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.     

Knotted periodic orbits in suspensions of Smale's horseshoe: Torus knots and bifurcation sequences

We construct a suspension of Smale's horseshoe diffeomorphism of the two-dimensional disc as a flow in an orientable three manifold. Such a suspension is natural in the sense that it occurs frequently in periodically forced nonlinear oscillators such as the Duffing equation. From this suspension we construct a knot-hòlder or template—a branched two-manifold with a semiflow—in such a way that the periodic orbits are isotopic to those in the full three-dimensional flow. We discuss some of the families of knotted periodic orbits carried by this template. In particular we obtain theorems of existence, uniqueness and non-existence for families of torus knots. We relate these families to resonant Hamiltonian bifurcations which occur as horseshoes are created in a one-parameter family of area preserving maps, and we also relate them to bifurcations of families of one-dimensional quadratic like maps which can be studied by kneading theory. Thus, using knot theory, kneading theory and Hamiltonian bifurcation theory, we are able to connect a countable subsequence of one-dimensional bifurcations with a subsequence of area-preserving bifurcations in a two parameter family of suspensions in which horseshoes are created as the parameters vary. One implication is that infinitely many bifurcation sequences are reversed as one passes from the one dimensional to the area-preserving family: there are no universal routes to chaos!     

Poincaré Mapping Method for Hydrodynamic Systms. Dynamic Chaos in a Fluid Layer between Eccentrically Rotating Cylinders

Investigation of the plane–parallel motion of particles of an incompressible medium reduces to investigation of a Hamiltonian system. The Hamiltonian function is a stream function. The timeperiodic mixing of an incompressible medium is described by a timeperiodic Hamiltonian function. The mixing of the medium is associated with dynamic chaos. Transition to dynamic chaos is studied by analysis of the positions of Lagrangian particles at times divisible by the period — Poincaré recurrence points. The set of Poincaré recurrence points is studied with the use of Poincaré mapping on the phase flow. A method for constructing Poincaré maps in parametric form is proposed. A map is constructed as a series in a small parameter. It is shown that the parametric method has a number of advantages over the generating function method is shown. The proposed method is used to examine the motion of particles of an incompressible viscous fluid layer between two circular cylinders. The outer cylinder is immovable, and the inner cylinder rotates about a point that does not coincide with the centers of both cylinders. An optimal mode for the motion is established, in which the area of the chaotic region is maximal.     

Sensitive Dependence on Initial Conditions of Strongly Nonlinear Periodic Orbits of the Forced Pendulum

We employ nonsmooth transformations of the independent coordinate to analytically construct families of strongly nonlinear periodic solutions of the harmonically forced nonlinear pendulum. Each family is parametrized by the period of oscillation, and the solutions are based on piecewise constant generating solutions. By examining the behavior of the constructed solutions for large periods, we find that the periodic orbits develop sensitive dependence on initial conditions. As a result, for small perturbations of the initial conditions the response of the system can jump from one periodic orbit to another and the dynamics become unpredictable. An analytical procedure is described which permits the study of the generation of periodic orbits as the period increases. The periodic solutions constructed in this work provide insight into the sensitive dependence on initial conditions of chaotic trajectories close to transverse intersections of invariant manifolds of saddle orbits of forced nonlinear oscillators.     

Construction of exact numerical solutions of the stationary traveling wave type for viscous thin films

An effective numerical procedure, based on the Galerkin method, for finding solutions of the stationary traveling wave type in the complete formulation is proposed for the case of viscous liquid films. Examples of a viscous film flowing freely down a vertical surface have been calculated. The calculations have been made for various values of the dimensionless surface tension , including =0. The method makes it possible to predict a number of bifurcations that occur as decreases. The existence of numerous families of stationary traveling waves when 1 was demonstrated in [6]. The present study shows that as 1 all but one of these families of wave solutions disappear. The shape of the periodic and solitary waves and the pressure distribution in the film are found for various . When =0 and the wave number is fairly small, the periodic solution has a singularity, as predicted in [14]: at the crest of the wave a corner point appears; the angle between the tangents at this point =140–150. The method proposed can be used to calculate other wavy film flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 94–100, May–June, 1990.     

Travelling, non-periodic patterns in nonlinear stability problems

In this short note we present results on the existence of several classes of travelling, non-periodic solutions of the complex Ginzburg-Landau equation. First we give a very short introduction to the G-L equation and show its importance in nonlinear stability theory. We then study the G-L equation with complex coefficients and establish the existence of a 2-parameter family of quasi-periodic solutions and two different types of one-parameter families of heteroclinic orbits; all members of these families travel with a well-defined wave-speed. The heteroclinic solutions correspond to (travelling) soliton-like localized structures which connect different (stable) periodic patterns. Mathematically, these families of travelling solutions (quasi-periodic and heteroclinic) are continuations into the complex case of the stationary solutions of the real G-L equation.     

Optimum lifting wings with specified longitudinal trim characteristics

The case of an infinitely slender wing that slightly disturbs a supersonic ideal gas flow is considered. The plan form and the free-stream Mach number M are given. The optimum surface of the wing y=g(x, z) is determined as a result of finding a bounded function of the local angles of attack _M=g(x, z)/x that minimizes the drag coefficient c_x for given values of the lift coefficient c_y and the pitching moment coefficient m_z. The problem is solved in the class of piecewise-constant functions for wings of complex geometry [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 185–189, July–August, 1987.     

Parameter identification in a soil with constant diffusivity

We consider infiltration into a soil that is assumed to have hydraulic conductivity of the form K = K = K_se^h and water content of the form = K – _r. Here h denotes capillary pressure head while K_s, , and _r represent soil specific parameters. These assumptions linearize the flow equation and permit a closed form solution that displays the roles of all the parameters appearing in the hydraulic function K and . We assume K_s and _r to be known. A measurement of diffusivity fixes the product of and resulting in a parameter identification problem for one parameter. We show that this parameter identification problem, in some cases, has a unique solution. We also show that, in some cases, this parameter identification problem can have multiple solutions, or no solution. In addition it is shown that solutions to the parameter identification problem can be very sensitive to small changes in the problem data.     

Chaos and integrability in a nonlinear wave equation

We consider the parametrized family of equations _tt ,u- _xx u-au+u ₂ ² u=O,x(0,L), with Dirichlet boundary conditions. This equation has finite-dimensional invariant manifolds of solutions. Studying the reduced equation to a four-dimensional manifold, we prove the existence of transversal homoclinic orbits to periodic solutions and of invariant sets with chaotic dynamics, provided that =2, 3, 4,.... For =1 we prove the existence of infinitely many first integrals pairwise in involution.     

MHD flow in the vicinity of the stagnation point in a purely azimuthal magnetic field

Axisymmetric MHD flow in the vicinity of the stagnation point in the presence of a purely azimuthal nonhomogeneous magnetic field B {0, B, 0} is studied. This problem belongs to the class of MHD problems whose solutions are known as solutions of the layer type [1]. This class also includes, in particular, the classical exact solutions of the Navier-Stokes equations.The approximate solutions of the analogous MHD problems for the limiting cases of large and small values of the diffusion number ==/ have been considered in [2–5]. In this case it is possible to divide the flow into the so-called viscous and current layers, for each of which the approximate equations, simpler than the exact equations, are solved numerically or in quadratures. Using this technique it is possible to avoid the basic mathematical difficulty, which is that the sought solution of the boundary-value problem must be selected from a family of two-parameter solutions. The approximate method permits dividing the problem into two stages (corresponding to the two boundary layers) in each of which one unknown parameter is determined (in place of their simultaneous determination by direct integration of the basic equations).The drawback of the approximate methods [2–5] is their nonapplicability in the most interesting case, when the thicknesses of the current and viscous layers are of comparable magnitude, i. e., when the kinematic and magnetic viscosities ( and ) are quantities of the same order. We should also note the poor accuracy of the methods in the framework of the considered approximations for a comparatively large volume of the calculations required, which, in turn, prevents obtaining more exact solutions.The present paper presents a numerical integration of the equations describing MHD flow in the vicinity of the stagnation point over a wide range of S and numbers (Alfvén and diffusion numbers), without the assumption of their smallness, with preliminary determination of the unknowns at the zero of the derivatives of the sought functions with the aid of the method of asymptotic integration.A critical value of the Alfvén number is found, for which the retardation of the fluid by the magnetic field (for the first considered configuration of the magnetic field) at the wall is so intense that the friction vanishes everywhere on the surface of the solid body. It is also found that with further increase of the number S a region of reverse flow appears near the wall, which is separated from the remaining flow by a plane on which the z-component of the velocity is equal to zero.     

Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system

This is a further study of the set of homoclinic solutions (i.e., nonzero solutions asymptotic to 0 as ¦x¦) of the reversible Hamiltonian systemu ^iv +Pu +u–u ²=0. The present contribution is in three parts. First, rigorously for P –2, it is proved that there is a unique (up to translation) homoclinic solution of the above system, that solution is even, and on the zero-energy surface its orbit coincides with the transverse intersection of the global stable and unstable manifolds. WhenP=–2 the origin is a node on its local stable and unstable manifolds, and whenP(–2,2) it is a focus. Therefore we can infer, rigorously, from the discovery by Devaney of a Smale horseshoe in the dynamics on the zero energy set, there are infinitely many distinct infinite families of homoclinic solutions forP(–2, –2+) for some>0. Buffoni has shown globally that there are infinitely many homoclinic solutions for allP(–2,0], based on a different approach due to Champneys and Toland. Second, numerically, the development of the set of symmetric homoclinic solutions is monitored asP increases fromP=–2. It is observed that two branches extend fromP=–2 toP=+2 where their amplitudes are found to converge to 0 asP 2. All other symmetric solution branches are in the form of closed loops with a turning point betweenP=–2 andP=+2. Numerically it is observed that each such turning point is accompanied by, though not coincident with, the bifurcation of a branch of nonsymmetrical homoclinic orbits, which can, in turn, be followed back toP=–2. Finally, heuristic explanations of the numerically observed phenomena are offered in the language of geometric dynamical systems theory. One idea involves a natural ordering of homoclinic orbits on the stable and unstable manifolds, given by the Horseshoe dynamics, and goes some way to accounting for the observed order (in terms ofP-values) of the occurrence of turning points. The near-coincidence of turning and asymmetric bifurcation points is explained in terms of the nontransversality of the intersection of the stable and unstable manifolds in the zero energy set on the one hand, and the nontransversality of the intersection of the same manifolds with the symmetric section in ⁴ on the other. Some conjectures based on present understanding are recorded.     

Cascades of homoclinic orbits to,and chaos near,a Hamiltonian saddle-center

We consider a class of reversible, two-degree of freedom Hamiltonian systems possessing homoclinic orbits to a saddle-center: an equilibrium having two non-zero real and two nonzero imaginary eigenvalues. Under mild nondegeneracy conditions, we construct a two-parameter unfolding and show that there is a countable infinity of secondary homoclinic bifurcations in any neighborhood of the original system. We also demonstrate the existence of families of periodic orbits and of shifts on two symbols (horseshoes). The lack of hyperbolicity and the presence of conserved quantities make the analysis somewhat delicate. We discuss specific examples for which the nondegeneracy conditions can be explicitly checked but indicate that this is not always possible. We illustrate our results with numerical work.     

Analytical and Experimental Investigation of Squeeze-Film Dampers Executing Circular Orbits

We have referred here the first results obtained in the course of an experimental research intended to examine oil film pressure of a squeeze film damper (with inertialess lubricant) executing circular orbits. An analytical expression for pressure arising into the gap was obtained before, for the short bearing SFD, in the general case of offset circular orbits. The expression obtained has permitted, in a simple way, the examination of the influence of the offset on pressure. Two approximated expressions for pressure (the one for pressure as a function of time at a fixed measuring section, the other for pressure along the circumference of the bearing at a given time), corresponding to centered orbits, were compared with the exact ones. We observed that the limits within which the nondimensional offset can be considered negligible, change according to comparisons between circumferential pressure diagrams or between pressure diagrams as functions of time. In the second case they are much wider. The experimental results concerned offset orbits with nominal radius r=0.5C and offset <0.1. Good accordance between theoretical and experimental pressure trends has been found, as concerns the positive part of dynamic pressure. As concerns the negative part of dynamic pressure, tests have shown that cavitation takes place for a more or less wide length, depending on feeding groove pressure, other conditions being equal. Complete effective pressure diagram coinciding with the theoretical one has been found for particular feeding conditions. Oil film tensile stresses have been frequently detected with both complete and incomplete diagrams owing to cavitation. Maximum measured tensile stress was about 220kN/m². It was finally found that theoretical and experimental trends are coincident only if the SFD is not starved. Analytical expression for pressure in the long bearing SFD executing offset circular orbits is referred in the appendix.     

Exact Solutions of the Hydrodynamic Equations Derived from Partially Invariant Solutions

The paper proposes a heuristic approach to constructing exact solutions of the hydrodynamic equations based on the specificity of these equations. A number of systems of hydrodynamic equations possess the following structure: they contain a reduced system of n equations and an additional equation for an extra function w. In this case, the reduced system, in which w = 0, admits a Lie group G. Taking a certain partially invariant solution of the reduced system with respect to this group as a seed:rdquo; solution, we can find a solution of the entire system, in which the functional dependence of the invariant part of the seed solution on the invariants of the group G has the previous form. Implementation of the algorithm proposed is exemplified by constructing new exact solutions of the equations of rotationally symmetric motion of an ideal incompressible liquid and the equations of concentrational convection in a plane boundary layer and thermal convection in a rotating layer of a viscous liquid.     

Resonance Capture in a Three Degree-of-Freedom Mechanical System

We study the phenomena of resonance capture in a three degree-of-freedom dynamical system modeling the dynamics of an unbalanced rotor, subject to a small constant torque, supported by orthogonal, linearly elastic supports, which is constrained to move in the plane. In the physical system the resonance exists between translational motions of the frame and the angular velocity of the unbalanced rotor. These equations, valid in the neighborhood of the resonance, possess a small parameter which is related to the imbalance. In the limit 0, the unperturbed system possesses a homoclinic orbit which separates bounded periodic motion corresponding to resonant solutions from unbounded motion which corresponds to solutions passing through the resonance. Using a generalized Melnikov integral, we characterize the splitting distance between the invariant manifolds which govern capture and escape from resonance for 0. It is shown that as certain slowly varying parameters evolve, the separation distance alternates sign, indicating that both capture into, and escape from resonance occur. We find that although a measurable set of initial conditions enter into a sustained resonance, as the system further evolves the orientation of the manifolds reverses and many of these captured solutions will subsequently escape.     

Unique Periodic Orbits for Delayed Positive Feedback and the Global Attractor

The delay differential equation, (t)=–x(t)+f(x(t–1)), with >0 and a real function f satisfying f(0)=0 and f>0 models a system governed by delayed positive feedback and instantaneous damping. Recently the geometric, topological, and dynamical properties of a three-dimensional compact invariant set were described in the phase space C=C([–1, 0], ) of initial data for solutions of the equation. In this paper, for a set of and f which include examples from neural network theory, we show that this three-dimensional set is the global attractor, i.e., the compact invariant set which attracts all bounded subsets of C. The proof involves, among others, results on uniqueness and absence of periodic orbits.     

STABILITY ANALYSIS OF LINEAR AND NONLINEAR PERIODIC CONVECTION IN THERMOHALINE DOUBLE－DIFFUSIVE SYSTEMS

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The Hopf bifurcation with symmetry for the Navier-Stokes equations in (Lp(Ω)) n,with application to plane poiseuille flow

The existence of periodic solutions of the Navier-Stokes equations in function spaces based upon (L _p())ⁿis proved. The paper has three parts, (a) A proof of the existence of strong solutions of the evolution equation with initial data in a solenoidal subspace of (L _p())ⁿ. (b) The evolution equation is restricted to a space of time periodic functions and a Fredholm integral equation on this space is formed. The Lyapunov-Schmidt method is applied to prove the existence of bifurcating time periodic solutions in the presence of symmetry. (c) The theory is applied to the bifurcation of periodic solutions from planar Poiseuille flow in the presence of symmetry (SO(2) x O(2) x S ¹) yielding new results for this classic problem. The O(2) invariance is in the spanwise direction. With the periodicity in time and in the streamwise direction we find that generically there is a bifurcation to both oblique travelling waves and to travelling waves that are stationary in the spanwise direction. There are however points of degeneracy on the neutral surface. A numerical method is used to identify these points and an analysis in the neighborhood of the degenerate points yields more complex periodic solutions as well as branches of quasi-periodic solutions.     

Spatial Stationary Long Waves in Shear Flows

The system of integrodifferential equations describing the spatial stationary freeboundary shear flows of an ideal fluid in the shallowwater approximation is considered. The generalized characteristics of the model are found and the hyperbolicity conditions are formulated. A new class of exact solutions of the governing equations is obtained which is characterized by a special dependence of the desired functions on the vertical coordinate. The system of equations describing this class of solutions in the hyperbolic case is reduced to Riemann invariants. New exact solutions of the equations of motion are found.     

Critical layer formation in a stratified medium with mean shear flows

The problem of the excitation of internal waves with a given wave number k and frequency in a stratified medium with shear flows is considered. The internal wave field of the form v(z)exp(–it+ikx) established as t in a medium without dissipation has a singular point at the level z=z₀ (critical level), at which the flow velocity U(z) coincides with the phase velocity /k. Dissipative effects (viscosity and heat conduction) smooth out this singularity. An exact solution of the model equation describing as t and zz₀ the field excited by oscillating sources activated at t=0 is constructed with allowance for dissipation. This makes it possible to describe the limiting steady-state field, determine the critical layer as the neighborhood of the critical level in which dissipation effects are important, and to estimate its width and the rate of convergence to the limiting steady-state regime. The asymptotic behavior of the fields is examined for Ri1, where Ri is the Richardson number. It is shown that when the well-known Miles stability condition Ri>1/4 is satisfied there are no natural oscillations with a critical level.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 82–93, May–June, 1990.