Smooth torus

Bifurcations of this type in diffeomorphisms have been the object of nuioericaJ. (8) amd theoretical (9-10) work. We learn from these studies that in addition to a smooth torus, a saddle-node bifurcation may yield other equally robust situations. In particular two of them appeax to be relevant to our work Either (i) the limit cycle (in the Poincare map) is destroyed when the saddle-node bifurcation occurs, and the disappearance of the saddle-node corresponds to the transition from a regulau , periodic, state to a chaotic one, or (ii) the limit cycle is destroyed before the disappearance of the saddle-node and there exints chaos concurrently with the initial periodic orbit (see fig 5c)... 

It is customary to put both variables on the unit torus that is, q and p are taken to be periodic variables with period equal to one. The sole control parameter, Q, determines the extent of the nonlinearity. As a increases, there is a dramatic transition between regular smooth orbits and trajectories that are almost completely chaotic. 

Let X be a smooth projective variety over k with an action of a torus H which has only finitely many fixed points. A one-parameter subgroup Gm —> H of H which does not lie in a finite set of given hyperplanes in the lattice of one-parameter groups of H will have the same fixed points as H. In future we call such a one-parameter group general . Thus the induced action of a general one-parameter group Gm — G has only finitely many fixed points on P. ... 

Small-order resonance horns (p, q small) and particularly those with 1 p, <7 4 are comparatively wide and easier to locate computationally through algorithms that will locate the periodic entrained trajectories. These algorithms, however, will be inadequate for a complete analysis of our systems since (at least as FA — 0) periodic trajectories appear in disconnected isolas. The motivation behind the construction of our torus-computing algorithm is to provide a means of study of this two-parameter bifurcation diagram that can continue smoothly both within the resonance horns and in the region of quasi-periodicity that separates or—from another point of view—unites them. 

Sheaf in fpqc topology 117 Smooth group scheme 88 Solvable group scheme 73 Spec A 41 Split torus 56 Strictly upper triangular 62 Subcomodule 23 Symplectic group 99... 

Suppose that U is smooth with periodic boundary conditions 0 = 7 an H -dimensioruil torus). Then, with respect to the space Cl(p,p) with inner product (f, g)p as defined above, we have ... 

Assume that the system (2.1) possesses a sufficiently smooth quasiperiodic solution jc (r) with frequency basis co. Then there exists a function (f) given on the torus such that A (r) = u (o>r) this function is a solution of the equation... 

A is a constant which characterizes the lag in the system. Suppose that (4.1) possesses a sufficiently smooth invariant torus... 

In the present subsection, the manifold M is assumed to be compact. Definition 3.4.1 The symplectic structure (Af,o ) is called completely inte-grable if there exists a proper surjective morphism of smooth complex manifolds f M N whose general fibre is a disconnected union of several completely isotropic n-dimensional tori (in particular, a general fibre may appear to be a torus). 

Hui = — 1. The exact meaning of the assertion of Proposition 3.4.4 is as follows. Let P be a smooth manifold of all complex structures J C End Hu (it is readily seen that they form a manifold). Then there exists an everywhere dense set P C P, such that for any complex structure J G P, a complex torus Mj = (Pr/Pi, J) does not have proper subtori. 

If the surface and the Hamiltonian F are analytic, then both conditions 1 and 2 of Theorem 5.2.3 are automatically fulfilled (the property 1 requires special proof), and therefore in an analytic case Theorem 5.2.3 immediately implies Theorem 5.2.1. More generally, if a compact orient able surface M is nonhomeo-morphic to a sphere and to a torus then the above-mentioned equations of system motion do not have a new integral which is a smooth function on T M analytic for fixed x M on cotangent two-dimensional planes T M and having only a finite number of distinct critical values. The number of critical points is not necessarily finite. Functions polynomial in momenta are an example of integrals analytic in the momenta... 

First we discuss Critical Points and the Topology of a torus abstractly. Then we apply the discussion to carbon and boron tori duals. Critical points, defined as the first derivative equals zero, 3 f/3x = 0, are fundamental in calculus in the method used to find minima, maxima, and inflection points of a simple function. This concept may be extended to describe topology. Let a smooth real valued function f represent the height of a point on a torus standing on its edge (see Figure 9), and solve for the critical points of f Also, require that the second derivatives not be zero. Mathematically, this means that the Hessian matrix, Hf, of second derivatives is invertible, or det Hf(p) 0 where Hf(p) = 3 f /3xi3xj. ... 

We will present in Sec. 12.2 a summary of results for the case where the unstable manifold of the saddle-node is homeomorphic to a torus along with the proof of a theorem on the persistence of the invariant torus in the smooth case. There, we will also develop a general theory for an effective reduction of the problem to a study of some family of endomorphisms (smooth non-invertible maps) of a circle. 

If the mapping (11.6.2) is the Poincare map of an autonomous system of differential equations, then the invariant curve corresponds to a two-dimensional smooth invariant torus (see Fig. 11.6.3). It is stable if L < 0, or it is saddle with a three-dimensional unstable manifold and an (m -h 2)-dimensional stable manifold if L > 0. Recall from Sec. 3.4, that the motion on the torus is determined by the Poincare rotation number if the rotation number v is irrational, then trajectories on the torus are quasiperiodic with a frequency rate u] otherwise, if the rotation number is rational, then there are resonant periodic orbits on a torus. 

As /i increases within a resonant zone other periodic orbits with the same rotation number M/N may appear. In some cases, the boundary of the resonant zone can lose its smoothness at some points, like in the example shown in Fig. 11.7.4 here, the resonant zone consists of the union of two regions D and Z>2 corresponding to the existence of, respectively, one and two pairs of periodic orbits on the torus. The points C and C2 in Fig. 11.7.4 correspond to a cusp-bifurcation. At the point S corresponding to the existence of a pair of saddle-node periodic orbits the boundary of the resonant zone is non-smooth. 

Fig. 11.7.5. The typical scenario of the breakdown of a two-dimensional torus due to a loss of its smoothness.

The closed invariant curve Wq for the Poincare map on the cross-section is the loci of intersection of an invariant two-dimensional torus W with the cross-section. The torus is smooth if the invariant curve is smooth, and it is non-smooth otherwise. If the original non-autonomous system does not have a global cross-section, then other configurations of W are also possible, as... 

Theorem 12.3. (Afraimovichr-Shilnikov [3, 6]) If the global unstable set of the saddle-node L is a smooth compact manifold a torus or a Klein bottle) at fi = Oy then a smooth closed attractive invariant manifold 7 (fl torus or a Klein bottle, respectively) exists for all small fi. 

If m = 1, then is a manifold. It is homeomorphic to a torus if m 1 and to a Klein bottle if m = — 1. As already mentioned, this manifold may be smooth or non-smooth, depending on whether intersects the strong-stable foliation transversely everywhere or not. When x and (p are... 

So, the results of Theorems 12.3, 12.5 and 12.7 are summarized as follows IfW is a smooth toruSy then a smooth attracting invariant torus persists after the disappearance of the saddle-node L. If is homeomorphic to a torus but it is non-smoothy then chaotic dynamics appears after the disappearance of L, Herey either the torus is destroyed and chaos exists for all small /i > 0 the big lobe condition is sufficient for that)y or chaotic zones on the parameter axis alternate with regions of simple dynamics. 

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