Orbital stable

If C is orbitally stable and, in addition, the distance between B and C tends to zero as t - oo, this form of stability is called asymptotic orbital stability. 

The donor ability (nucleophilicity) of R2M , including the stability of complexes being formed, increases as M becomes heavier. The acceptor properties of M in R2M (such as the ability to form the adducts with Lewis bases, for example, with pyridine and piperidine at — 30°C) are determined by the low-lying unoccupied atomic d- and p -orbitals °. Stable free radicals R3M (M = Ge, Sn) are obtained by a photochemical disproportionation reaction of R2M in a hydrocarbon solvent medium see equation 26, R = (Me3Si)2CH. 

Theorem 4.2. Let n — of the Floquet multipliers of (4.6) lie inside the unit circle in the complex plane. Then 7 is an asymptotically orbitally stable trajectory of (4.5). 

Corollary 5.2. If (3.4) holds then T is asymptotically orbitally) stable. 

Proof. The quantity under the integral sign in the definition of A in (5.2) is the trace of the Jacobian matrix for the system (5.1) evaluated along the periodic orbit. Theorem 4.2 then applies. A periodic orbit for an autonomous system has one Floquet multiplier equal to 1. Since there are only two multipliers and one of them is 1, is the remaining one. The periodic orbit is asymptotically orbitally stable because, in view of Lemma 5.1, A<0. ... 

When the inequality in (3.4) is reversed, there will be a periodic orbit (by an application of the Poincare-Bendixson theorem). By our assumption of hyperbolicity, this orbit must be asymptotically orbitally stable since it is so from the inside. These comments establish the next result. 

Theorem 3.2. Suppose that the parameters ai, mi, and T[ are chosen so that (3.2) has a (linearly) asymptotically orbitally stable periodic solution (S(t), x(t)) with period T>0. Fix ai and ti > tj. Then there exist a critical value ml and a branch of periodic orbits of (3.1), with positive Xi component, bifurcating from the hypothesized orbit for m2 near ml-... 

For each subcritical temperature T < 1, dynamic System 3 has a simple equilibrium point, i.e. dp/dA = 0 = dP/dA, at (pG(T), Pa(T)), (pd(T), Pa(T)), and (pL(T), P<7(T)). Using well-established methods of dynamic system theory—for simple equilibrium points it suffices to examine the eigenvalues of System 3—one then determines that both the saturated vapor point (po(T), Po-(T)) and the saturated liquid point (Pl(T), PV(T)) are (orbitally stable with respect to A) dicritical nodes, i.e. that each solution path p(A), P(A) approaches the equilibrium point from a definite direction and, conversely, each direction corresponds to exactly one path (see Figure 2). 

Certain mathematical-physical considerations and the subsequent fitting of f (p T) allow us to conclude that the coexistence envelope diameter point (pd(T), pa(T)) is an (orbitally unstable) improper node, i.e. that all solution paths leaving (pD(T), pequilibrium points (pG(T), p (T)), (pD(T), P (T)), and (pL(T), Pa(T)) converge to the critical point (1, 1). This multiple equilibrium point is an orbitally stable, but structurally (topologically) unstable, multiple node. The parameter T thus can be considered as a bifurcation parameter, and T = 1 as a bifurcation value of dynamic System 3. 

Definition 14.2. A point eo on the stability boundary of a periodic trajectory Le is said to be safe if L q is asymptotically orbitally stable. 

VII. MOLECULAR SPECTRA AND PERIODIC ORBITS (STABLE AND UNSTABLE)... 

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