Autonomous Vector Research Articles

Diverse partially nonlocal bright–dark Peregrine “three sisters” excitations in a (3+1)-dimensional vector nonlinear Schrödinger equation

This paper aims to analyze diverse excitations of partially nonlocal bright–dark Peregrine “three sisters” based on a nonautonomous (3+1)-dimensional vector partially nonlocal nonlinear Schrödinger equation with different diffractions in two horizontal directions by simplifying into an autonomous vector nonlinear Schrödinger equation, whose solutions are exploited to produce analytical solutions of the nonautonomous vector equation via the nonrecursive Darboux method. Diverse excitations of partially nonlocal bright–dark Peregrine “three sisters” are discussed in an exponential diffraction system by comparing two values of maximum accumulated time with the excited time for Peregrine “three sisters”. When the value of the x or y-directional diffraction is fixed, with the increase of the magnitude of the initial y or x-directional diffraction, the amplitude of fully or nascently excited bright–dark Peregrine structures increases and their widths both decrease in the exponential system. This investigation gives a deeper knowledge for partially nonlocal solitons in the versatile fields of optical engineering, ocean engineering and other engineering disciplines.

Non-uniqueness of integral curves for autonomous Hamiltonian vector fields

In this work we prove the existence of an autonomous Hamiltonian vector field in $W^{1,r}(\mathbb T^d; \mathbb R^d)$ with $r < d-1$ and $d \geq 4$ for which the associated transport equation has non-unique positive solutions. As a consequence of Ambrosio's superposition principle [2], we show that this vector field has non-unique integral curves with a positive Lebesgue measure set of initial data and moreover, we show that the Hamiltonian is not constant along these integral curves.

On symplectomorphisms and Hamiltonian flows

We propose the construction of a sequence of time one flows of autonomous Hamiltonian vector fields, converging to a fixed near the identity C^1 symplectic diffeomorphism psi . This convergence is proved to be uniformly exponentially fast, in a non analytic symplectic topology framework.

Chaotic switching in driven-dissipative Bose-Hubbard dimers: When a flip bifurcation meets a T-point in $ \mathbb{R}^4 $

&lt;p style='text-indent:20px;'&gt;The Bose-Hubbard dimer model is a celebrated fundamental quantum mechanical model that accounts for the dynamics of bosons at two interacting sites. It has been realized experimentally by two coupled, driven and lossy photonic crystal nanocavities, which are optical devices that operate with only a few hundred photons due to their extremely small size. Our work focuses on characterizing the different dynamics that arise in the semiclassical approximation of such driven-dissipative photonic Bose-Hubbard dimers. Mathematically, this system is a four-dimensional autonomous vector field that describes two specific coupled oscillators, where both the amplitude and the phase are important. We perform a bifurcation analysis of this system to identify regions of different behavior as the pump power &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ f $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and the detuning &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \delta $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; of the driving signal are varied, for the case of fixed positive coupling. The bifurcation diagram in the &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ (f, \delta) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-plane is organized by two points of codimension-two bifurcations——a &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \mathbb{Z}_2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-equivariant homoclinic flip bifurcation and a Bykov T-point——and provides a roadmap for the observable dynamics, including different types of chaotic behavior. To illustrate the overall structure and different accumulation processes of bifurcation curves and associated regions, our bifurcation analysis is complemented by the computation of kneading invariants and of maximum Lyapunov exponents in the &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ (f, \delta) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-plane. The bifurcation diagram displays a menagerie of dynamical behavior and offers insights into the theory of global bifurcations in a four-dimensional phase space, including novel bifurcation phenomena such as degenerate singular heteroclinic cycles.&lt;/p&gt;

Dynamics of an autonomous spacecraft control system at initial transition to a tracking mode

The problems of autonomous digital control of the information satellites and space robots during their initial transition to a tracking mode, namely in the initial orientation modes, are considered. Autonomous angular guidance and modularly limited vector digital control using a vector of the modified Rodrigues parameters are applying to bring the spacecraft’s orientation from completely arbitrary to the required one. The developed methods, algorithms and simulation results for a mini-satellite in a sun-synchronous orbit are presented.

On the accumulation of separatrices by invariant circles

AbstractLet f be a smooth symplectic diffeomorphism of ${\mathbb R}^2$ admitting a (non-split) separatrix associated to a hyperbolic fixed point. We prove that if f is a perturbation of the time-1 map of a symplectic autonomous vector field, this separatrix is accumulated by a positive measure set of invariant circles. However, we provide examples of smooth symplectic diffeomorphisms with a Lyapunov unstable non-split separatrix that are not accumulated by invariant circles.

Differentiability properties of the flow of 2d autonomous vector fields

We investigate under which assumptions the flow associated to autonomous planar vector fields inherits the Sobolev or BV regularity of the vector field. We consider nearly incompressible and divergence-free vector fields, taking advantage in both cases of the underlying Hamiltonian structure. Finally, we provide an example of an autonomous planar Sobolev divergence-free vector field, such that the corresponding regular Lagrangian flow has no bounded variation.

Dissecting a Resonance Wedge on Heteroclinic Bifurcations

This article studies routes to chaos occurring within a resonance wedge for a 3-parametric family of differential equations acting on a 3-sphere. Our starting point is an autonomous vector field whose flow exhibits a weakly attracting heteroclinic network made by two 1-dimensional connections and a 2-dimensional separatrix between two equilibria with different Morse indices. After changing the parameters, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We derive the first return map near the ghost of the attractor and we reduce the analysis of the system to a 2-dimensional map on the cylinder. Complex dynamical features arise from a discrete-time Bogdanov-Takens singularity, which may be seen as the organizing center by which one can obtain infinitely many attracting tori, strange attractors, infinitely many sinks and non-trivial contracting wandering domains. These dynamical phenomena occur within a structure that we call resonance wedge. As an application, we may see the "classical" Arnold tongue as a projection of a resonance wedge. The results are general, extend to other contexts and lead to a fine-tuning of the theory.

Regularity estimates for the flow of BV autonomous divergence-free vector fields in

We consider the regular Lagrangian flow X associated to a bounded divergence-free vector field b with bounded variation. We prove a Lusin-Lipschitz regularity result for X and we show that the Lipschitz constant grows at most linearly in time. As a consequence we deduce that both geometric and analytical mixing have a lower bound of order as

Smoothing does not give a selection principle for transport equations with bounded autonomous fields

We give an example of a bounded divergence free autonomous vector field in \({\mathbb {R}}^3\) (and of a nonautonomous bounded divergence free vector field in \({\mathbb {R}}^2\)) and of a smooth initial data for which the Cauchy problem for the corresponding transport equation has 2 distinct solutions. We then show that both solutions are limits of classical solutions of transport equations for appropriate smoothings of the vector fields and of the initial data.

Complex dynamics of a vibration machine caused by a relay feedback control

In this paper we describe some nonlinear phenomena observed in a vibration machine with an unbalanced rotor driven by a DC motor controlled by a relay system. We demonstrate that this type of control leads to the appearance of several complex kinds of behavior caused by interactions between slow and fast dynamics. Our numerical results are verified through comparison with an experimental vibration system. In order to understand the mechanism leading to these phenomena, we apply a modeling approach based on the reduction of a 5D autonomous non-smooth vector field to a 1D discontinuous map for the phase of the switching events. We demonstrate that there are two different reasons causing this map to be discontinuous, so that depending on parameters it may be either a gap map or a circle diffeomorphism. In both parameter domains the map exhibits a period adding structure. We explain also which features of the behavior of the modeled vibration machine are captured by the 1D map and which are not.

Critical homoclinics in a restricted four-body problem: numerical continuation and center manifold computations

The present work studies the robustness of certain basic homoclinic motions in an equilateral restricted four body problem. The problem can be viewed as a two parameter family of conservative autonomous vector fields. The main tools are numerical continuation techniques for homoclinic and periodic orbits, as well as formal series methods for computing normal forms and center stable/unstable manifold parameterizations. After careful numerical study of a number of special cases we formulate several conjectures about the global bifurcations of the homoclinic families.

Ergodicity effects on transport-diffusion equations with localized damping

The main objective of this paper is to study the time decay of transport-diffusion equation with inhomogeneous localized damping in the multi-dimensional torus. The drift is governed by an autonomous Lipschitz vector field and the diffusion by the standard heat equation with small viscosity parameter $\nu$. In the first part we deal with the inviscid case and show some results on the time decay of the energy using in a crucial way the ergodicity and the unique ergodicity of the flow generated by the drift. In the second part we analyze the same problem with small viscosity and provide quite similar results on the exponential decay uniformly with respect to the viscosity in some logarithmic time scaling of the \mbox{type $t\in [0,C_0\ln(1/\nu)]$}.

Automated two-dimensional localization of underwater acoustic transient impulses using vector sensor image processing

An algorithm is presented for automatically localizing transient impulsive sounds collected on several autonomous underwater vector sensors, spaced 15 to 23 m apart. The procedure, which does not require precise time synchronization, exploits transient signals of interest arriving from different azimuthal directions on each sensor. For each sensor the method first constructs time-frequency representations of both the squared acoustic pressure (spectrogram) and dominant directionality of the active intensity (azigram). Within each azigram sets of time-frequency cells associated with transient energy arriving from a consistent azimuthal sector are identified. Standard image processing techniques then link sets that share similar duration and bandwidth between different sensors, after which the algorithm triangulates the source location using the azimuths associated with the detection set. Data collected from shallow coral reef environments demonstrate the algorithm's ability to detect SCUBA bubble plumes and humpback whale song, and reveal consistent spatial distributions of somniferous fish activity. Analytical estimates and direct evaluations both yield false transient localization rates of 3%–6% in the coral reef environment. Many localized pulses have low signal-to-noise ratios, whose distribution has a median of 7.7 dB and an IQR of 7.1 dB. [Work sponsored by DARPA PALS.]

Machine Learning Modeling of Aerobic Biodegradation for Azo Dyes and Hexavalent Chromium

The present study emphasizes the efficacy of a biosurfactant-producing bacterial strain Klebsiella sp. KOD36 in biodegradation of azo dyes and hexavalent chromium individually and in a simultaneous system. The bacterial strain has exhibited a considerable potential for biodegradation of chromium and azo dyes in single and combination systems (maximum 97%, 94% in an individual and combined system, respectively). Simultaneous aerobic biodegradation of azo dyes and hexavalent chromium (SBAHC) was modeled using machine learning programming, which includes gene expression programming, random forest, support vector regression, and support vector regression-fruit fly optimization algorithm. The correlation coefficient includes the dispersion index, and the Willmott agreement index was employed as statistical metrics to assess the performance of each model separately. In addition, the Taylor diagram was used to further investigate the methods used. The findings of the present study were that the support vector regression-fruitfly optimization algorithm (SVR-FOA) with correlation coefficient (CC) of 0.644, (scattered index) SI of 0.374, and (Willmott’s index of agreement) WI of 0.607 performed better than the autonomous support vector regression (SVR), gene expression programming (GEP), and random forest (RF) methods. In addition, the standalone SVR model with CC of 0.146, SI of 0.473, and WI of 0.408 ranked the second best. In summary, the SBAHC can be accurately estimated using the hybrid SVR-FOA method. In other words, FOA has proven to be a powerful optimization algorithm for increasing the accuracy of the SVR method.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

The Least Distance between Extrema and the Minimum Period of Solutions of Autonomous Vector Differential Equations

The Least Distance between Extrema and the Minimum Period of Solutions of Autonomous Vector Differential Equations

Study on the Geometrical Properties and Existence of Orbit Homoclinic to a Saddle Point in n-Dimensional Autonomous Vector Field with Polynomials

According to the theory of stable and unstable manifolds of an equilibrium point, we firstly find out some geometrical properties of orbits on the stable and unstable manifolds of a saddle point under some brief conditions of nonlinear terms composed of polynomials for [Formula: see text]-dimensional time continuous system. These properties show that the orbits on stable and unstable manifolds of the saddle point will stay on the corresponding stable and unstable subspaces in the [Formula: see text]-neighborhood of the saddle point. Furthermore, the necessary conditions of existence for orbit homoclinic to a saddle point are exposed. Some examples including homoclinic bifurcation are given to indicate the application of the results. Finally, the conclusions are presented.

Spectral spread and non-autonomous Hamiltonian diffeomorphisms

For any symplectic manifold $${(M,\omega )}$$ , the set of Hamiltonian diffeomorphisms $${{\text {Ham}}^c(M,\omega )}$$ forms a group and $${{\text {Ham}}^c(M,\omega )}$$ contains an important subset $${{\text {Aut}}(M,\omega )}$$ which consists of time one flows of autonomous(time-independent) Hamiltonian vector fields on M. One might expect that $${{\text {Aut}}(M,\omega )}$$ is a very small subset of $${{\text {Ham}}^c(M,\omega )}$$ . In this paper, we estimate the size of the subset $${{\text {Aut}}(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric which was introduced by Hofer. Polterovich and Shelukhin proved that the complement $${{\text {Ham}}^c\backslash {\text {Aut}}(M,\omega )}$$ is a dense subset of $${{\text {Ham}}^c(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric if $${(M,\omega )}$$ is a closed symplectically aspherical manifold where Conley conjecture is established (Polterovich and Schelukhin in Sel Math 22(1):227–296, 2016). In this paper, we generalize above theorem to general closed symplectic manifolds and general conv! ex symplectic manifolds. So, we prove that the set of all non-autonomous Hamiltonian diffeomorphisms $${{\text {Ham}}^c\backslash {\text {Aut}}(M,\omega )}$$ is a dense subset of $${{\text {Ham}}^c(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric if $${(M,\omega )}$$ is a closed or convex symplectic manifold without relying on the solution of Conley conjecture.

Phase-plane geometries in coupled enzyme assays

The determination of a substrate or enzyme activity by coupling one enzymatic reaction with another easily detectable (indicator) reaction is a common practice in the biochemical sciences. Usually, the kinetics of enzyme reactions is simplified with singular perturbation analysis to derive rate or time course expressions valid under the quasi-steady-state and reactant stationary state assumptions. In this paper, the dynamical behavior of coupled enzyme catalyzed reaction mechanisms is studied by analysis of the phase-plane. We analyze two types of time-dependent slow manifolds – Sisyphus and Laelaps manifolds – that occur in the asymptotically autonomous vector fields that arise from enzyme coupled reactions. Projection onto slow manifolds yields various reduced models, and we present a geometric interpretation of the slow/fast dynamics that occur in the phase–planes of these reactions.