Elliptic fixed point

Increasing a leads to the effective double-well potential shown earlier with two elliptic (stable) and one hyperbolic (unstable) fixed points. The elliptic fixed points become unstable for parameter values below... 

M 8] [P 7] The effect of switching between the flow fields (a) and (d) given in Figure 1.18 at various periods T was analyzed (see Figure 1.19) [28], For very high alterations, a simple superposition of the flow fields is achieved. Elliptic fixed points surrounded by closed orbits (tori) of various periods are found. 

Figure 4. Phase portrait of an elliptic fixed point, for a 1-DOF linear Hamiltonian.

According to Chirikov [23J, the onset of chaos is associated with the overlap of neighboring nonlinear resonances. The overlap criterion, which bears the qualitative significance, uses the model of isolated resonances. Each resonance is characterized by its width, the maximum distance (in the action variable) from the elliptic fixed point The overlap means that the sum of the widths of two neighboring resonances is equal to the distance between two fixed points of these isolated resonances. We start with the pendulum Hamiltonian, which describes an isolated 1 N resonance under the periodic perturbation of frequency Q ... 

Thus there are two qualitatively different cases. When the eigenvalues are complex, Ag2 = iuj, x is an elliptic fixed point corresponding to a local maximum or minimum of the streamfunction. Such elliptic points are surrounded by closed circular orbits that stay in the close vicinity of x. Therefore the area around the elliptic points remains isolated from the rest of the flow, thus preventing efficient mixing. 

Figure 10 Poincare map for two noniinearly coupled oscillators at fixed energy. The energetic periphery is labeled X. (A) The elliptic fixed point of an orbit with a),/o)2 = /i. (B, C) Elliptic fixed points of two distinct orbits with mjuiz - Vj. (D) Sep-aratrix surrounding the Wj/wj = Vi resonance zone. The separatrix pinches off at four hyperbolic fixed points, three of which are more or less clearly visible. (E) One of four elliptic fixed points of an orbit with V (the other three are not visi-...

Trajectories initiated close to an elliptic fixed point behave in a qualitatively different manner from those near a hyperbolic fixed point. For one thing, elliptic fixed points are invariably surrounded by invariant tori, with frequency ratios not far from that of the fixed point all motion on each torus stays on the same torus. Hyperbolic fixed points may or may not be surrounded by tori. However, they are always associated with a single unique set of manifolds composed of motion asymptotic to them in positive and negative time. These manifolds are called stability manifolds or separatrix manifolds, and their continued existence in the presence of a coupling term is guaranteed by the Stable Manifold theorem. The nature of the asymptotic manifolds will be seen to be of special interest and importance, and we discuss them at length in the following section. 

For now, we note that the elliptic fixed points are separated from one another by hyperbolic fixed points, which meet along a line generally known as... 

Figure 11 Surface of section plots for the De Leon—Berne Hamiltonian as a function of energy for energies below the barrier. Note the destruction of quasiperiodic motions (KAM tori) as a function of energy. Also note the persistence of ICAM tori near certain elliptic fixed points. Reprinted with permission from Ref. 119.

In the case of an elliptic fixed point (which would correspond to a stable... 

In such a case the iterates of the two maps will also be conjugate and, if they are numerical methods, they will have similar stability properties and performance (e.g. the same effective order). It is difficult to separate the relevance of the two properties in cases where symplectic and reversible maps are conjugate. This is however rarely the case and certainly does not hold generically for discrete maps in many dimensions [209]. There is no direct correspondence between reversible and symplectic maps, however each class of maps admits certain theorems of dynamical systems which are in many ways analogous (for example, the Kolmogorov-Arnold-Moser, or KAM, theory for symplectic maps near elliptic fixed points [386] has an analogue for reversible maps [97]). 

We see that the phase-plane is broken up into a sequence of fixed points and a series of both open and closed constant-energy curves. The origin (0= =0) and its periodic equivalents (0 27rn, = 0), are stable fixed points (or elliptic... 

For larger T (T = 1.6), chaotic behavior arises, the hyperbolic fixed point is disrupted and the tori are perturbed (see Figure 1.20) [28]. A chaotic region appears with homoclinic tangle and formation of new hyperbolic and elliptic points. 

Circular polarization of electromagnetic radiation is a polarization such that the tip of E, at a fixed point in space, describes a circle as time progresses. E, at one point in time, describes a helix along the direction of wave propagation k. The magnitude of the electric field vector is constant as it rotates. Circular polarization is a limiting case of elliptical polarization. The other special case is the easier-to-understand linear polarization. Circular (and elliptical) polarization is possible because the propagating E and El fields... 

The mathematical structure of the models is their unifying background systems of nonlinear coupled differential equations with eventually nonlocal terms. Approximate analytic solutions have been calculated for linearized or reduced models, and their asymptotic behaviors have been determined, while various numerical simulations have been performed for the complete models. The structure of the fixed points and their values and stability have been analyzed, and some preliminary correspondence between fixed points and morphological classes of galaxies is evident—for example, the parallelism between low and high gas content with elliptical and spiral galaxies, respectively. 

Note that an unstable limit cycle surrounds the stable fixed point, just as we expect in a subcritical bifurcation. Furthermore, the cycle is nearly elliptical and surrounds a gently winding spiral—these are typical features of either kind of Hopf bifurcation. ... 

A resonant n/n = p/q elliptic periodic orbit is a multiple fixed point on the resonant invariant curve. The angle i) changes during one iteration by... 

All points on this invariant curve are fixed points. A linear analysis shows that the two eigenvalues of a fixed point are equal to 1. This means that the periodic elliptic orbits in the rotating frame have two pairs of unit eigenvalues. 

We remark that all the fixed points on a resonant invariant curve of the Poincare map correspond to elliptic motion of the small body, with the same semimajor axis a, such that n/n is rational, and the same eccentricity e. They differ only in the orientation, which means that all these orbits have different values of w, as shown in Figure (15). 

If the system has symmetries (as is the case with the restricted problem), usually the symmetric periodic orbits survive (but not always ). The resonant fixed points that survive correspond to monoparametric families of elliptic periodic orbits, in the rotating frame. These families bifurcate from the circular family, at the corresponding circular resonant orbits. From the above analysis we come to the conclusion that out of the infinite set of resonant elliptic periodic orbits of the two-body problem, with the same semimajor axes and the same eccentricities, but different orientations, as shown in Figure 15, only a finite number survive as periodic orbits in the rotating frame, and in most cases only two, usually, but not always, are symmetric. 

According to the Poincare-Birkhoff fixed point theorem all resonant tori break up for arbitrarily small perturbations. If the rotation number is p/q the perturbation leaves q pairs of hyperbolic and elliptic periodic orbits. The unstable hyperbolic orbits are embedded in a layer filled by aperiodic, chaotic orbits that do not stay on an invariant surface, but cover a finite non-zero volume of the phase space in a chaotic layer around the original resonant torus. The elliptic points, however, are wrapped around by new concentric tori that form islands of regular orbits within the chaotic band (Fig. 2.5). 

It is very important to stress that changes in the geometrical shape of the integration domain can induce bifurcations in the drift velocity field [31, 47, 50, 52]. Let us consider, for example, the drift velocity field computed for an elliptical domain with major axis o = 3A and minor axis b = a/1.1. As shown in Fig. 9.13(a), instead of the stable limit cycle of the resonance attractor in the circular domain of radius Rg, = 1.5A we have two pairs of fixed points where the drift velocity vanishes. In each pair, one fixed point is a saddle and the other one is a stable node. Depending on the initial conditions, the spiral wave approaches one of the two stable nodes. Trajectories of the spiral center obtained by numerical integration of the Oregonator model (9.1) are in perfect agreement with the predicted drift... 

The theoretically predicted destruction of the resonance attractor in response to deviations from the circular shape of the integration domain has been confirmed experimentally within the light-sensitive BZ medium. A spiral wave was exposed to uniform illumination proportional to the total gray level obtained in an elliptical integration domain. Fig. 9.13(b) shows the resonant drift mediated during global feedback control. The spiral wave drifts towards a stable node of the drift velocity field. Close to this fixed point the drift velocity becomes very slow. Thus, the experimentally observed termination of the spiral drift at certain positions in a uniform medium is explained in the framework of the developed theory of feedback-mediated resonant drift. 

If the distance apart of the fixed points Fx and Fa is 2c, the elliptic co-ordinates of a point f, ij, distant rx and ra from the fixed points, are given by the equations... 

Exact forms (83,81,82) for the integrals, J, K and S for the two-centre problems of dihydrogen and other homogeneous diatomic molecular species were achieved in early calculations by transforming to the elliptical coordinates system defined in Figure 6.3. In this coordinate system the nuclear positions are the two fixed points of the coordinate system the foci of the ellipses separated by the bond length R with a point P defined by the three coordinates... 

Figure 8.1 Ellipse described by the tip of the real electric vector at a fixed point O in space (top panel) and particular cases of elliptical, linear, and circular polarization. The...

The polarization ellipse along with a designation of the rotation direction (right- or left-handed) fully describes the temporal evolution of the real electric vector at a fixed point in space. This evolution can also be visualized by plotting the curve in (9,tp,t) coordinates described by the tip of the electric vector as a function of time. For example, in the case of an elliptically polarized plane wave with right-handed polarization, the curve is a right-handed helix with an elliptical projection onto the Ocp - plane centered around... 

We have discussed the typical manifestation of periodic orbits on a Poincare map as fixed points that are either elliptic or hyperbolic. Let us now consider the properties of motion nearby these fixed points in terms of their stability properties. This is accomplished by a straightforward linear stability analysis about the fixed point. We can carry out such an analysis on any fixed point, whether or not the surrounding phase space is chaotic (as long as we can find the fixed point). 

Because of the existence of first-derivative discontinuities when internal boundary conditions are specified, flows past multi-element airfoils in the aerospace industry, as cases in point, are simulated using singly connected computational domains such as that shown in Figure 9-12 which displace the source of the discontinuity to a computational boundary. Whereas the modeling by Sharpe and Anderson (1991) of wells as internal fixed points produces undesired discontinuities, aerospace methods produce meshes where all metrics and derivatives are continuous. Sharpe and Anderson also embed their elliptic operators in first-order, time-like systems. The complete process yields shocks in some instances, perhaps because the embedded system possesses nonlinear hyperbolic properties. Jameson (1975) has shown how various transient diffusive systems can be derived to host relaxation-based techniques these methods are further optimized for computational speed. 

Fig. 10.7.1. A fixed point when 71 = 2. The sectors between the separatrices are called elliptic.

When 6=1, the Henon map becomes conservative, as its Jacobian equals -1-1. At 6 = 1 and a = —1, it has an unstable parabolic fixed point with two multipliers +1 at 6 = 1 and a = 3, it is a stable parabolic fixed point with two multipliers —1. In between these points, for —1 < a < 3 (i.e. (a, 6) G T), the map has a fixed point with multipliers where cos > = 1 y/a -h 1. This is a generic elliptic point for tp 7r/2,27r/3,arccos(—1/4) [167]. Since the Henon map is conservative when 6=1, the Lyapunov values are all zero. When we cross the curve AH, the Jacobian becomes different from 1, hence the map either attracts or expands areas which, obviously, prohibits the existence of invariant closed curves. Thus, no invariant curve is born upon crossing the curve AH. ... 

Analogous to the spherical filler of radius R in the Kraus model, Bhattacharya and Bhowmick [31] consider an elliptical filler represented by R(1 + e cos 0), in the polar coordinate. The swelling is completely restricted at the surface and the restriction diminishes radially outwards (Fig. 40 where, qt and q, are the tangential and the radial components of the linear expansion coefficient, q0). This restriction is experienced till the hypothetical sphere of influence of the restraining filler is existent. One can designate rapp [> R( 1 + e cos 0)] as a certain distance away from the center of the particle where the restriction is still being felt. As the distance approaches infinity, the swelling assumes normality, as in a gum compound. This distance, rapp, however, is not a fixed or well-defined point in space and in fact is variable and is conceived to extend to the outer surface of the hypothetical sphere of influence. 

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