Solution Branches Research Articles

A local ROM for Rayleigh–Bénard bifurcation problems

This work presents a local reduced-order method for computing bifurcation diagrams in 2D Rayleigh–Bénard convection problems. The proposed method is based on Proper Orthogonal Decomposition, and employs a reduced-order study of the regularity of solutions to detect new solution branches within the bifurcation diagram. The locality of the method is achieved through k-means clustering, and the selection of local problems in the online stage is done in the solution space. All hyperparameters of the reduced method, such as the number of clusters or the number of POD modes, are estimated by deterministic criteria. Furthermore, the offline–online splitting strategy for online calculations is explicitly outlined. The method is applied to a single-parameter problem and a two-parameter problem, showing its ability to rapidly compute bifurcation diagrams with small errors. Compared to a standard approach that samples each branch separately, the local approach produces more accurate results in less computational time.

Stability and bifurcation diagram for a shadow Gierer–Meinhardt system in one spatial dimension

We are concerned with a Neumann problem of a shadow system of the Gierer–Meinhardt model in an interval I=(0,1) . A stationary problem is studied, and we consider the diffusion coefficient ɛ > 0 as a bifurcation parameter. Then a complete bifurcation diagram of the stationary solutions is obtained, and a stability of every stationary solution is determined. In particular, for each n⩾1 , two branches of n-mode solutions emanate from a trivial branch. All 1-mode solutions are stable for small τ > 0, and all n-mode solutions, n⩾2 , are unstable for all τ > 0, where τ > 0 is a time constant. The system is known for having stationary spiky patterns with large amplitude for small ɛ > 0. Then, asymptotic expansions of maximum and minimum values of a stationary solution as ɛ → 0 are also obtained.

Identification of Real and Complex Solution Varieties and Their Singularities Defined by Loop Constraints of Linkages Using the Kinematic Tangent Cone

Abstract The configuration space (c-space) of a mechanism is the real-solution variety of a set of loop closure constraints, which is therefore the chief object in kinematic analysis. Singularities of this variety (referred to as c-space singularities) are singular configurations of the mechanism. In addition, a mechanism may exhibit other kinematic singularities that are not visible from the differential geometry of the c-space (referred to as hidden singularities). Such situations were analyzed by investigating the local geometry of the c-space and its corank stratification. It has been shown recently that hidden singularities and shakiness can be attributed to the fact that complex solution branches intersect with the c-space, i.e., with real-solution branches. This paper employs the kinematic tangent cone to identify local solution branches. While the kinematic tangent cone is an established generally applicable concept, which gives rise to a computational (numeric and symbolic) algorithm, it has yet only been applied to analyzing the real-solution set. Application of the method is shown for several examples. Further, the algebraic aspects are briefly elaborated.

The non-spherical ground state of Proca stars

Spherical Proca Stars (PSs) are regarded as the ground state amongst the family of PSs. In accordance, spherical PSs are thought to have a fundamental branch of stable solutions. In this Letter, we provide energetic, morphological and dynamical evidence that spherical PSs are actually excited states. The ground state is shown to be a family of static, non-spherical, in fact prolate, PSs. The spherical stars in the fundamental branch, albeit stable against spherical perturbations, turn out to succumb to non-spherical dynamics, undergoing an isometry breaking into prolate PSs. We also provide evidence for the dynamical formation of prolate PSs, starting from spherical dilute initial data, via gravitational cooling. Consequently, PSs provide a remarkable example of (possibly compact) relativistic stars, in General Relativity minimally coupled to a simple, physical, field theory model, where staticity plus stability implies non-sphericity.

On elementary four-wave interactions in dispersive media

The cubic interactions in a discrete system of four weakly nonlinear waves propagating in a conservative dispersive medium are studied. By reducing the problem to a single ordinary differential equation governing the motion of a classical particle in a quartic potential, the complete explicit branches of solutions are presented, either steady, periodic, breather or pump, thus recovering or generalizing some already published results in hydrodynamics, nonlinear optics and plasma physics, and presenting some new ones. Various stability criteria are also formulated for steady equilibria. Theory is applied to deep-water gravity waves for which models of isolated quartets are described, including bidirectional standing waves and quadri-directional travelling waves, steady or not, resonant or not.

Lie group and stability analysis of tangent hyperbolic fluid flow driven by a shrinking sheet

This study elaborates on the mathematical modeling of tangent hyperbolic fluid flow driven by a shrinking sheet in a porous medium. The roles of Soret-Dufour and thermal radiation are considered to model the energy and concentration equations. The primary objective is to find an appropriate similarity transformation through Lie group analysis, which helps to produce the self-similar equations from the governing nonlinear partial systems. The similarity solutions are calculated numerically using the “bvp4c” solver in Matlab. Additionally, adequate suction is required to obtain the similarity solutions. As expected, dual solutions are found within a certain range of the suction parameter. The skin friction coefficient, heat, and mass transfer rates at the sheet are observed for pertinent flow constraints of physical interest. It reveals that the dual solutions bifurcate from a critical point, and beyond this point, no solution exists. Furthermore, the branches of dual solutions on the nondimensional velocity, temperature, and concentration profiles depict different characteristics. Thus, a linear temporal stability analysis has been employed to test the flow stability, revealing that the upper branch is physically reliable. The heat transfer rate increases by 7.69% and 9.55% in the upper and lower branches, respectively, for Soret parameter changes from 0.6 to 0.9. Meanwhile, the mass transfer rate decreases by 85.30% and 324.60% in the upper and lower branches, respectively. The consequences of Weissenberg and suction parameters show an opposite character on the velocity, temperature, and concentration profiles in the upper branch.

Thermal characteristics of MHD [formula omitted] water nanofluids flow past a stretching/shrinking wedge in the view of Cattaneo-Christov heat flux

Wedge flow has a variety of potential uses, including thermal systems, geothermal power plants, aerodynamics and cooling of nuclear power plants. In this framework, the modified Cattaneo-Christov heat flux is implemented to examine the thermal characteristics in the dissipative flow of a viscous nanoliquid over a shrinking/stretching wedge. The laminar and two-dimensional flow of CoFe2O4/ water over a wedge domain is scrutinized. The self-similar solutions using bvp4c together with the stability analysis are estimated to demonstrate the physical existence of first and second solutions under relevant parameters used in the study. The findings reveals that the reverse flow phenomena, the novel form of decreasing flow exhibits physical existence which is significantly different from those in the stretching flow scenario. Thermal relaxation has a substantial impact on energy flux, which results in oscillatory heat conduction with a narrower temperature. When the volumetric friction of nanoparticles grows from 0.0 to 0.02, the local skin friction coefficients rises. The upper branch elucidation is stable and physically realistic, while the second branch solution is imaginary and having no physical existence. The rise and fall in temperature and velocity is perceived with suction parameter. However, the dropping of velocity is seen with the nanoparticle volume percentage.

Minimal Branches of Solutions of Nonlinear Operator Equations in Banach Spaces

Minimal Branches of Solutions of Nonlinear Operator Equations in Banach Spaces

Influence of magnetic field on dynamic behavior of double-diffusive convection

Double-diffusive convection under an external magnetic field has several industrial applications, and how the magnetic field affects flow and heat and mass transfer is a research focus. This paper studies the dynamical behavior of two-dimensional double-diffusive convection in a uniform magnetic field is investigated using a high-accuracy numerical method. The effects of the magnetic field strength on the flow and on the heat and mass transfer are analyzed, and the process of flow transition in the presence of a magnetic field is studied. The results show that, as the strength of the magnetic field increases, the flow is gradually inhibited, and most of the heat transfer occurs through conduction. For a fixed magnetic field strength, the temperature-dominated flow changes from periodic to chaotic, and finally back to periodic, as the orientation of the magnetic field increases. Two types of periodic flow occur in the transition route, namely, a simple periodic flow with an integer multiple of the fundamental frequency and a complex periodic flow with multiple lower-frequency subharmonic waves. In the presence of a strong magnetic field, the effect of the magnetic field orientation on the flow is significant. At high Rayleigh numbers and a fixed magnetic field strength, the heat transfer efficiency of a vertical magnetic field surpasses that of a horizontal magnetic field by approximately 50% when temperature is dominant, achieving a mass transfer efficiency of about 100%. In cases when concentration is dominant, the heat transfer efficiency of a vertical magnetic field is 20% higher than that of a horizontal magnetic field, and the mass transfer efficiency is approximately 60% higher for the vertical magnetic field compared to the horizontal magnetic field. The flow transitions from chaotic or periodic to steady as the Prandtl number increases. The solution branch bifurcates at certain Prandtl numbers, and hysteresis appears when Prandtl number is equal to 0.023. For low Rayleigh numbers, the heat and mass transfer efficiencies increase monotonically with increasing Prandtl number under a weak magnetic field but remain constant for various Prandtl numbers under a strong magnetic field.

Dual solutions of Darcy-Forchheimer radiative-convective magneto-nanofluid flow over an exponentially stretching/shrinking surface: Stability and entropy generation analysis

A numerical investigation was conducted to examine the existence of dual solutions over exponentially stretching or shrinking permeable surfaces. The study aimed to analyze the impact of thermal and second-order velocity slip on magnetohydrodynamic (MHD) mixed convective radiative nanofluid flow within a Darcy-Forchheimer porous medium while considering heat generation. Two distinct types of nanofluids, namely Cu-water and TiO 2/water, were employed in the investigation. To address the nonlinear partial differential equations (PDEs), similarity transformations were utilized to transform them into a system of ordinary differential equations (ODEs). The resulting coupled higher-order nonlinear ODEs were then solved using a MATLAB-based boundary value algorithm known as bvp4c. Two distinct solutions are observed within a restricted range of the stretching/shrinking parameter ξ ( ξ > ξ c ) and suction parameter S ( S > S c ), terminating respectively at ξ = ξ c or S = S c . The critical values are identified as ξ c = − 1.1952 ( Cu − water ) , − 0.9304 ( Ti O 2 − water ) and S c = 1.6437 ( Cu − water ) , 1.8900 ( Ti O 2 − water ) , maintaining the fixed values of the remaining physical parameters. Notably, the critical value for Cu-water is lower than that of TiO 2-water. Moreover, in both solution branches, the range of skin-friction coefficient and local Nusselt number for Cu-water surpasses that of TiO 2-water. An evaluation of the temporal stability of these solutions reveals that only the first solution remains stable. This is supported by the positivity of the smallest eigenvalues ( γ 1 > 0 ), indicating stability. Conversely, for the second solution, the smallest eigenvalues ( γ 1 < 0 ) are negative, signifying its instability. Also, Cu-water nanofluid exhibits higher fluid velocity and lower fluid temperature than TiO 2-water nanofluid for stable solutions and the opposite behavior for unstable solution branches. Significantly, within the initial solution branch, there is an escalation in velocity profiles attributed to the magnetic parameter, Grashof number, porous parameter, and first- and second-order velocity slip parameter. Conversely, a decrease is observed for the Forchheimer number, while the opposite holds true for the second solution branch. For both solution branches, the temperature profiles experience a reduction due to thermal slip and stretching/shrinking parameters, whereas enhancements are noted for thermal radiation and heat generation parameters. Furthermore, an increase in the suction parameter leads to elevated skin friction coefficient and local Nusselt number in the first stable solution, thereby promptly improving heat transfer efficiency. Moreover, improvements in entropy generation arise from factors such as thermal radiation, magnetic field, and nanoparticle solid volume fraction, presenting opportunities to optimize heat transmission in cooling and heating systems.

Theoretical and experimental study on the secondary heat recovery cycle of the mixed working fluid in ocean thermal energy conversion

The small ocean temperature difference causes the power generation efficiency to not be high while the power generation cost is high; this method has not been commercially applied. Therefore, it is particularly important to find a new thermal cycle to improve the cycle efficiency. In this paper, a new cycle of the ocean temperature difference using a non-azeotropic mixed as working fluid is proposed, and a secondary heat recovery device is added to the solution branch separated by the separator. Theoretical analysis and numerical calculation were used to analyze the relationships between cycle efficiency and working fluid concentration, turbine inlet pressure and cold seawater temperature were obtained. The results show that the maximum cycle efficiency can reach 5.25% by adding a secondary heat recovery device. A 10 kW OTEC experimental system was constructed and analyzed; the system performance was obtained. The efficiency of the experimental system was found to be as high as 3.8%, while the experimental value of the test system was lower than the theoretical calculation value. Finally, the heat transfer coefficients of an evaporator and condenser under these conditions were obtained, which provides basic data and technical support for the engineering application of an OTEC plant.

A new paradigm for multi-fidelity continuation using parallel model refinement

Numerical path continuation is commonly applied to determine how limit states of dynamical systems evolve with a free parameter. For high-fidelity models of complicated nonlinear systems, the sequential nature inherent to continuation can become an unsurmountable obstacle. In the present work, we propose a new paradigm for obtaining the targeted solution curves in a parallelized way. The point of departure is a low-fidelity model, which is of lower order and/or neglects certain nonlinear/coupling terms. The simplifications must be as drastic as necessary to obtain an approximate solution curve with reasonable effort. A subset of relevant points along this approximate solution curve is then selected, and, departing there, points on the targeted solution branch of the high-fidelity model are then iteratively computed. It is important to note that the iterative correction towards the high-fidelity solution branch can be applied independently for each point, and is thus predestined for parallel computation. The proposed generic concept is exemplified for a selection of nonlinear vibration problems. Different types of system models, nonlinearities and analyses are considered, and the Harmonic Balance method is used in all cases to compute periodic limit states. In particular, it is shown that the proposed concept is applicable to modal and harmonic order refinement. Finally, it is shown that the concept is also interesting when instead of a model refinement a second system parameter (besides the continuation parameter) is varied, and it permits to robustly reach parameter ranges that are extremely difficult to obtain with conventional path continuation. The method is available, along with the presented numerical examples, as a branch PEACE (Parallelized Re-analysis Of Solution Curves) of the open source tool NLvib.

Fast–slow dynamics related to sharp transition behaviors in the Rayleigh oscillator with two slow square wave excitations

This paper aims to report the fast–slow dynamics in the Rayleigh oscillator with two slow square wave excitations. Typically, a square wave excitation is a periodic excitation that has instantaneous alternation between maximum and minimum amplitude levels, which may trigger a transient response in the system. We show that, by introducing two square wave excitations with multiple frequencies, fast–slow oscillations with catastrophic jumps can be observed in the Rayleigh oscillator. Then, we explain the generation mechanisms of these fast–slow oscillations based on the fast–slow analysis, in which the two slow square wave excitations with different frequencies are transformed based on their frequency ratio, and subsequently treated as a single slow variable. Our study shows that square wave excitations play a crucial role in determining the fast–slow dynamics of the system. Under the control of square wave excitations, the equilibrium can exhibit rapid jumps in different levels, and the variation of amplitude of the limit cycle is not gradual, characterized by instantaneous increase and decrease. Such phenomenon can be referred to as the step-shaped sharp transition of attractor, which accounts for the generation of fast–slow oscillations. Furthermore, with the increase of the frequency ratios, the rectangular-pulse-shaped explosion can be created in the solutions branches of the fast subsystem, which thus leads to fast–slow oscillations with multiple catastrophic jumps. Based on this, the fast–slow oscillations of point–point type and fast–slow oscillations of cycle–cycle type are revealed.

Continuous and discontinuous compressible flows in a converging–diverging channel solved by physics-informed neural networks without exogenous data

Physics-informed neural networks (PINNs) are employed to solve the classical compressible flow problem in a converging–diverging nozzle. This problem represents a typical example described by the Euler equations, a thorough understanding of which serves as a guide for solving more general compressible flows. Given a geometry of the channel, analytical solutions for the steady states do indeed exist, and they depend on the ratio between the back pressure of the outlet and the stagnation pressure of the inlet. Moreover, in the diverging region, the solution may branch into subsonic flow, supersonic flow, or a mixture of both with a discontinuous transition where a normal shock occurs. Classical numerical schemes with shock fitting and capturing methods have been developed to solve this type of problem effectively, whereas the original PINNs are unable to predict the flows correctly. We make a first attempt to exploit the power of PINNs to solve this problem directly by adjusting the weights of different components of the loss function to acquire physical solutions and in the meantime, avoid trivial solutions. With a universal setting yet no exogenous data, we are able to solve this problem accurately; that is, for different given pressure ratios, PINNs provide different branches of solutions at both steady and unsteady states, some of which are discontinuous in nature. For an inverse problem such as unknown specific-heat ratio, it works effectively as well.

Steady-state bifurcations and patterns formation in a diffusive toxic-phytoplankton–zooplankton model

In this paper, we study a diffusive toxic-phytoplankton–zooplankton model with prey-taxis under Neumann boundary condition. By analyzing the characteristic equation, we discuss the local stability of the positive constant solutions and show the repulsive prey-taxis is the key factor that destabilizes the solutions. By means of the abstract bifurcation theorem, we investigate the existence of non-constant positive steady-state solutions bifurcating from the constant coexistence equilibrium. Furthermore, we obtain the criterion for the stability of the branching solutions near the bifurcation point. Numerical simulations support our theoretical results, together with some interesting phenomena, stable heterogeneous periodic solutions emerge when prey-tactic sensitivity coefficient is well below the critical value, and zooplankton populations present extinction and continued transitions as habitat size increases.

Landau singularities of the 7-point ziggurat. Part II

We solve the Landau equations to find the singularities of nine three-loop 7-point graphs that arise as relaxations of the graph studied in [22]. Along the way we establish that Y − ∆ equivalence fails for certain branches of solutions to the Landau equations. We find two graphs with singularities outside the heptagon symbol alphabet; in particular they are not cluster variables of Gr(4, 7). We compare maximal residues of scalar graphs exhibiting these singularities to those in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super-Yang-Mills theory in order to probe their cancellation from its amplitudes.

Entropy optimization and stability analysis in flow of quadratic radiative Ag−MgO/water hybrid micropolar nanofluid over an exponentially contracting permeable Riga surface

ABSTRACT Driven by the significant thermal relevance of low-conductivity electrically active fluids and their impact on controlling flows across multiple industrial and engineering domains, this study aims to thoroughly investigate the primary behavior of electro-magneto-non-Newtonian micropolar A g − M g O / H 2 O hybridized nanofluid, specifically focusing on its flow dynamics and heat transfer characteristics over an exponentially contracting porous Riga surface. The Riga surface, an exceptional electromagnetic sensor surface, holds potential applications in micro-coolers, submarines, and nuclear reactor heat management systems. These findings hold significant relevance due to the prevalent use of micropolar hybridized nanofluids in cooling minute electronic components such as microchips and associated equipment. Our innovative objective is to emphasize the influence of an electromagnetic Riga plate in conjunction with micropolar characteristics, quadratic thermal radiant energy, convective conditions, and velocity slip at the boundary on various key parameters. These parameters include the coefficient of shear stress, coefficient of couple-stress, heat transfer factor (Nusselt number), entropy generation, velocity distribution, micro-rotation or angular velocity, and temperature profiles. We successfully tackled the highly intricate coupled partial differential equations by employing numerical techniques within the MATLAB programming platform. Specifically, we utilized a fourth-order method known as bvp4c to address the challenging boundary value problem. The outcomes revealed the existence of dual solutions characterized by a stable upper solution branch and an unstable lower solution branch within the region of the contracting sheet. Notably, this observation occurred under specific conditions related to the mass suction parameter. The identification of critical values in cases with dual solutions prompted a subsequent stability analysis to discern the nature and characteristics of these solutions. Two distinct solutions have been identified for λ S < λ , reaching termination at λ = λ S within the contracting region. Notably, the first solution yields a positive minimum eigenvalue ( β 1 > 0 ), indicating its stability, while the second solution results in a nonpositive eigenvalue ( β 1 < 0 ), implying instability. Graphical representations illustrate a substantial enhancement in nanofluid temperature for both solution branches, notably influenced by the quadratic thermal radiating parameter, Eckert number, and Biot number. Moreover, the magnetic interaction number and velocity slip parameter notably affect nanofluid motion within the first solution branch. Interestingly, it has been observed that the shear stress and couple stress coefficients exhibit an upward trend with increasing suction value. However, the heat transfer factor (Nusselt number) might decrease in the case of a stable solution. This research direction aims to optimize nanofluid compositions, aiming to minimize entropy generation while maximizing desired thermal properties. Such endeavors are crucial steps towards fostering more efficient and sustainable industrial practices, aligning with the overarching objectives of enhancing energy utilization and promoting industrial growth.

Searching nontrivial magnetic equilibria using the deflated Newton method

Nonlinear systems that model physical experiments often have many equilibrium configurations, and the number of these static solutions grows with the number of degrees of freedom and the presence of symmetries. It is impossible to know a priori how many equilibria exist and which ones are stable or relevant, therefore from the modeler’s perspective, an exhaustive search and symmetry classification in the space of solutions are necessary. With this purpose in mind, the method of deflation (introduced by Farrell as a modification of the classic Newton iterative method) offers a systematic way of finding every possible solution of a set of equations. In this contribution we apply deflated Newton and deflated continuation methods to a model of macroscopic magnetic rotors, and find hundreds of new equilibria that can be classified according to their symmetry. We assess the benefits and limitations of the method for finding branches of solutions in the presence of a symmetry group, and explore the high-dimensional basins of attraction of the method in selected 2-dimensional sections, illustrating the effect of deflation on the convergence.

Multi-state Dirac stars

In this paper, we construct the multi-state Dirac stars (MSDSs) consisting of two pairs of Dirac fields. The two pairs of Dirac fields are in the ground state and the first excited state, respectively, with opposite spins to ensure that the system possesses spherical symmetry. We discuss the solutions of the MSDSs under synchronized and nonsynchronized frequencies. By varying the ratio of masses between the two sets of Dirac fields, different branches of solutions can be obtained. Furthermore, we analyze the characteristics of the various MSDSs solutions and analyze the relationship between the ADM mass M of the MSDSs and the synchronized and nonsynchronized frequencies. Subsequently, we calculate the binding energy E_B of the MSDSs and discuss the stability of the solutions. Then, we investigated the solutions of the MSDSs under the single particle condition. Finally, we discuss the feasibility of simulating the dark matter halos using MSDSs.

Global bifurcation of positive solutions for a superlinear p -Laplacian system

We are concerned with the principal eigenvalue of (P) { − Δ p u = λ θ 1 φ p ( v ) , x ∈ Ω , − Δ p v = λ θ 2 φ p ( u ) , x ∈ Ω , u = 0 = v , x ∈ ∂ Ω and the global structure of positive solutions for the system (Q) { − Δ p u = λ f ( v ) , x ∈ Ω , − Δ p v = λ g ( u ) , x ∈ Ω , u = 0 = v , x ∈ ∂ Ω , where φ p ( s ) = | s | p − 2 s , Δ p s = div ( | ∇ s | p − 2 ∇ s ) , λ &gt; 0 is a parameter, Ω ⊂ R N , N &gt; 2 , is a bounded domain with smooth boundary ∂ Ω , f , g : R → ( 0 , ∞ ) are continuous functions with p -superlinear growth at infinity. We obtain the principal eigenvalue of ( P ) by using a nonlinear Krein–Rutman theorem and the unbounded branch of positive solutions for ( Q ) via bifurcation technology.