Invariant Curves Research Articles

Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game

This paper considers a Bertrand competition between two firms whose decision variables are derived from a quadratic utility function. The first firm produces new products with their own prices while the second firm re-manufactures returned products and sells them in the market at prices that may be less than or equal to the price of the first firm. Dynamically, this competition is constructed on which boundedly rational firms apply a gradient adjustment mechanism to update their prices in each period. According to this mechanism and the nature of the competition, a two-dimensional piecewise smooth discrete dynamic map was constructed in order to study the complex dynamic characteristics of the game. The phase plane of the map was divided into two different regions, separated by border curve. The equilibrium points of the map, in each region on where they are defined, were calculated, and their stability conditions were investigated. Furthermore, we conducted a global analysis to investigate the complex structure of the map, such as closed invariant curves, periodic cycles, and chaotic attractors and their basins, which cause qualitative changes as some parameters are allowed to vary.

Solid-liquid phase equilibria of the quaternary system Na+//Cl−, NO3−, SO42−-H2O at 358.15 K

The stable (solid + liquid) phase equilibrium and the solution density of the quaternary system Na+//Cl−, NO3−, SO42−-H2O and its subsystems at 358.15 K were determined by the isothermal solution saturation method and moist residues. All phase diagrams of ternary systems for Na+//Cl−, SO42−-H2O, Na+//Cl−, NO3−-H2O, Na+// NO3−, SO42−-H2O at 358.15 K contain one invariant point, two invariant curves, and three crystallization regions. Neither double salt nor solid solution is found in the three ternary systems at 358.15 K. When NaCl, NaNO3, and Na2SO4 are analyzed as (7.02, 48.62, 1.58) wt %, respectively, the quaternary system Na+//Cl−, NO3−, SO42−-H2O reach saturation at 358.15 K. The phase diagram of the quaternary system at 358.15 K includes one invariant point, three invariant curves, and three crystallization regions. The three crystallization regions correspond to NaCl, NaNO3, and Na2SO4 and the Na2SO4 crystallization region is much bigger than that of the other two salts. Moreover, using the “translation technology”, the phase diagram of Na+//Cl−, SO42−, NO3−-H2O system can be predicted and constructed. All experimental data, especially the crystalline regions, can provide the fundamental basis for the design and optimization of the crystallization process of NaCl, NaNO3, and Na2SO4 in coal chemical high salinity wastewater.

Rotating spacetimes generalizing Lifshitz black holes

We present a spinning black hole solution in d dimensions with a maximal number of rotation parameters in the context of the Einstein–Maxwell-Dilaton theory. An interesting feature of such a solution is that it accommodates Lifshitz black holes when the rotation parameters are set to zero. We verify the rotating nature of the black hole solution by performing the quasi-local analysis of conserved charges and defining the corresponding angular momenta. In addition, we perform the thermodynamical analysis of the black hole configuration, show that the first law of thermodynamics is completely consistent, and obtain a Smarr-like formula. We further study the thermodynamic stability of the constructed solution from a local viewpoint, by computing the associated specific heats, and from a global perspective, by using the so-called new thermodynamic geometry. We finally make some comments related to a pathology found in the causal structure of the obtained rotating black hole spacetime and compute some of its curvature invariants.

Tubular Surfaces Around a Null Curve and Its Spherical Images

In this study, we define tubular surfaces whose center curves are null curves and their spherical images in Minkowski 3􀀀space. Firstly, we give the interior properties of the surfaces and calculate their invariant curvatures. Then, we obtain some special characterizations for the parameter curves of the surfaces. Finally, we demonstrate the theory via example and give their visualizations with the help of Mathematica.

Analytic classification of germs of parabolic antiholomorphic diffeomorphisms of codimension k

AbstractWe investigate the local dynamics of antiholomorphic diffeomorphisms around a parabolic fixed point. We first give a normal form. Then we give a complete classification including a modulus space for antiholomorphic germs with a parabolic fixed point under analytic conjugacy. We then study some geometric applications: existence of real analytic invariant curves, existence of holomorphic and antiholomorphic roots of holomorphic and antiholomorphic parabolic germs, and commuting holomorphic and antiholomorphic parabolic germs.

Complex dynamical behaviors in a Bertrand game with service factor and differentiated products

In this paper, taking the factor of product service provided by the manufacturers into consideration, a static Bertrand duopoly game with service factor is studied first, in which these two oligarchs produce differentiated products. A dynamic Bertrand duopoly game with bounded rationality is established by using the gradient mechanism. Keeping the adjustment speeds in a relatively small range may help the long-term stable operation of the market. It is found that there is another 2-cycle, different from the one flip bifurcated from the fixed point, which may appear through a saddle-node bifurcation. The unstable set of the saddle cycle, connecting the saddle to the node, gives a closed invariant curve. In addition, the emergence of intermittent chaos implies that the established economic system has the capability of self-regulating, where PM-I intermittency and crisis-induced intermittency have been studied. With the help of the critical curves, the qualitative changes on the basin of attraction are investigated.

Nonlocal deformations of autonomous invariant curves for Liénard equations with quadratic damping

We consider a family of nonlinear oscillators with quadratic damping, that generalizes the Liénard equation. We show that certain nonlocal transformations preserve autonomous invariant curves of equations from this family. Thus, nonlocal transformations can be used for extending known classification of invariant curves to the whole equivalence class of the corresponding equation, which includes non-polynomial equations. Moreover, we demonstrate that an autonomous first integral for one of two non-locally related equations can be constructed in the parametric form from the general solution of the other equation. In order to illustrate our results, we construct two integrable subfamilies of the considered family of equations, that are non-locally equivalent to two equations from the Painlevé–Gambier classification. We also discuss several particular members of these subfamilies, including a traveling wave reduction of a nonlinear diffusion equation, and construct their invariant curves and first integrals.

Bi-slant submanifolds of an S manifold

PurposeBi-slant submanifolds of S-manifolds are introduced, and some examples of these submanifolds are presented.Design/methodology/approachSome properties of Di-geodesic and Di-umbilical bi-slant submanifolds are examined.FindingsThe Riemannian curvature invariants of these submanifolds are computed, and some results are discussed with the help of these invariants.Originality/valueThe topic is original, and the manuscript has not been submitted to any other journal.

On exact analytical solution of Einstein–Maxwell-scalar field equations

The exact analytical solution of Einstein–Maxwell-scalar (EMS) field equations has been explored which covers several well-known solutions such as the Reissner–Nordström, Schwarzschild-MOG (Modified Gravity), Janis–Newman–Winicour, and Schwarzschild solutions. It has been assumed that the interactions between the tensor, vector, and scalar fields are negligible. The newly obtained solution is characterized by three free parameters as the total mass M , electric (magnetic) charge Q e ( Q m ), and scalar charge C (or n parameter) of the gravitational compact object. It is also shown that dual solution for the vector potential A ϕ = Q m cos θ is satisfied by the EMS field equations and the electric charge can be safely replaced by magnetic charge Q m . Finally, we have studied curvature invariants and test particle motion around the gravitational source described the obtained new spacetime metric. We have also provided analysis of degeneracy values of spin parameter of the rotating Kerr black hole and charge parameter of compact object described by the new spacetime metric through comparison of radii of ISCO, photonsphere and energy efficiency. It is shown that new black hole solution in Einstein–Maxwell-scalar field theory can mimic spin parameter of the Kerr black hole up to a ∗ ≲ 0 . 6 while the astrophysical black hole’ observations that it reaches up to a ∗ ≲ 0 . 99 . Consequently, one may conclude that the obtained new black hole solution can be considered as realistic candidate for the astrophysical black holes with a ∗ ≲ 0 . 6 .

Craniofacial growth and SITAR growth curve analysis.

SITAR (SuperImposition by Translation And Rotation) is a shape invariant growth curve model that effectively summarizes somatic growth in puberty. To apply the SITAR model to longitudinal mandibular growth data to clarify its suitability to facial growth analysis. 2D-cephalometric data on two mandibular measurements (AP: articulare-pogonion; CP: condylion-pogonion) were selected from the Denver Growth Study, consisting of longitudinal records (age range: 7.9-19.0 years) of females (sample size N: 21; number of radiographs n: 154) and males (N: 18; n: 137). The SITAR mixed effects model estimated, for each measurement and gender separately, a mean growth curve versus chronological age, along with mean age at peak velocity (APV) and peak velocity (PV), plus subject-specific random effects for PV and mean size. The models were also fitted versus Greulich-Pyle bone age. In males, mean APV occurred at 14.6 years (AP) and 14.4 years (CP), with mean PV 3.1 mm/year (AP) and 3.3 mm/year (CP). In females, APV occurred at 11.6 years (AP and CP), with mean PV 2.3 mm/year (AP) and 2.4 mm/year (CP). The models explained 95-96 per cent of the cross-sectional variance for males and 92-93 per cent for females. The random effects demonstrated standard deviations (SDs) in size of 5.6 mm for males and 3.9 mm for females, and SDs for PV between 0.3 and 0.5 mm/year. The bone age results were similar. The SITAR model is a useful tool to analyse epidemiological craniofacial growth based on cephalometric data and provides an array of information on pubertal mandibular growth and its variance in a concise manner.

On Two-Frequency Oscillations of a DC Electric Drive with Pulse Control

Purpose of research is of the paper is to analyze bifurcations of two-frequency oscillations of a DC electric drive with pulse-width control.Methods. The research is based on the construction of a stroboscopic Poincare map, the calculation of saddle periodic orbits and their stable and unstable invariant manifolds.Results. The study of the mechanisms of the occurrence of two-frequency oscillations from a periodic motion that loses stability in a DC electric drive with pulse-width control was carried out. A non-local saddle-node bifurcation leading to resonance (synchronization) on a torus characterized by a pair of independent frequencies when their ratio becomes a rational number, was studied.Conclusion. A bifurcation analysis of the control system of a DC electric drive, the dynamics of which is described by non-smooth nonautonomous differential equations, was carried out. The research was conducted on an iterable map obtained from the specified vector field in an analytical form. It is shown that the system under consideration demonstrates two-frequency oscillations that occur through the Neimark-Sacker bifurcation. In the phase space of the discrete model, a closed invariant curve corresponds to oscillations with two independent frequencies. It is shown that if these frequencies are correlated multiply, then a resonance occurs when the dynamics becomes periodic. But at the same time, the closed curve remains invariant, and the limit points of the orbit form a pair of periodic cycles – stable and saddle, corresponding to a rational frequency ratio. A closed invariant curve is formed by unstable manifolds of a saddle cycle. If the frequency ratio is irrational, then the dynamics is quasi-periodic. The orbits of such motion fill the closed curve everywhere densely.

Palatini gravity with nonmetricity, torsion, and boundaries in metric and connection variables

We prove the equivalence in the covariant phase space of the metric and connection formulations for Palatini gravity, with nonmetricity and torsion, on a spacetime manifold with boundary. To this end, we will rely on the cohomological approach provided by the relative bicomplex framework. Finally, we discuss some of the physical implications derived from this equivalence in the context of singularity identification through curvature invariants.

Quadratic systems with an invariant algebraic curve of degree 3 and a Darboux invariantq

Let \(QS\) be the class of non-degenerate planar quadratic differential ystems and \(QS_3\) its subclass formed by the systems possessing an invariant cubic \(f(x,y)=0\). In this article, using the action of the group of real affine transformations and time rescaling on \(QS\), we obtain all the possible normal forms for the quadratic systems in \(QS_3\). Working with these normal forms we complete the characterization of the phase portraits in \(QS_3\) having a Darboux invariant of the form \(f(x, y)e^{st}\), with \(s \in \mathbb R\).
 For more information see https://ejde.math.txstate.edu/Volumes/2021/69/abstr.html

Regular Schwarzschild-like spacetime embedded in a five dimensional bulk

We take advantage of the Shiromizu et al. covariant formalism to find out the brane properties originating from the five dimensional bulk spacetime. Making a different choice for the conformal factor e−2b(z) compared to Estrada (0000), we reach a new solution with a lot of interesting properties, where b(z)=ln(1/cosh2μz). The non-local tensor Eab rooted from the 5-dimensional Riemann tensor gives an anisotropic stress energy tensor on the brane with positive energy density and negative radial pressure. The BH on the brane looks like a black string in the 5-dimensional space, with no singularities of the curvature invariants.

A critical assessment of black hole solutions with a linear term in their redshift function

Different theories of gravity can admit the same black hole solution, but the parameters usually have different physical interpretations. In this work we study in depth the linear term beta r in the redshift function of black holes, which arises in conformal gravity, de Rham–Gabadadze–Tolley (dRGT) massive gravity, f(R) gravity (as approximate solution) and general relativity. Geometrically we quantify the parameter beta in terms of the curvature invariants. Astrophysically we found that beta can be expressed in terms of the cosmological constant, the photon orbit radius and the innermost stable circular orbit (ISCO) radius. The metric degeneracy can be broken once black hole thermodynamics is taken into account. Notably, we show that under Hawking evaporation, different physical theories with the same black hole solution (at the level of the metric) can lead to black hole remnants with different values of their physical masses with direct consequences on their viability as dark matter candidates. In particular, the mass of the graviton in massive gravity can be expressed in terms of the cosmological constant and of the formation epoch of the remnant. Furthermore the upper bound of remnant mass can be estimated to be around 0.5 times 10^{27} kg.

Existence of invariant curves with prescribed frequency for degenerate area preserving mappings

Existence of invariant curves with prescribed frequency for degenerate area preserving mappings

Chaotic Motion in the Breathing Circle Billiard

We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves in the phase space. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map has positive topological entropy. The proof relies on variational techniques based on the Aubry–Mather theory.

Positivity of vector bundles and Hodge theory

Differential geometry, especially the use of curvature, plays a central role in modern Hodge theory. The vector bundles that occur in the theory (Hodge bundles) have metrics given by the polarizations of the Hodge structures, and the sign and singularity properties of the resulting curvatures have far reaching implications in the geometry of families of algebraic varieties. A special property of the curvatures is that they are [Formula: see text] order invariants expressed in terms of the norms of algebro-geometric bundle mappings. This partly expository paper will explain some of the positivity and singularity properties of the curvature invariants that arise in the Hodge theory with special emphasis on the norm property.

Regular model of magnetized black hole with rational nonlinear electrodynamics

A modified Hayward metric of magnetically charged black hole space–time based on rational nonlinear electrodynamics with the Lagrangian [Formula: see text] is considered. We introduce the fundamental length, characterizing quantum gravity effects. If the fundamental length vanishes the general relativity coupling to rational nonlinear electrodynamics is recovered. We obtain corrections to the Reissner–Nordström solution as the radius approaches infinity. The metric possesses a de Sitter core without singularities as [Formula: see text]. The Hawking temperature and the heat capacity are calculated. It was shown that phase transitions occur and black holes are thermodynamically stable at some event horizon radii. We demonstrate that curvature invariants are bounded and the limiting curvature conjecture takes place.

Firewall from Effective Field Theory

For an effective field theory in the background of an evaporating black hole with spherical symmetry, we consider non-renormalizable interactions and their relevance to physical effects. The background geometry is determined by the semi-classical Einstein equation for an uneventful horizon where the vacuum energy–momentum tensor is small for freely falling observers. Surprisingly, after Hawking radiation appears, the transition amplitude from the Unruh vacuum to certain multi-particle states grows exponentially with time for a class of higher-derivative operators after the collapsing matter enters the near-horizon region, despite the absence of large curvature invariants. Within the scrambling time, the uneventful horizon transitions towards a firewall, and eventually the effective field theory breaks down.