisting a triplet of U(1) vector fields with the parity violating interaction which is non-minimally coupled to inflaton. The vector field sector enjoys global O(3) symmetry which admits isotropic configuration and provides not only vector modes but also scalar and tensor modes. We decompose the scalar perturbations into the adiabatic, entropy and isocurvature perturbations and compute all power spectra and cross correlations of the scalar and the tensor sectors. The tensor modes associated with the vector fields contribute to the power spectrum of gravitational waves while the parity violating term generates chirality in gravitational power spectra and bispectra. We study nonlinear scalar and tensor perturbations and compute all bispectra and three-point cross-correlations. In particular, it is shown that the non-Gaussianity of curvature perturbations and gravitational waves are enhanced by the vector field perturbations. We show that non-Gaussianities put strong constraints on the model parameters such as the parity violating coupling and the fractional energy of the vector fields.

Θ-SEIHRD mathematical model of Covid19-stability analysis using fast-slow decomposition.

In general, a mathematical model that contains many linear/nonlinear differential equations, describing a phenomenon, does not have an explicit hierarchy of system variables. That is, the identification of the fast variables and the slow variables of the system is not explicitly clear. The decomposition of a system into fast and slow subsystems is usually based on intuitive ideas and knowledge of the mathematical model being investigated. In this study, we apply the singular perturbed vector field (SPVF) method to the COVID-19 mathematical model of to expose the hierarchy of the model. This decomposition enables us to rewrite the model in new coordinates in the form of fast and slow subsystems and, hence, to investigate only the fast subsystem with different asymptotic methods. In addition, this decomposition enables us to investigate the stability analysis of the model, which is important in case of COVID-19. We found the stable equilibrium points of the mathematical model and compared the results of the model with those reported by the Chinese authorities and found a fit of approximately 96 percent.

Limit cycle bifurcations by perturbing a class of planar quintic vector fields

This paper first investigates all possible phase portraits of a class of planar quintic vector field given byx˙=(1+2bx2+2dx4)y,y˙=−2(2ax2+by2+3cx4+2dx2y2)x,a,b,c,d∈R. Then, we study the bifurcation problem of its small perturbed vector field with polynomial perturbations of arbitrary degree n,n∈N by the corresponding Abelian integral, and prove that the lower bound for the maximal number of limit cycles bifurcating from the periodic orbits is 3[n−12]−1,n≥5.

Analysis of a breast cancer mathematical model by a new method to find an optimal protocol for [formula omitted]-positive cancer

Treatment of breast cancer (positive for HER2, i.e., ERBB2) is described by a mathematical model involving non-linear ordinary differential equations with a hidden hierarchy. To reveal the hierarchy of dynamical variables of the system being considered, we applied the singular perturbed vector field (SPVF) method, where a system of equations can be decomposed to fast and slow sub-systems with explicit small parameters. This new form of the model, which is called a singular perturbed system, enables us to apply a semi-analytical method called the method of directly defining inverse mapping (MDDiM), which is based on the homotopy analysis asymptotic method. We introduced the treatment protocol in explicit form, through an analytical function that describes the exact dose and intervals between treatments in a cyclical manner. In addition, a new algorithm for the optimal dosage that causes tumour shrinkage is presented in this study. Furthermore, we took the concept of protocol optimisation a step further and derived a differential equation that represents vaccination depending on tumour size and yields an optimal protocol of different doses at every time point. We introduced the treatment protocol in explicit form, through an analytical function that describes the exact dose and intervals between treatments in a cyclical manner. In addition, a new algorithm for finding the optimal dosage that causes tumour shrinkage is presented in this study. Additionally, we took the concept of protocol optimisation a step further and derived a differential equation that represents vaccination depending on tumour size and yields an optimal protocol of different doses at every time point.

On Generalised Single-Heading Navigation

Introducing the notion of a pseudoloxodrome, we generalise a single-heading navigation to conformally flat Riemannian manifolds, under the action of a perturbing vector field (wind, current) of arbitrary force. The findings are applied to time-optimal navigation with the use of the Euler–Lagrange equations. We refer to the Zermelo navigation problem admitting space and time dependence of both a perturbation and a ship's speed. The necessary conditions for single-heading time-optimal navigation are obtained and the pseudoloxodromes of minimum and maximum time are discussed. Furthermore, we describe winds which yield the pseudoloxodromic and loxodromic time extremals. Our research is also illustrated with the examples in dimension two emphasising the single-heading solutions among the time-optimal trajectories in the presence of some space-dependent winds.

Spectral stability and semiclassical measures for renormalized KAM systems**The author has been supported by La Caixa, Severo Ochoa ICMAT, International PhD Programme, 2014, and MTM2017-85934-C3-3-P (MINECO, Spain).

An exact semiclassical version of the classical KAM theorem about small perturbations of vector fields on the torus is given. Moreover, a renormalization theorem based on counterterms for some semiclassical systems that are close to being completely integrable is obtained. We apply these results to characterize the sets of semiclassical measures and quantum limits for sequences of L2-eigenfunctions of these systems.

Fast–slow vector fields of reaction–diffusion systems

AbstractA geometrically invariant concept of fast–slow vector fields perturbed by transport terms describing molecular diffusion is proposed in this paper. It is an extension of our concept of singularly perturbed vector fields for ODEs to reaction–diffusion systems with chemical reactions having wide range of characteristic time scales, while transport processes remain comparatively slow. Under this assumption we developed a decomposition into a fast and slow subsystems. It is assumed that the transport terms for the fast subsystem can be neglected to the leading order. For the slow subsystem we modify a concept of singularly perturbed profiles proposed in our previous works. The results are used to justify and to modify an algorithm of reaction–diffusion manifolds (REDIMs). The modified REDIM method is applied to the Michaelis–Menten model to illustrate the suggested approach.

Convergence to Normal Forms of Integrable PDEs

In an infinite dimensional Hilbert space we consider a family of commuting analytic vector fields vanishing at the origin and which are nonlinear perturbations of some fundamental linear vector fields. We prove that one can construct by the method of Poincar\'e normal form a local analytic coordinate transformation near the origin transforming the family into a normal form. The result applies to the KdV and NLS equations and to the Toda lattice with periodic boundary conditions. One gets existence of Birkhoff coordinates in a neighbourhood of the origin. The proof is obtained by directly estimating, in an iterative way, the terms of the Poincar\'e normal form and of the transformation to it, through a rapid convergence algorithm.

Chaos in periodically forced reversible vector fields

We discuss the appearance of chaos in time-periodic perturbations of reversible vector fields in the plane. We use the normal forms of codimension~$1$ reversible vector fields and discuss the ways a time-dependent periodic forcing term of pulse form may be added to them to yield topological chaotic behaviour. Chaos here means that the resulting dynamics is semi-conjugate to a shift in a finite alphabet. The results rely on the classification of reversible vector fields and on the theory of topological horseshoes. This work is part of a project of studying periodic forcing of symmetric vector fields.

On the linearization of vector fields on a torus with prescribed frequency

In this paper we are mainly concerned with the linearization of the flow with prescribed frequency for analytic perturbation of constant vector fields on a torus under weaker non-degeneracy condition and non-resonant condition. As is well known the perturbation of constant vector fields may induce a shift of frequency, when Kolmogorov's non-degeneracy condition is violated. By introducing external parameters and using the polynomial structure to truncate, we prove that if the frequency mapping has the nonzero Brouwer's topological degree at some non-resonant frequency, then the conjugated vector fields will have a linear flow with this frequency.