Planar Vector Fields with a Given Set of Orbits

We determine all the \({\mathcal{C}^1}\) planar vector fields with a given set of orbits of the form y ? y(x) = 0 satisfying convenient assumptions. The case when these orbits are branches of an algebraic curve is also study. We show that if a quadratic vector field admits a unique irreducible invariant algebraic curve \({g(x, y) = \sum_{j=0}^S a_j(x) y^{S-j}= 0}\) with S branches with respect to the variable y, then the degree of the polynomial g is at most 4S.     

Decay of Almost Periodic Solutions¶of Conservation Laws

We consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables, which are almost periodic in the space variables in a generalized sense introduced by Stepanoff and Wiener, which extends the original one of H. Bohr. We prove that if u(x,t) is such a solution whose inclusion intervals at time t, with respect to ?>0, satisfy l _epsiv;(t)/t→0 as t→∞, and such that the scaling sequence u ^T (x,t)=u(T x,T t) is pre-compact as t→∞ in L _loc ¹(? ^{d +1} ₊, then u(x,t) decays to its mean value \(\), which is independent of t, as t→∞. The decay considered here is in L ¹ _loc of the variable ξ≡x/t, which implies, as we show, that \(\) as t→∞, where M _x denotes taking the mean value with respect to x. In many cases we show that, if the initial data are almost periodic in the generalized sense, then so also are the solutions. We also show, in these cases, how to reduce the condition on the growth of the inclusion intervals l _?(t) with t, as t→∞, for fixed ? > 0, to a condition on the growth of l _?(0) with ?, as ?→ 0, which amounts to imposing restrictions only on the initial data. We show with a simple example the existence of almost periodic (non-periodic) functions whose inclusion intervals satisfy any prescribed growth condition as ?→ 0. The applications given here include inviscid and viscous scalar conservation laws in several space variables, some inviscid systems in chromatography and isentropic gas dynamics, as well as many viscous 2 × 2 systems such as those of nonlinear elasticity and Eulerian isentropic gas dynamics, with artificial viscosity, among others. In the case of the inviscid scalar equations and chromatography systems, the class of initial data for which decay results are proved includes, in particular, the L ^∞ generalized limit periodic functions. Our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws.     

Limit Cycles of Piecewise Smooth Differential Equations on Two Dimensional Torus

In this paper we study the limit cycles of some classes of piecewise smooth vector fields defined in the two dimensional torus. The piecewise smooth vector fields that we consider are composed by linear, Ricatti with constant coefficients and perturbations of these one, which are given in (3). Considering these piecewise smooth vector fields we characterize the global dynamics, studying the upper bound of number of limit cycles, the existence of non-trivial recurrence and a continuum of periodic orbits. We also present a family of piecewise smooth vector fields that posses a finite number of fold points and, for this family we prove that for any 2k number of limit cycles there exists a piecewise smooth vector fields in this family that presents k number of limit cycles and prove that some classes of piecewise smooth vector fields presents a non-trivial recurrence or a continuum of periodic orbits.     

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 Poincaré 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.     

Stability of Periodic Solutions¶of Conservation Laws with Viscosity:¶Analysis of the Evans Function

Nonclassical conservation laws with viscosity arising in multiphase fluid and solid mechanics exhibit a rich variety of traveling-wave phenomena, including homoclinic (pulse-type) and periodic solutions along with the standard heteroclinic (shock, or front-type) solutions. Here, we investigate stability of periodic traveling waves within the abstract Evans-function framework established by R. A. Gardner. Our main result is to derive a useful stability index analogous to that developed by Gardner and Zumbrun in the traveling-front or -pulse context, giving necessary conditions for stability with respect to initial perturbations that are periodic on the same period T as the traveling wave; moreover, we show that the periodic-stability index has an interpretation analogous to that of the traveling-front or -pulse index in terms of well-posedness of an associated Riemann problem for an inviscid medium, now to be interpreted as allowing a wider class of measure-valued solutionsor, alternatively, in terms of existence and nonsingularity of a local “mass map” from perturbation mass to potential time-asymptotic T-periodic states. A closely related calculation yields also a complementary long-wave stability criterion necessary for stability with respect to periodic perturbations of arbitrarily large period NT, N → ∞. We augment these analytical results with numerical investigations analogous to those carried out by Brin in the traveling-front or -pulse case, approximating the spectrum of the linearized operator about the wave.The stability index and long-wave stability criterion are explicitly evaluable in the same planar, Hamiltonian cases as is the index of Gardner and Zumbrun, and together yield rigorous results of instability similar to those obtained previously for pulse-type solutions; this is established through a novel dichotomy asserting that the two criteria are in certain cases logically exclusive. In particular, we obtain results bearing on the nature and mechanism for formation of highly oscillatory Turing-like patterns observed numerically by Frid and Liu and ?ani? and Peters in models of multiphase flow. Specifically, for the van der Waals model considered by Frid and Liu, we show instability of all periodic waves such that the period increases with amplitude in the one-parameter family of nearby periodic orbits, and in particular of large- and small-amplitude waves; for the standard, double-well potential, this yields instability of all periodic waves.Likewise, for a quadratic-flux model like that considered by ?ani? and Peters, we show instability of large-amplitude waves of the type lying near observed patterns, and of all small-amplitude waves; our numerical results give evidence that intermediate-amplitude waves are unstable as well. These results give support for an alternative mechanism for pattern formation conjectured by Azevedo, Marchesin, Plohr, and Zumbrun, not involving periodic waves.     

Hydrodynamic properties of squirmer swimming in power-law fluid near a wall

Hydrodynamic properties of squirmer swimming in power-law fluid near a wall considering the interaction between squirmer and wall are numerically studied with an immersed boundary-lattice Boltzmann method. The power-law index, Reynolds number, initial orientation angle of squirmer, and initial distance of squirmer from the wall are all taken into account to investigate the swimming characteristics for pusher (β?<?0), neutral squirmer (β?=?0), and puller (β?>?0) (three kinds of swimmer types) near the no-slip boundary. Four new kinds of swimming modes are found. Results show that, for the pushers and pullers, the wall displays an increasing attraction with increasing power-law index n, which differs from the neutral squirmer who always departs from the wall after the first collision with the wall. Both the initial orientation angle and initial distance from the wall only affect the moving situations rather than the moving modes of the squirmers. However, the squirmers depart from the wall as the Reynolds number increases and chaotic orbits appear for some squirmers at Re?=?5. Several typical flow fields are analyzed and the power consumption and torque for different kinds of flows are also studied. It is found that, as the absolute value of β increases, the power consumption generally increases in shear-thinning (n?=?0.4), Newtonian (n?=?1), and shear-thickening (n?=?1.6) fluids. Moreover, the pushers (β?<?0) and the pullers (β?>?0) expend almost the same power if the absolute value of β remains the same. In addition, the power consumption of the squirmers is highly dependent on the power-law index n.     

Travelling Breathers in Klein-Gordon Lattices as Homoclinic Orbits to <Emphasis Type="BoldItalic">p</Emphasis>-Tori

Klein-Gordon chains are one-dimensional lattices of nonlinear oscillators in an anharmonic on-site potential, linearly coupled with their first neighbors. In this paper, we study the existence in such networks of spatially localized solutions, which appear time periodic in a referential in translation at constant velocity. These solutions are called travelling breathers. In the case of travelling wave solutions, the existence of exact solutions has been obtained by Iooss and Kirchgässner. Formal multiscale expansions have been used by Remoissenet to derive approximate solutions of travelling breathers in the form of modulated plane waves. James and Sire have studied the existence of specific travelling breather solutions, consisting in pulsating travelling waves which are exactly translated of 2 lattice sites after a fixed propagation time T. In this paper, we generalize this approach to pulsating travelling waves which are exactly translated of p≥ 3 sites after a given time T p being arbitrary. By formulating the problem as a dynamical system, one is able to reduce the system locally to a finite dimensional set of ordinary differential equations (ODE), whose dimension depends on the parameter values of the problem. We prove that the principal part of this system of ODE admits homoclinic connections to p-tori under general conditions on the potential. One can obtain leading order approximations of these homoclinic connections and these orbits should correspond, for the oscillator chain, to small amplitude travelling breather solutions superposed on an exponentially small quasi-periodic tail.     

From linear viscoelasticity to the architecture of highly branched polyethylene

In this work, the linear viscoelastic behavior of some low-density polyethylene in the melt is used to obtain their architecture. In this way, the number of branches per molecule and long chain branching (LCB) content is determined. For this purpose, a method based on the molecular dynamics of simple star-shaped molecules is presented. It allows one to infer the topology of an average molecule through a set of 2N _c parameters {C ⁿ _i , the number concentration of a level i} and {M _bi , the mass of a segment of level i} representing an irregular Cayley tree with N _c levels. The inverse problem uses the complex shear modulus as a function of the frequency data along with a minimization algorithm. Results from the present method are compared with NMR and SEC measurements of the level of branching. It appears that SEC and rheology leads to similar results on the determination of LCB while NMR overestimate the number of branch points per molecule. Moreover, rheology allows one to go further than the basic evaluation of LCB content and shows a picture of the structure of the molecules that is in agreement with the kinetics of free radical polymerization of polyethylene.     

Beltrami Fields with a Nonconstant Proportionality Factor are Rare

We consider the existence of Beltrami fields with a nonconstant proportionality factor f in an open subset U of \({\mathbb{R}^3}\). By reformulating this problem as a constrained evolution equation on a surface, we find an explicit differential equation that f must satisfy whenever there is a nontrivial Beltrami field with this factor. This ensures that there are no nontrivial regular solutions for an open and dense set of factors f in the C^k topology, \({k\geqq 7}\). In particular, there are no nontrivial Beltrami fields whenever f has a regular level set diffeomorphic to the sphere. This provides an explanation of the helical flow paradox of Morgulis et al. (Commun Pure Appl Math 48:571–582, 1995).     

Determination of boundary layers in the plane problem for three-layer strips. Part 2

In the first part of this paper, we considered the exact statement of the plane elasticity problem in displacements for strips made of various materials (problem A, an isotropic material; problem B, an orthotropic material with 2G ₁₂ < √E ₁ E ₂; problem C, an orthotropic material with 2G ₁₂ > √E ₁ E ₂). Further, we stated and solved the boundary layer problem (the problem on a solution decaying away from the boundary) for a sandwich strip of regular structure consisting of isotropic layers (problem AA). In the present paper, we use the solution of the plane problem to consider the problem for sandwich strips of regular structure with isotropic face layers and orthotropic filler (problem AB).     

The stress state of a plate subjected to a piecewise homogeneous load

We consider the stress-strain state of a plate having a doubly connected domain S bounded from the outside by a circle of radius R and from the inside by an ellipse with two rectilinear cuts. The cuts lie symmetrically on the x-axis. The plate is subjected to various forces: the hole contour (the ellipse) is under the action of uniformly distributed forces of intensity q, and the cut shores are free of loads; at the points ±ib of the imaginary axis, the plate is under the action of a lumped force P.The solution of the problem is reduced to determining two analytic functions φ(z) and ψ(z) satisfying certain boundary conditions (depending on the type of the acting loads).We use the Kolosov-Muskhelishvili method to reduce the problem to a system of linear algebraic equations for the coefficients in the expansions of the functions φ(z) and ψ(z). The solution thus obtained is illustrated by numerical examples.     

On the Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes Equation on the Whole Space

The existence of a time periodic solution of the compressible Navier–Stokes equation on the whole space is proved for a sufficiently small time periodic external force when the space dimension is greater than or equal to 3. The proof is based on the spectral properties of the time-T-map associated with the linearized problem around the motionless state with constant density in some weighted L ^∞ and Sobolev spaces. The time periodic solution is shown to be asymptotically stable under sufficiently small initial perturbations and the L ^∞ norm of the perturbation decays as time goes to infinity.     

Brinkman Flow of a Viscous Fluid Through a Spherical Porous Medium Embedded in Another Porous Medium

An analytical investigation for a two-dimensional steady, viscous, and incompressible flow past a permeable sphere embedded in another porous medium is presented using the Brinkman model, assuming a uniform shear flow far away from the sphere. Semi-analytical solutions of the problem are derived and relevant quantities such as velocities and shearing stresses on the surface of the sphere are obtained. The streamlines inside and outside the sphere and the radial velocity are shown in several graphs for different values of the porous parameters \({\sigma _1 =(\mu /\tilde {\mu }) (a/\sqrt{K_1 })}\) and \({\sigma _2 =(\mu /\tilde {\mu }) (a/\sqrt{K_2 })}\) , where a is the radius of the sphere, μ is the dynamic viscosity of the fluid, \({\tilde {\mu }}\) is an effective or Brinkman viscosity, while K ₁ and K ₂ are the permeabilities of the two porous media. It is shown that the dimensionless shearing stress on the sphere is periodic in nature and its absolute value increases with an increase of both porous parameters σ ₁ and σ ₂.     

Pre-Chamber Ignition Mechanism: Simulations of Transient Autoignition in a Mixing Layer Between Reactants and Partially-Burnt Products

The structure of autoignition in a mixing layer between fully-burnt or partially-burnt combustion products from a methane-air flame at ? = 0.85 and a methane-air mixture of a leaner equivalence ratio has been studied with transient diffusion flamelet calculations. This configuration is relevant to scavenged pre-chamber natural-gas engines, where the turbulent jet ejected from the pre-chamber may be quenched or may be composed of fully-burnt products. The degree of reaction in the jet fluid is described by a progress variable c (c = taking values 0.5, 0.8, and 1.0) and the mixing by a mixture fraction ξ (ξ = 1 in the jet fluid and 0 in the CH₄-air mixture to be ignited). At high scalar dissipation rates, N₀, ignition does not occur and a chemically-frozen steady-state condition emerges at long times. At scalar dissipation rates below a critical value, ignition occurs at a time that increases with N₀. The flame reaches the ξ = 0 boundary at a finite time that decreases with N₀. The results help identify overall timescales of the jet-ignition problem and suggest a methodology by which estimates of ignition times in real engines may be made.     

Effects of Lewis Number on Head on Quenching of Turbulent Premixed Flames: A Direct Numerical Simulation Analysis

The head on quenching of statistically planar turbulent premixed flames by an isothermal inert wall has been analysed using three-dimensional Direct Numerical Simulation (DNS) data for different values of global Lewis number Le(0.8, 1.0 and 1.2) and turbulent Reynolds number Re_t. The statistics of head on quenching have been analysed in terms of the wall Peclet number Pe (i.e. distance of the flame from the wall normalised by the Zel’dovich flame thickness) and the normalised wall heat flux Φ. It has been found that the maximum (minimum) value of Φ(Pe) for the turbulent Le=0.8 cases are greater (smaller) than the corresponding laminar value, whereas both Pe and Φ in turbulent cases remain comparable to the corresponding laminar values for Le=1.0 and 1.2. Detailed physical explanations are provided for the observed Le dependences of Pe and Φ. The existing closure of mean reaction rate \(\overline {\dot {\omega }}\) using the scalar dissipation rate (SDR) in the near wall region has been assessed based on a-priori analysis of DNS data and modifications to the existing closures of mean reaction rate and SDR have been suggested to account for the wall effects in such a manner that the modified closures perform well both near to and away from the wall.     

The Viscosity Method for the Homogenization of Soft Inclusions

In this paper, we consider periodic soft inclusions T _ε with periodicity ε, where the solution, u _ε , satisfies semi-linear elliptic equations of non-divergence in \({\Omega_{\epsilon}=\Omega\setminus \overline{T}_\epsilon}\) with Neumann data on \({\partial T^{\mathfrak a}}\). The difficulty lies in the non-divergence structure of the operator where the standard energy method, which is based on the divergence theorem, cannot be applied. The main object is to develop a viscosity method to find the homogenized equation satisfied by the limit of u _ε , referred to as u, as ε approaches to zero. We introduce the concept of a compatibility condition between the equation and the Neumann condition on the boundary for the existence of uniformly bounded periodic first correctors. The concept of a second corrector is then developed to show that the limit, u, is the viscosity solution of a homogenized equation.     

Miscible Thermo-Viscous Fingering Instability in Porous Media. Part 1: Linear Stability Analysis

The development of the thermo-viscous fingering instability of miscible displacements in homogeneous porous media is examined. In this first part of the study dealing with stability analysis, the basic equations and the parameters governing the problem in a rectilinear geometry are developed. An exponential dependence of viscosity on temperature and concentration is represented by two parameters, thermal mobility ratio β _T and a solutal mobility ratio β _C , respectively. Other parameters involved are the Lewis number Le and a thermal-lag coefficient λ. The governing equations are linearized and solved to obtain instability characteristics using either a quasi-steady-state approximation (QSSA) or initial value calculations (IVC). Exact analytical solutions are also obtained for very weakly diffusing systems. Using the QSSA approach, it was found that an increase in thermal mobility ratio β _T is seen to enhance the instability for fixed β _C , Le and λ. For fixed β _C and β _T , a decrease in the thermal-lag coefficient and/or an increase in the Lewis number always decrease the instability. Moreover, strong thermal diffusion at large Le as well as enhanced redistribution of heat between the solid and fluid phases at small λ is seen to alleviate the destabilizing effects of positive β _T . Consequently, the instability gets strictly dominated by the solutal front. The linear stability analysis using IVC approach leads to conclusions similar to the QSSA approach except for the case of large Le and unity λ flow where the instability is seen to get even less pronounced than in the case of a reference isothermal flow of the same β _C , but β _T  = 0. At practically, small value of λ, however, the instability ultimately approaches that due to β _C only.     

A simple criterion for finite time stability with application to impacted buckling of elastic columns

The existing theories of finite-time stability depend on a prescribed bound on initial disturbances and a prescribed threshold for allowable responses. It remains a challenge to identify the critical value of loading parameter for finite time instability observed in experiments without the need of specifying any prescribed threshold for allowable responses. Based on an energy balance analysis of a simple dynamic system, this paper proposes a general criterion for finite time stability which indicates that finite time stability of a linear dynamic system with constant coefficients during a given time interval [0, t _f ] is guaranteed provided the product of its maximum growth rate (determined by the maximum eigen-root p₁ >0) and the duration t _f does not exceed 2, i.e., p₁t _f <2. The proposed criterion (p₁t _f =2) is applied to several problems of impacted buckling of elastic columns: (i) an elastic column impacted by a striking mass, (ii) longitudinal impact of an elastic column on a rigid wall, and (iii) an elastic column compressed at a constant speed (“Hoff problem”), in which the time-varying axial force is replaced approximately by its average value over the time duration. Comparison of critical parameters predicted by the proposed criterion with available experimental and simulation data shows that the proposed criterion is in robust reasonable agreement with the known data, which suggests that the proposed simple criterion (p₁t _f =2) can be used to estimate critical parameters for finite time stability of dynamic systems governed by linear equations with constant coefficients.     

Intermittent observer for nonlinear systems with state delays

Kolmogorov–Sinai entropy-based irregularity measures such as approximate entropy (ApEn), sample entropy and fuzzy entropy are widely used for short-term heart rate variability analysis. These entropy statistics are estimated for a specific value of the tolerance parameter (r) that is mostly chosen from a common recommended range. Entropy measurement on short-term signals is highly sensitive to the choice of r. An incorrect selection of r results in an inaccurate entropy value, thereby leading to unreliable information retrieval. By addressing this inaccuracy due to r selection, the quality and reliability of information retrieval can be improved. Thus, we hypothesize that generating a complete entropy profile using all potential r values will give a more complete and useful information about signal irregularity in contrast to the case of finding entropy at a single selected value of r. In order to do so, one must be able to accurately select all potential r candidates. In this paper, we use a data-driven algorithm based on cumulative histograms to automatically select potential r values for an individual signal based on its dynamics. An appropriate set of r values is designated by the algorithm for generating a series of ApEn values (ApEn profile) instead of a single value of ApEn. ApEn profile- based secondary measures such as TotalApEn and SDApEn have been used as features to classify sets of synthetic and physiologic data. Our study proves that secondary measures obtained from an ApEn profile are more efficient in indicating irregularity levels in comparison with the traditional measure of ApEn evaluated at a single r value, specially in the case of short length data.