Boundedness of Fatou Components of Holomorphic Maps

We shall discuss the topological properties of the Fatou sets for holomorphic maps. Let f ₁,f ₂,....,f _N be non-constant holomorphic maps in the plane, each having order less than 1/2. It is shown that if the lower order of f _j , is greater than 0 for some j{1,2,...,N}, then the Fatou set of the map h=f _N f _N-1 ... f ₁ has no unbounded components     

Intermittency as a codimension-three phenomenon

We analyze the interaction of three Hopf modes and show that locally a bifurcation gives rise to intermittency between three periodic solutions. This phenomenon can occur naturally in three-parameter families. Consider a vector fieldf with an equilibrium and suppose that the linearization off about this equilibrium has three rationally independent complex conjugate pairs of eigenvalues on the imaginary axis. As the parameters are varied, generically three branches of periodic solutions bifurcate from the steady-state solution. Using Birkhoff normal form, we can approximatef close to the bifurcation point by a vector field commuting with the symmetry group of the three-torus. The resulting system decouples into phase amplitude equations. The main part of the analysis concentrates on the amplitude equations in R³ that commute with an action ofZ ₂+Z ₂+Z ₂. Under certain conditions, there exists an asymptotically stable heteroclinic cycle. A similar example of such a phenomenon can be found in recent work by Guckenheimer and Holmes. The heteroclinic cycle connects three fixed points in the amplitude equations that correspond to three periodic orbits of the vector field in Birkhoff normal form. We can considerf as being an arbitrarily small perturbation of such a vector field. For this perturbation, the heteroclinic cycle disappears, but an invariant region where it was is still stable. Thus, we show that nearby solutions will still cycle around among the three periodic orbits.     

A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam

This paper delineates a class of time-periodically perturbed evolution equations in a Banach space whose associated Poincaré map contains a Smale horseshoe. This implies that such systems possess periodic orbits with arbitrarily high period. The method uses techniques originally due to Melnikov and applies to systems of the form x=f _o(X)+f ₁(X,t), where f _o(X) is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics: sinusoidally forced vibrations of a buckled beam.     

Dynamics of Relative Phases: Generalised Multibreathers

For small Hamiltonian perturbation of a Hamiltonian systemof arbitrary number of degrees of freedom with anormally non-degenerate submanifold of periodic orbits we construct a nearbysubmanifold and an `effective Hamiltonian' on it such that the differencebetween the two Hamiltonian vector fields is small. The effective Hamiltonianis independent of one coordinate, the `overall phase', and hence thecorresponding action is preserved. Unlike standard averaging approaches,critical points of our effective Hamiltonian subject to given actioncorrespond to exact periodic solutions. We prove there has to be at least acertain number of these critical points given by global topological principles.The linearisation of the effective Hamiltonian about critical points is provedto give the linearised dynamics for the full system to leading order in theperturbation. Hence in the case of distinct eigenvalues which move at non-zerospeed with ,the linear stability type of the periodic orbit can be read offfrom the effective Hamiltonian. Our principal application is to networks ofoscillators or rotors where many such submanifolds of periodic orbits occurat the uncoupled limit – simply excite a number N 2 of the units inrational frequency ratio and put the others on equilibria, subject to anon-resonance condition. The resulting exact periodic solutions for weakcoupling are known as multibreathers. We call the approximate solutions givenby the effective Hamiltonian dynamics, `generalised multibreathers'. Theycorrespond to solutions which look periodic on a short time scale but therelative phases of the excited units may evolve slowly. Extensions aresketched to travelling breathers and energy exchange between degrees offreedom.     

Bifurcation and chaotic behavior of a discrete-time Ricardo–Malthus model

The dynamics of a discrete-time Ricardo–Malthus model obtained by numerical discretization is investigated, where the step size δ is regarded as a bifurcation parameter. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of $R^{2}_{+}$ by using the theory of center manifold and normal form. Numerical simulations are presented not only to illustrate our theoretical results, but also to exhibit the system’s complex dynamical behavior, such as the cascade of period-doubling bifurcation in orbits of period 2, 4, 8 16, period-11, 22, 28 orbits, quasiperiodic orbits, and the chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors.     

Experimental study of the induced flow birefringence of a suspension of rigid particles at the transition from Couette flow to Taylor vortex flow

The flow birefringence induced in solutions of rigid particles is studied experimentally in the region of the axisymmetrical Taylor vortex flow which arises once the velocity gradient G in the annular gap of a conventional Couette cell reaches a critical value G _c .The measurements are performed for several values of G > G _c and for 10 radial observation points in the annular gap. Solutions of two types of rigid particles are investigated: the first is a suspension of flattened clay particles like bentonite, while the second contains rod-like particles of tobacco mosaic virus (TMV). The variations of the birefringence intensity n and of the extinction angle measured in the domain of the axisymmetrical flow show a different behavior according to the shape of the particle in solution. This fact is confirmed theoretically with a good agreement for the measurements performed with solutions of flat particles.     

Computational methods for global analysis of homoclinic and heteroclinic orbits: A case study

In earlier paper we have developed a numerical method for the computation of branches of heteroclinic orbits for a system of autonomous ordinary differential equations in ⁿ . The idea of the method is to reduce a boundary value problem on the real line to a boundary value problem on a finite interval by using linear approximation of the unstable and stable manifolds. In this paper we extend our algorithm to incorporate higher-order approximations of the unstable and stable manifolds. This approximation is especially useful if we want to compute center manifolds accurately. A procedure for switching between the periodic approximation of homoclinic orbits and the higher-order approximation of homoclinic orbits provides additional flexibility to the method. The algorithm is applied to a model problem: the DC Josephson Junction. Computations are done using the software package AUTO.     

Fragile networks and rheology of concentrated suspensions

Suspensions consisting of particles of colloidal dimensions have been reported to form connected structures. When attractive forces act between particles in suspension they may flocculate and, depending on particle concentration, shear history and other parameters, flocs may build-up in a three-dimensional network which spans the suspension sample. In this paper a floc network model is introduced to interpret the elastic behavior of flocculated suspensions at small deformations. Elastic percolation concepts are used to explain the variation of the elastic modulus with concentration. Data taken from the suspension rheology literature, and new results with suspensions of magnetic -Fe₂O₃ and non-magnetic -Fe₂O₃ particles in mineral oil are interpreted with the model proposed.Non-zero elastic modulus appeared at threshold particle concentrations of about 0.7 vol.% and 0.4 vol.% of the magnetic and non-magnetic suspensions, respectively. The difference is attributed to the denser flocs formed by magnetic suspensions. The volume fraction of particles in the flocs was estimated from the threshold particle concentration by transforming this concentration into a critical volume concentration of flocs, and identifying this critical concentration with the theoretical percolation threshold of three-dimensional networks of different coordination numbers. The results obtained indicate that the flocs are low-density structures, in agreement with cryo-scanning electron micrographs. Above the critical concentration the dynamic elastic modulus G was found to follow a scaling law of the type G ( _f - _f ^c ) ^f , where _f is the volume fraction of flocs in suspension, and _f ^c is its threshold value. For magnetic suspensions the exponent f was found to rise from a low value of about 1.0 to a value of 2.26 as particle concentration was increased. For the non-magnetic a similar change in f was observed; f changed from 0.95 to 3.6. Two other flocculated suspension systems taken from the literature showed a similar change in exponent. This suggests the possibility of a change in the mechanism of stress transport in the suspension as concentration increases, i.e., from a floc-floc bond-bending force mechanism to a rigidity percolation mechanism.     

Equivalent Conditions for Ω-Inverse Limit Stability

Recently we obtained sufficient conditions for an endomorphism to be -inverse limit stable. That is, if an endomorphism f satisfies weak Axiom A and the no-cycles condition, then f is -inverse limit stable. In this paper we give alternative conditions for -inverse limit stability. The following are equivalent: (a) f satisfies weak Axiom A and the no-cycles condition; (b) the chain recurrent set is prehyperbolic; and (c) the closure of the set of -limit points of f, L ⁺(f), is prehyperbolic with no cycles.     

Analytic Tools to Bound the Criticality at the Outer Boundary of the Period Annulus

In this paper we consider planar potential differential systems and we study the bifurcation of critical periodic orbits from the outer boundary of the period annulus of a center. In the literature the usual approach to tackle this problem is to obtain a uniform asymptotic expansion of the period function near the outer boundary. The novelty in the present paper is that we directly embed the derivative of the period function into a collection of functions that form a Chebyshev system near the outer boundary. We obtain in this way explicit sufficient conditions in order that at most \(n\geqslant 0\) critical periodic orbits bifurcate from the outer boundary. These theoretical results are then applied to study the bifurcation diagram of the period function of the family \(\ddot{x}=x^p-x^q,\) \(p,q\in {\mathbb {R}}\) with \(p>q\).     

Excess thermal noise generated during Poiseuille flow of certain polymer solutions

The previously reported results concerning the generation of excess thermal noise induced by capillary flow of aqueous solutions of poly(ethylene oxide) (PEO) are supplemented by measurements on the following solutions: PEO/DMF, PEO/i-PrOH, PS/THF, PVAC/cyclohexanone, and poly(acrylamide)/water. Similarly to the previous findings, a noise level increasing with the flow rate is recorded, the noise exhibiting a l/f -frequency spectrum. Within a certain flow range, distinct peaks are recorded in the spectrum (harmonics of a fundamental frequency,f ₀). Thef ₀-values of the various solutions under varying flow conditions arrange themselves along a commonf ₀-shear rate curve. They appear to be associated with transversal oscillations of the solution upstream the capillary entrance.     

Stability transitions for periodic orbits in hamiltonian systems

The paper considers one-parameter families of periodic solutions of real analytic Hamiltonian systems with two degrees of freedom, the parameter being the energy h. Conditions are given which guarantee that this family will undergo infinitely many changes in stability status as h tends to some finite value h ₀. First considered is the case of a critical point (with eigenvalues ±, ±i, and >0) of the Hamiltonian at energy h ₀ with the property that the family limits to a homoclinic orbit asymptotic to this point. Some generalizations of this case are given, and applications are made to examples such as the Hénon-Heiles Hamiltonian. We obtain an infinite sequence of distinct energy intervals converging to h ₀ on which the periodic orbits are elliptic. Requirements for the elliptic stability of the orbits are then given. The additional conditions for an infinite sequence of distinct energy intervals converging to h ₀, on which the orbits are hyperbolic, involve the coexistence problem for an associated Hill's equation that appears when the relevant Poincaré maps along the orbits are computed in coordinates. The results are compared to the case where the critical point has eigenvalues (±±i), and >0, investigated by Henrard and Devaney.     

Flow structure of wake behind a rotationally oscillating circular cylinder

Flow around a circular cylinder oscillating rotationally with a relatively high forcing frequency has been investigated experimentally. The dominant parameters affecting this experiment are the Reynolds number (Re), oscillation amplitude (θ_A), and frequency ratio F_R=f_f/f_n, where f_f is the forcing frequency and f_n is the natural frequency of vortex shedding. Experiments were carried out under conditions of Re=4.14×10³, 0°θ_A60° and 0.0F_R2.0. Rotational oscillation of the cylinder significantly modified the flow structure in the near-wake. Depending on the frequency ratio F_R, the cylinder wake showed five different flow regimes, each with a distinct wake structure. The vortex formation length and the vortex shedding frequency were greatly changed before and after the lock-on regime where vortices shed at the same frequency as the forcing frequency. The lock-on phenomenon always occurred at F_R=1.0 and the frequency range of the lock-on regime expanded with increasing oscillation amplitude θ_A. In addition, the drag coefficient was reduced when the frequency ratio F_R was less than 1.0 (F_R<1.0) while fixing the oscillation amplitude at θ_A=30°. When the oscillation amplitude θ_A was used as a control parameter at a fixed frequency ratio F_R=1.0 (lock-on regime), the drag reduction effect was observed at all oscillation amplitudes except for the case of θ_A=30°. This type of active flow control method can be used effectively in aerodynamic applications while optimizing the forcing parameters.     

Non-iterative Rauscher method for 1-DOF system: a new approach to studying non-autonomous system via equivalent autonomous one

Rauscher method becomes the matter of interest because in combination with the method of nonlinear normal vibration modes it allows to calculate steady forced vibrations in the system with multiple degrees of freedom (DOF) via reduction in the number of DOFs. However, modern realizations of that approach have drawbacks such as iterative nature and the need to have initial approximation for the solution. The primary principle of Rauscher method is in obtaining periodic solutions of a non-autonomous system via studying some equivalent autonomous one. In the paper, a new non-iterative variant of Rauscher method is considered. In its current statement, the method can be used in analysis of forced harmonic oscillations in a nonlinear system with one degree of freedom. The primary goals of the study were to find out what kind of equivalent autonomous systems could be built for a given non-autonomous one and how they can be used for the construction of periodic solutions and/or periodic phase plane orbits of the initial system. It is shown that three different types of equivalent autonomous dynamical systems can be built for a given 1-DOF non-autonomous one. The system of 1st type is a fourth-order dynamical system. Technically it can be considered as a 2-DOF system where additional “DOF” is explicitly “responsible” for forced oscillations. The system of 2nd type is a third-order dynamical system. Its periodic orbits are exactly the same as in the initial system. Using the invariant manifold of the system of 1st type, the system of 2nd type can be reduced to the form \(W(x,x')=0\) (which is called here the equivalent system of the 3rd type). It is important that the function \(W(x,x')\) can be built a priori. Once \(W(x,x')\) is found: (i) one can obtain different periodical orbits corresponding to forced oscillations in the initial system; (ii) one can estimate amplitudes of vibrations for these regimes; (iii) one can track bifurcations of periodical regimes of the initial system with respect to change in amplitude of external excitation f. As shown in the paper, periodical orbits of the initial non-autonomous system can be obtained via two different approaches: (i) as set of points on phase plane satisfying the condition \(W(x,x')=0\); (ii) via the application of harmonic balance method to the equivalent system of 1st type using system’s energy level as a continuation parameter. This approach has advantage over application of harmonic balance method to initial system because the latter requires good initial guess for expansion coefficients, while the new approach does not and always starts from zero initial guess.     

Isoenergetic Families of Planar Orbits Generated by Homogeneous Potentials

In the light of the inverse problem of dynamics, we study in some detail the monoparametric isoenergetic families of planar orbits f (x, y) = c, created by homogeneous potentials V (x, y). For any preassigned family of orbits and for any degree of homogeneity m of the potential, we offer the criteria which the family has to satisfy so that it can result from such a potential. When the criteria are fulfilled, Szebehelys first order partial differential equation for the unknown potential V (x, y) is substantially simplified and, in most of the cases, can be solved to completion and uniqueness.     

Convergence of biorthogonal series of biharmonic eigenfunctions by the method of titchmarsh

Canonical edge problems for the biharmonic equation can be solved by separating variables. The eigenvalues and eigenvectors arising in this separation are derived from a reduced system of ordinary differential equations along lines suggested in the excellent work of R. C. Smith (1952). We study the reduced system which is governed by a vector ordinary differential equation. A solution of the biharmonic problem, governed by a partial differential equation, can be found only if the prescribed data is restricted to a subspace of the space spanned by the eigenfunctions of the reduced problem. The theory leads to problems in generalized harmonic analysis which seek conditions under which arbitrary vector fields f(y) with values in ² can be represented in terms of eigenvectors of the reduced problem. This paper adds new theorems and conjectures to the theory. We extend Smith's generalization to fourth-order problems of the methods introduced by Titchmarsh (1946) to study eigenfunction expansions associated with second-order problems. We use this method to prove that, if f(y)=[(f ₁(y), f ₂y)], -1y1, f(y) C¹[-1, 1], f L₂[-1, 1], then the series expressing f(y) converges uniformly to f(y) in the open interval (-1, 1), uniformly in [-1, 1] if f ₁(±1)=0 and, in any case, to [0, f ₂(±1)-f ₁(±1)] at y=±1. This is unlike Fourier series, which converge to the mean value of the periodic extension of a function. The series exhibits a Gibbs phenomenon near the end points of discontinuity when f ₁(±1) 0.The Gibbs undershoot and overshoot for the step function vector [1, 0] and ramp function vector [y, 0] are computed numerically. The undershoot and overshoot are much larger than in the case of Fourier series and, unlike Fourier series, the Gibbs oscillations do not appear to be entirely suppressed by Féjer's method of summing Cesaro sums. We show that, when f(y) has interior points of discontinuity, the series for f(y) diverges and we present numerical results which indicate that, in this divergent case, the Cesaro sums converge to f(y) apparently with Gibbs oscillations near the point of discontinuity.     

Large Time Behavior and Stability of Equilibria of Degenerate Parabolic Equations

The article deals with positive solutions of the Dirichlet problem for

A simplified v2–f model for near‐wall turbulence

A simplified version of the v²–f model is proposed that accounts for the distinct effects of low‐Reynolds number and near‐wall turbulence. It incorporates modified C_ε(1,2) coefficients to amplify the level of dissipation in non‐equilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. Unlike the conventional v²–f, it requires one additional equation (i.e. the elliptic equation for the elliptic relaxation parameter f_µ) to be solved in conjunction with the k–ε model. The scaling is evaluated from k in collaboration with an anisotropic coefficient C_v and f_µ. Consequently, the model needs no boundary condition on and avoids free stream sensitivity. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2007 John Wiley & Sons, Ltd.     

Excess thermal noise and low-frequency oscillations during capillary flow of aqueous solutions of poly(ethylene oxide)

The excess thermal noise generated in polymer solutions through narrow capillaries is studied in detail for aqueous solutions of poly(ethylene oxide), , of varying concentration. With increasing flow rate, the excess noise level increases, the noise spectrum assuming a 1/f -form with 1.5. Within a critical flow range, distinct peaks appear in the spectrum, their frequencies being multiples of a fundamental frequency. The latter frequency (f ₀) is found to increase with the flow rate; this variation, as well as that brought about by varying concentration and capillary dimensions, can be accommodated in a single curve correlatingf ₀ with the shear rate at the capillary wall. No such correlation was found for the total noise level. The value off ₀ appeared to be determined by transversal oscillations of the liquid stream entering the capillary. Addition of small amounts of silica particles (Aerosil) led to the disappearance of the peaks in the spectrum.     

Two Dimensional Subsonic Euler Flows Past a Wall or a Symmetric Body

The existence and uniqueness of two dimensional steady compressible Euler flows past a wall or a symmetric body are established. More precisely, given positive convex horizontal velocity in the upstream, there exists a critical value \({\rho_{\rm cr}}\) such that if the incoming density in the upstream is larger than \({\rho_{\rm cr}}\), then there exists a subsonic flow past a wall. Furthermore, \({\rho_{\rm cr}}\) is critical in the sense that there is no such subsonic flow if the density of the incoming flow is less than \({\rho_{\rm cr}}\). The subsonic flows possess large vorticity and positive horizontal velocity above the wall except at the corner points on the boundary. Moreover, the existence and uniqueness of a two dimensional subsonic Euler flow past a symmetric body are also obtained when the incoming velocity field is a general small perturbation of a constant velocity field and the density of the incoming flow is larger than a critical value. The asymptotic behavior of the flows is obtained with the aid of some integral estimates for the difference between the velocity field and its far field states.