Single Real Parameter Research Articles

Nonautonomous Fold Bifurcations from Spectral Intervals of Higher Strict Multiplicity

Abstract We provide two sufficient criteria for the bifurcation of bounded entire or homoclinic solutions to nonautonomous difference equations depending on a single real parameter. Our analysis is based on a nonhyperbolic solution, whose variational equation possesses exponential dichotomies on semiaxes ensuring that the corresponding critical spectral interval of the dichotomy spectrum has strict multiplicity > 1. This extends earlier results on the fold bifurcation.

Imaging phase plane models

In application-oriented mathematics, particularly in the context of nonlinear system analysis, phase plane analysis through SageMath offers a visual display of the qualitative behaviour of solutions to differential equations without inundating students with cumbersome calculations of the plane-phase. A variety of examples is usually given to illustrate phase-plane behaviour. We approach these problems by considering a problem containing a single real parameter that exemplifies the various situations clearly and simply. We developed two computer programs in SageMath: one program calculates aspects of the Jacobian matrix and displays the phase plane portraits, the other determines the centre manifold. Computations and images generated with computer codes are useful in understanding dynamic models in biology, physics and engineering that involve planar non-linear autonomous differential equations. This paper covers analytical and computational skills that are helpful for students and teachers.

Nonlinear automorphism of the conformal algebra in 2D and continuous sqrt{Toverline{T}} deformations

The conformal algebra in 2D (Diff(S1)⨁Diff(S1)) is shown to be preserved under a nonlinear map that mixes both chiral (holomorphic) generators T and overline{T} . It depends on a single real parameter and it can be regarded as a “nonlinear SO(1, 1) automorphism.” The map preserves the form of the momentum density and naturally induces a flow on the energy density by a marginal sqrt{Toverline{T}} deformation. In turn, the general solution of the corresponding flow equation of the deformed action can be analytically solved in closed form, recovering the nonlinear automorphism. The deformed CFT2 can also be described through the original theory on a field-dependent curved metric whose lapse and shift functions are given by the variation of the deformed Hamiltonian with respect to the energy and momentum densities, respectively. The conformal symmetries of the deformed theories can then also be seen to arise from diffeomorphisms that fulfill suitably deformed conformal Killing equations. Besides, Cardy formula is shown to map to itself under the nonlinear automorphism. As a simple example, the deformation of N free bosons is briefly addressed, making contact with recent related results and the dimensional reduction of the ModMax theory. Furthermore, the nonlinear map between the conformal algebra in 2D and its ultra/non-relativistic versions (BMS3≈CCA2≈GCA2), including the corresponding finite sqrt{Toverline{T}} deformation, are recovered from a limiting case of the nonlinear automorphism. The extension to a three-parameter nonlinear ISO(1, 1) automorphism of the conformal algebra, and a discrete nonlinear automorphism of BMS3 are also briefly discussed.

Compact-like vortices in isotropic curved spacetime

We introduced a generalized Maxwell–Higgs model in a (3+1) isotropic spacetime, and we found their stationary solutions using the BPS approach in curved spacetime. In order to investigate the compact-like vortices, we assume a particular choice for the generalization term. The model is controlled by a potential driven by a single real parameter that can be used to change the vortex solutions profile as they approach their bound values. Resembling some flat spacetime vortex solutions, our model tends to compress the vortices when the parameter l increases. Through numerical analysis, we also show the energy behavior and the magnetic field of the model.

Deformed hopfion-Rañada knots in ModMax electrodynamics

Source-free so-called ModMax theories of nonLinear electrodynamics (NLE) in the four dimensional Minkowski spacetime vacuum are the only possible continuous deformations—and as a function of a single real and positive parameter—of source-free Maxwell linear electrodynamics (MLE) in the same vacuum, which preserve all the same Poincaré and conformal spacetime symmetries as well as the continuous duality invariance of Maxwell’s theory. Null field configurations of the latter however, including null electromagnetic knots, are singular for the Lagrangian formulation of any spacetime Poincaré and conformal invariant theory of NLE. In particular null hopfion-Rañada (HR) knots are a distinguished and fascinating class on their own of topologically nontrivial solutions to Maxwell’s equations. This work addresses the fate of these configurations within ModMax theories. A doubled class of ModMax deformed HR knots is thereby identified, each of which coalescing back in a continuous fashion to the original HR knot when the nonlinear deformation parameter is turned off.

Classical field theory limit of many-body quantum Gibbs states in 2D and 3D

We provide a rigorous derivation of nonlinear Gibbs measures in two and three space dimensions, starting from many-body quantum systems in thermal equilibrium. More precisely, we prove that the grand-canonical Gibbs state of a large bosonic quantum system converges to the Gibbs measure of a nonlinear Schrodinger-type classical field theory, in terms of partition functions and reduced density matrices. The Gibbs measure thus describes the behavior of the infinite Bose gas at criticality, that is, close to the phase transition to a Bose–Einstein condensate. The Gibbs measure is concentrated on singular distributions and has to be appropriately renormalized, while the quantum system is well defined without any renormalization. By tuning a single real parameter (the chemical potential), we obtain a counter-term for the diverging repulsive interactions which provides the desired Wick renormalization of the limit classical theory. The proof relies on a new estimate on the entropy relative to quasi-free states and a novel method to control quantum variances.

New results on asymmetric thick branes

This work deals with the presence and stability of thick brane solutions in the warped five dimensional braneworld scenario with a single extra spatial dimension of infinite extent. We combine two distinct procedures that give rise to new possibilities, allowing that we describe models of asymmetric thick branes, with the asymmetry being controlled by a single real parameter. We illustrate the main results with some distinct models, which show that the method works for both standard and generalized models, and the solutions are gravitationally stable against small perturbations of the metric.

Existence and stability of marginally trapped surfaces in black-hole spacetimes

Marginally outer trapped surfaces (MOTSs, or marginal surfaces in short) are routinely used in numerical simulations of black hole spacetimes. They are an invaluable tool for locating and characterizing black holes quasi-locally in real time while the simulation is ongoing. It is often believed that a MOTS can behave unpredictably under time evolution; an existing MOTS can disappear, and a new one can appear without any apparent reason. In this paper we show that in fact the behavior of a MOTS is perfectly predictable and its behavior is dictated by a single real parameter, the \emph{stability parameter}, which can be monitored during the course of a numerical simulation. We demonstrate the utility of the stability parameter to fully understand the variety of marginal surfaces that can be present in binary black hole initial data. We also develop a new horizon finder capable of locating very highly distorted marginal surfaces and we show that even in these cases, the stability parameter perfectly predicts the existence and stability of marginal surfaces.

On the existence of periodic orbits and KAM tori in the Sprott A system: a special case of the Nosé–Hoover oscillator

We consider the well-known Sprott A system, which is a special case of the widely studied Nosé–Hoover oscillator. The system depends on a single real parameter a, and for suitable choices of the parameter value, it is shown to present chaotic behavior, even in the absence of an equilibrium point. In this paper, we prove that, for $$a\ne 0,$$ the Sprott A system has neither invariant algebraic surfaces nor polynomial first integrals. For $$a>0$$ small, by using the averaging method we prove the existence of a linearly stable periodic orbit, which bifurcates from a non-isolated zero-Hopf equilibrium point located at the origin. Moreover, we show numerically the existence of nested invariant tori surrounding this periodic orbit. Thus, we observe that these dynamical elements and their perturbation play an important role in the occurrence of chaotic behavior in the Sprott A system.

Fractional-order PWC systems without zero Lyapunov exponents

In this paper, it is shown numerically that a class of fractional-order piece-wise continuous systems, which depend on a single real bifurcation parameter, have no zero Lyapunov exponents but can be chaotic or hyperchaotic with hidden attractors. Although not analytically proved, this conjecture is verified on several systems including a fractional-order piece-wise continuous hyperchaotic system, a piece-wise continuous chaotic Chen system, a piece-wise continuous variant of the chaotic Shimizu-Morioka system and a piece-wise continuous chaotic Sprott system. These systems are continuously approximated based on results of differential inclusions and selection theory, and numerically integrated with the Adams-Bashforth-Moulton method for fractional-order differential equations. It is believed that the obtained results are valid for many, if not most, fractional-order PWC systems.

Edge states and spin-valley edge photocurrent in transition metal dichalcogenide monolayers

We develop an analytical theory for edge states in monolayers of transition metal dichalcogenides based on a general boundary condition for a two-band $\mathbf{kp}$ Hamiltonian in the case of uncoupled valleys. Taking into account edge spin-orbit interaction, we reveal that edge states, in general, have linear dispersion that is determined by three real phenomenological parameters in the boundary condition. In the absence of the edge spin-orbit interaction, edge states are described by a single real parameter whose sign determines whether their spectra intersect the bulk gap or not. In the former case we show that illumination by circularly polarized light results in spin- and valley-polarized photocurrent along the edge. Flow direction and spin and valley polarization of the edge photocurrent are determined by the direction of circular polarization of the illuminated light.

Compact Chern–Simons vortices

We introduce and investigate new models of the Chern–Simons type in the three-dimensional spacetime, focusing on the existence of compact vortices. The models are controlled by potentials driven by a single real parameter that can be used to change the profile of the vortex solutions as they approach their boundary values. One of the models unveils an interesting new behavior, the tendency to make the vortex compact, as the parameter increases to larger and larger values. We also investigate the behavior of the energy density and calculate the total energy numerically.

Boundary conditions of Weyl semimetals

We find that generic boundary conditions of Weyl semimetal is dictated by only a single real parameter, in the continuum limit. We determine how the energy dispersions (the Fermi arcs) and the wave functions of edge states depend on this parameter. Lattice models are found to be consistent with our generic observation. Furthermore, the enhanced parameter space of the boundary condition is shown to support a novel topological number.

On the formation of hidden chaotic attractors and nested invariant tori in the Sprott A system

We consider the well-known Sprott A system, which depends on a single real parameter a and, for $$a=1$$ , was shown to present a hidden chaotic attractor. We study the formation of hidden chaotic attractors as well as the formation of nested invariant tori in this system, performing a bifurcation analysis by varying the parameter a. We prove that, for $$a=0$$ , the Sprott A system has a line of equilibria in the z-axis, the phase space is foliated by concentric invariant spheres with two equilibrium points located at the south and north poles, and each one of these spheres is filled by heteroclinic orbits of south pole–north pole type. For $$a\ne 0$$ , the spheres are no longer invariant algebraic surfaces and the heteroclinic orbits are destroyed. We do a detailed numerical study for $$a>0$$ small, showing that small nested invariant tori and a limit set, which encompasses these tori and is the $$\alpha $$ - and $$\omega $$ -limit set of almost all orbits in the phase space, are formed in a neighborhood of the origin. As the parameter a increases, this limit set evolves into a hidden chaotic attractor, which coexists with the nested invariant tori. In particular, we find hidden chaotic attractors for $$a<1$$ . Furthermore, we make a global analysis of Sprott A system, including the dynamics at infinity via the Poincare compactification, showing that for $$a>0$$ , the only orbit which escapes to infinity is the one contained in the z-axis and all other orbits are either homoclinic to a limit set (or to a hidden chaotic attractor, depending on the value of a), or contained on an invariant torus, depending on the initial condition considered.

Three-body scattering problem in the fixed center approximation: The case of attraction

We study the scattering of a light particle on a bound pair of heavy particles (e.g., the deuteron) within the fixed center approximation in the case of light-heavy attraction, solving the integral equation for the three-body Green's function both in the coordinate and in the momentum space. The results for the three-body scattering amplitude appear to be ambiguous -- they depend on a single real parameter. This parameter may be fixed by a three-body input, e.g., the three-body scattering length. We also solve the integral equation for the three-body Green function in the momentum space, introducing a finite cut-off. We show that all three approaches are equivalent. We also discuss how our approach to the problem matches with the introduction of three-body contact interaction as done by other authors.

Energy projection and modified Laughlin states

We develop a method to efficiently calculate trial wave functions for quantum Hall systems which involve projection onto the lowest Landau level. The method essentially replaces the lowest Landau level projection by projection onto the $M$ lowest eigenstates of a suitably chosen Hamiltonian acting within the lowest Landau level. The resulting ``energy projection'' is a controlled approximation to the exact lowest Landau level projection which improves with increasing $M$. It allows us to study the projected trial wave functions for system sizes close to the maximal sizes that can be reached by exact diagonalization and can be straightforwardly applied in any geometry. As a first application and test case, we study a class of trial wave functions first proposed by Girvin and Jach [Girvin and Jach, Phys. Rev. B 29, 5617 (1984)], which are modifications of the Laughlin states involving a single real parameter. While these modified Laughlin states probably represent the same universality class exemplified by the Laughlin wave functions, we show by extensive numerical work for systems on the sphere and torus that they provide a significant improvement of the variational energy, overlap with the exact wave function and properties of the entanglement spectrum.

Hierarchical clustering of time series data with parametric derivative dynamic time warping

Dynamic Time Warping (DTW) is a popular and efficient distance measure used in classification and clustering algorithms applied to time series data. By computing the DTW distance not on raw data but on the time series of the (first, discrete) derivative of the data, we obtain the so-called Derivative Dynamic Time Warping (DDTW) distance measure. DDTW, used alone, is usually inefficient, but there exist datasets on which DDTW gives good results, sometimes much better than DTW. To improve the performance of the two distance measures, we can combine them into a new single (parametric) distance function. The literature contains examples of the combining of DTW and DDTW in algorithms for supervised classification of time series data. In this paper, we demonstrate that combination of DTW and DDTW can also be applied in a method of time series clustering (unsupervised classification). In particular, we focus on a hierarchical clustering (with average linkage) of univariate (one-dimensional) time series data. We construct a new parametric distance function, combining DTW and DDTW, where a single real number parameter controls the contribution of each of the two measures to the total value of the combined distances. The parameter is tuned in the initial phase of the clustering algorithm. Using this technique in clustering methods requires a different approach (to address certain specific problems) than for supervised methods. In the clustering process we use three internal cluster validation measures (measures which do not use labels) and three external cluster validation measures (measures which do use clustering data labels). Internal measures are used to select an optimal value of the parameter of the algorithm, where external measures give information about the overall performance of the new method and enable comparison with other distance functions. Computational experiments are performed on a large real-world data base (UCR Time Series Classification Archive: 84 datasets) from a very broad range of fields, including medicine, finance, multimedia and engineering. The experimental results demonstrate the effectiveness of the proposed approach for hierarchical clustering of time series data. The method with the new parametric distance function outperforms DTW (and DDTW) on the data base used. The results are confirmed by graphical and statistical comparison.

Note on a parameter switching method for nonlinear ODEs

Abstract In this paper we study analytically a parameter switching (PS) algorithm applied to a class of systems of ODE, depending on a single real parameter. The algorithm allows the numerical approximation of any solution of the underlying system by simple periodical switches of the control parameter. Near a general approach of the convergence of the PS algorithm, some dissipative properties are investigated and the dynamical behavior of solutions is investigated with the Lyapunov function method. A numerical example is presented.

Spontaneous generation of singularities in paraxial optical fields.

In nonrelativistic quantum mechanics, the spontaneous generation of singularities in smooth and finite wave functions is a well understood phenomenon also occurring for free particles. We use the familiar analogy between the two-dimensional Schrödinger equation and the optical paraxial wave equation to define a new class of square-integrable paraxial optical fields that develop a spatial singularity in the focal point of a weakly focusing thin lens. These fields are characterized by a single real parameter whose value determines the nature of the singularity. This novel field enhancement mechanism may stimulate fruitful research for diverse technological and scientific applications.

A tournament-based competitive-cooperative multiagent architecture for real parameter optimization

Real parameter optimization is an important task in almost all engineering applications. This paper introduces a novel multiagent architecture and agent interaction mechanism for the solution of single objective type real-parameter optimization problems. The proposed multiagent system includes several metaheuristics as problem-solving agents that act on a common population containing the frontiers of search process and a common archive keeping the promising solutions extracted so far. Each session of the proposed architecture includes two phases: a tournament among all agents to determine the currently best performing agent and a search procedure conducted by the winner. In the tournament phase, each agent performs a fixed number of fitness evaluations over the common population and gets a success score in terms the fitness improvements it achieved by itself. The agent with the best score is the winner of the tournament. Then, the winner agent is allowed to conduct its search algorithm using the common population until its procedure gets stuck at a locally optimal solution or maximum fitness evaluations per session is reached. Afterwards, the procedure restarts with another tournament to determine the next winner. In all phases and iterations of the proposed framework, all agents use the same population and archive in conducting their search procedures. This way, agents compete with each other in terms of their fitness improvements achieved over a fixed number of fitness evaluations in tournaments, and they cooperate by sharing their search experiences through accumulating them in a common population and a common archive. The proposed multiagent system is experimentally evaluated using the well-known CEC2005 benchmark problems set. Analysis of the obtained results exhibited that the proposed framework performs significantly better than its state-of-the-art competitors in almost all problem instances.