Stability analysis of orbital modes for a generalized Lane-Emden equation
aa r X i v : . [ m a t h - ph ] S e p Stability analysis of orbital modes for a generalized Lane-Emden equation
Ronald Adams ∗ and Stefan C. Mancas † Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114-3900, USA
Haret C. Rosu ‡ IPICYT, Instituto Potosino de Investigacion Cientifica y Tecnologica,Camino a la presa San Jos´e 2055, Col. Lomas 4a Secci´on, 78216 San Luis Potos´ı, S.L.P., Mexico
We present a stability analysis of the standard nonautonomous systems type for a recently in-troduced generalized Lane-Emden equation which is shown to explain the presence of some of thestructures observed in the atomic spatial distributions of magnetically-trapped ultracold atomicclouds. A Lyapunov function is defined which helps us to prove that stable spatial structures in theatomic clouds exist only for the adiabatic index γ = 1 + 1 /n with even n . In the case when n isodd we provide an instability result indicating the divergence of the density function for the atoms.Several numerical solutions, which according to our stability analysis are stable, are also presented. Keywords: stability analysis, generalized Lane-Emden equation, nonautonomous system, Lyapunov func-tion, numerical orbital modes
I. THE GENERALIZED LANE-EMDEN EQUATION
In a recent article [13], Rodrigues et al. use the condition of hydrostatic equilibrium in the set of thefluid continuity and momentum (Navier-Stokes) equations and a Poisson-like equation to determine theequation of state of a laser-cooled gas within a magneto-optical trap. The authors obtain what they callthe generalized
Lane-Emden equation for the confined atomic profiles, which is also derived in the contextof astrophysical fluids [4], and is given by γ ζ ddζ (cid:18) ζ θ γ − dθdζ (cid:19) + 1 − Ω θ = 0 . (1)The adiabatic index γ relates the pressure and density of atoms by the relation p = C γ ρ γ , where C γ = p (0) ρ (0) γ , and p (0), ρ (0) are the pressure and density at the center of the atomic cloud while the densityat some distance r from the center is given by ρ ( r ) = ρ (0) θ ( r ). Using radial symmetry, θ ( ζ ) is thenondimensional density of atoms which depends on the nondimensional distance ζ from the center of thecloud given by ζ = ra γ , with a γ = q C γ mω ρ (0) γ − , and trap frequency ω = κm [13]. The nondimensionalconstant Ω is the ratio of multiple scattering induced radiation pressure to trapping forces, is defined byΩ = Qρ (0)3 mω , with Q being the square of effective electric charge [11], and is the equivalent of the plasmafrequency [10]. The stability of solutions depends on Ω, as we will see in the following section, withstable solutions found when 0 < Ω <
1. Ter¸cas et al. [16] derived the same equation which models thepolytropic equilibrium of a magneto-optical trap that describes the crossover between the two limitingcases: temperature-dominated (Ω → → γ = 2, (1) is linear θ ζζ + 2 ζ θ ζ + 1 − Ω θ , (2)and may be roughly considered as a repulsive electrostatic counterpart of the classic Lane-Emden equationin astrophysics with solutions representing the Newton-Poisson gravitational potential of stars, considered ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] as spheres filled with polytropic gas [6]. For other values of γ , (1) has an additional operatorial term ofthe form γ ( γ − θ ζ /θ , and also two nonoperatorial terms instead of one.When the effects of the multiple scattering can be neglected, and thermal effects dominate Ω →
0, andfor γ = 1 the atomic density is given by θ ( ζ ) = (cid:20) (1 − γ ) (cid:18) c + c ζ + ζ γ (cid:19)(cid:21) γ − . (3)For a bounded density at the origin c = 0, and using θ (0) = θ > θ ( ζ ) = (cid:20) θ γ − − γ − γ ζ (cid:21) γ − . (4)The case γ = 1 corresponds to an isothermal gas as discussed in [13], and yields the Maxwell-Boltzmannequilibrium where (1) reduces to the separable equation ddζ (cid:18) ζ d ln θdζ (cid:19) = − ζ . (5)By two quadratures, and assuming the same initial conditions, the atomic density has the Gaussian profile θ ( ζ ) = θ e − ζ . (6)This solution can also be found by taking the limit γ → →
1. For an isobaricprocess, γ = 0, the density of atoms is given by θ ( ζ ) = 1Ω H ( ξ − ξ ) , (7)where H ( ξ − ξ ) is the Heaviside step function, and ξ = √ π Ω is the Lane-Emden radius of the traps,which was also observed in the experiments of [7]. This solution is known as the water-bag equilibriumprofile [12, 18].For the aforementioned case γ = 2, which can be taken into account in the astrophysics of dark matterhalos with possible substantial pressure compared to the atom density of the dark atoms [3, 4], thepolytropic equation of state is p = C ρ . Then (2) can be written in self-adjoint form ddζ (cid:18) ζ dθdζ (cid:19) + ζ − Ω θ ψ = − Ω θ , and A = − Ω2 , it becomes ddζ (cid:18) ζ dψdζ (cid:19) + Aζ ψ = 0 . (9)In the latter format, this equation is close to the ‘electrostatic Lane-Emden equation’, known as theThomas-Fermi equation, associated with the field of the statistical distribution of electrons in heavyatoms which takes into account their repulsive electrostatic interactions [17] ddζ (cid:18) ζ dψdζ (cid:19) + Bζ ψ = 0 , (10)where B = − √ / π , see equation (1.2) in [17].By assuming initial conditions θ (0) = θ , and θ ζ (0) = 0, the atom density is given by θ ( ζ ) = 1Ω " θ Ω −
1) shc r Ω2 ζ ! , (11)where shc denotes the cardinal hyperbolic sinus function defined byshc( x ) := (cid:26) sinh( x ) x , for x = 0 , , for x = 0 . The boundary of the halo ζ M , can be found numerically by assuming that on the boundary we must havezero atom density, thus shc (cid:16)q Ω2 ζ M (cid:17) = − θ Ω , provided that 0 < Ω < θ .Since (1) cannot be solved analytically for other values of γ , an important issue that we address inthe rest of this paper is the stability analysis of solutions that are obtained numerically. We develop thisstability analysis using a generalized Lyapunov function for the corresponding nonautonomous systemof differential equations in Section 2. If the positive departure from unity of the adiabatic index isparametrized by 1 /n , where n ∈ N , then we find that stable solutions exist only for even n , while forodd n all the solutions are unstable. The numerical solutions are presented in Section 3 as a proof of theresults obtained in the previous section, and the paper ends with a conclusion section. II. THE STABILITY ANALYSIS
For any γ = 1, we write (1) in self-adjoint form as ddζ (cid:18) ζ ddζ θ γ − (cid:19) = γ − γ ζ (Ω θ − . (12)Letting θ = z γ − we obtain ddζ (cid:18) ζ dzdζ (cid:19) = γ − γ ζ (cid:16) Ω z γ − − (cid:17) , (13)and by using γ = 1 + 1 /n with n ∈ N , we get ddζ (cid:18) ζ dzdζ (cid:19) = ζ n + 1 (Ω z n − . (14)Eq. (14) can be written as a nonautonomous first order system dydζ = f ( ζ, y ) (15)using y = (cid:18) zz ζ (cid:19) , and f ( z, z ζ ) = (cid:18) z ζ n +1 (Ω z n − − z ζ ζ (cid:19) . There is only one real critical point of the system (15) for n odd given by ¯ y T = (cid:0) /n , (cid:1) while for n even there are two ¯ y T , = (cid:0) ∓ /n , (cid:1) . All the critical points are shifted to the origin by the substitution x T = y T − ¯ y T = ( x , x ), then (15) becomes x ζ = f ( ζ, x + ¯ y ) = (cid:18) x n +1 (cid:2) Ω (cid:0) x ∓ /n (cid:1) n − (cid:3) − x ζ (cid:19) . (16)First, we define the notion of stability for a general nonlinear nonautonomous system x ζ = f ( ζ, x ) , ζ ≥ ζ , (17)where ζ ≥ x ( ζ ) = x ∈ R n . We assume f : R + × R n → R n is continuous and satisfies f ( ζ,
0) = 0, ∀ ζ ≥ ζ . Definition II.1. ([9]) The system (17) is said to be:(i)
Lyapunov stable if for each ǫ > , there is δ ( ǫ, ζ ) > k x ( ζ ) k < δ ⇒ k x ( ζ ) k < ǫ, ∀ ζ ≥ ζ ≥ . (18)(ii) Uniformly stable if for each ǫ >
0, there is δ = δ ( ǫ ) >
0, independent of ζ , such that (18) issatisfied.(iii) Asymptotically stable if it is
Lyapunov stable , and there is a constant l = l ( ζ ) > x ( ζ ) → ζ → ∞ , for all k x ( ζ ) k < l .(iv) Uniformly asymptotically stable if it is uniformly stable , and there is a constant l > ζ , such that for all k x ( ζ ) k < l , x ( ζ ) → ζ → ∞ , uniformly in ζ . This means that for each η > T = T ( η ) > k x ( ζ ) k < η, ∀ ζ ≥ ζ + T ( η ) , ∀ k x ( ζ ) k < l. (19)(v) Exponentially stable if there exist positive constants l , k , and p such that k x ( ζ ) k ≤ k k x ( ζ ) k e − p ( ζ − ζ ) , ∀ k x ( ζ ) k < l. (20)Before attempting the stability problem for the nonlinear system (16), it behooves us to first study thestability of the linearization of (16) about ¯ x T = (0 , n is even, we use the left equilibrium¯ y T = (cid:0) − /n , (cid:1) , and we set A ( ζ ) = ∇ x f ( ζ, x ) | x =¯ x , then A ( ζ ) = Ω1+1 /n (cid:0) x − /n (cid:1) n − − ζ !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =¯ x = (cid:18) − /n Ω /n − ζ (cid:19) . (21)Using (21) we then consider the corresponding linear distance-varying system to (16), x ζ = A ( ζ ) x ( ζ ) , x ( ζ ) = x . (22)To establish the stability of (22), we make use of a linear matrix inequality developed in [5]. We restatethe result below for completeness. Theorem II.2. (Theorem 1 in [5].) Consider the system (22). Suppose there exists a positive definitedifferentiable matrix function P : R + → R n × n and a continuous function g : R + → R such that A T P + P A + P ζ ≤ g ( ζ ) P (23)with R ∞ ζ g ( ζ ) dζ = −∞ , then the system (22) is asymptotically stable. Theorem II.3.
The origin is asymptotically stable for the system (22).
Proof . To apply the above result to the linear system (22) let P ( ζ ) = (cid:0) a ( ζ ) b ( ζ ) b ( ζ ) c ( ζ ) (cid:1) , then A T P + P A + P ζ = a ζ − Ω /n /n b a − ζ b + b ζ − Ω /n /n ca − ζ b + b ζ − Ω /n /n c b − ζ c + c ζ ! . (24)By setting b = 0, a = ζ , c = /n Ω /n ζ , and g ( ζ ) = − ζ the matrix inequality (23) becomes − ζ − /n )Ω /n ζ ! ≤ − ζ − /n Ω /n ζ ! , (25)hence for ζ >
0, (25) is satisfied for ζ ≥ ζ . Furthermore, the matrix function P ( ζ ) = ζ /n Ω /n ζ ! , is differentiable. For the positive definiteness of P ( ζ ) we compute x T P x , x T P x = (cid:18) x x (cid:19) T ζ /n Ω /n ζ ! (cid:18) x x (cid:19) = 1 ζ (cid:18) x + 1 + 1 /n Ω /n x (cid:19) > , for x and x both nonzero. Therefore the distance-varying system (22) is locally asymptotically stable.It is only of interest to us to study the left equilibrium point ¯ y T = (cid:0) − /n , (cid:1) for stability, as theequilibrium point ¯ y T = (cid:0) /n , (cid:1) will be shown to be unstable (see Theorem II.10), for n ∈ N . Beforewe state the stability/instability results for the nonlinear system (16) we first need to introduce someterminology that is used in [15]. Definition II.4. ([15]) A scalar continuous function V ( x ) is said to be locally positive definite if V (0) = 0and, in a ball B r (0) x = 0 = ⇒ V ( x ) > . (26)If V (0) = 0 and the above property holds over the whole state space, then V ( x ) is said to be globallypositive definite .Related concepts can be defined analogously, in a local or global sense. A function V ( x ) is negativedefinite if − V ( x ) is positive definite; V ( x ) is positive semi-definite if V (0) = 0 and V ( x ) ≥ x = 0; V ( x ) is negative semi-definite if − V ( x ) is positive semi-definite. For our purposes we will need the notionof positive definiteness for a scalar valued distance-varying function. Definition II.5. ([15]) A scalar distance-varying function V ( x, ζ ) is locally positive definite if V (0 , ζ ) = 0,and there exists a distance-invariant positive definite function V ( x ) such that ∀ ζ ≥ ζ , V ( x, ζ ) ≥ V ( x ) . (27)Thus, a distance-variant function is locally positive definite if it dominates a distance-invariant locallypositive function. The notions of negative definite, semi-definite, and negative semi-definite can be definedsimilarly.Next, we introduce the notion of a decrescent function. Definition II.6. ([15]) A scalar function V ( x, ζ ) is said to be decrescent if V (0 , ζ ) = 0, and if thereexists a distance-invariant positive definite function W ( x ) such that ∀ ζ ≥ , V ( x, ζ ) ≤ W ( x ) . (28)In other words, V ( x, ζ ) is decrescent if it is dominated by a distance-invariant positive definite function.With the above definitions introduced we now state the classical Lyapunov stability theorem for non-autonomous systems. Theorem II.7. ([15]) If, in a ball B r (0) around the equilibrium point 0, there exists a scalar function V ( x, ζ ) with continuous partial derivatives such that(i) V is positive definite,(ii) V ζ is negative semi-definite.Then the equilibrium point 0 is stable in the sense of Lyapunov. If, furthermore,(iii) V is decrescent,then the origin is uniformly stable.With the stability of the linear system established, we turn our attention to the nonlinear system (16).We seek to prove the stability of the origin for (16) in the case where n is even and ¯ y T = (cid:0) − /n , (cid:1) ,leading to the system (16) with the minus sign ( − ).Defining the Lyapunov function V ( x ) = − n + 1) "(cid:18) x − /n (cid:19) n +1 + 1Ω /n + 2 n + 1 x + x , (29)we then calculate the derivative of VV ζ = − x n + 1 (cid:18) x − /n (cid:19) n + 2 x n + 1 + 2 x n + 1 (cid:20) Ω (cid:18) x − /n (cid:19) n − (cid:21) − x ζ (30)= − x ζ ≤ . If x < /n , then − x Ω /n ≥ −
1. This allows us to utilize the Bernoulli inequality (1 + u ) s ≥ su ,where u ≥ − s ∈ R + \ (0 , Theorem II.8.
The origin is a uniformly Lyapunov stable equilibrium point of system (16) for n even. Proof . We start by establishing that V ( x ) is a locally positive definite function for x ∈ B r (0) where r = /n , and ζ ≥ ζ . We first consider the case where x ≤ /n , and we apply the Bernoulli inequalityabove to obtain V ( x ) = 2Ω( n + 1) /n (cid:16) − x Ω /n (cid:17) n +1 − n + 1) Ω /n + 2 n + 1 x + x ≥ n + 1) /n (cid:16) − ( n + 1) x Ω /n (cid:17) − n + 1) Ω /n + 2 n + 1 x + x = − n + 1 x + 2 n + 1 x + x = x ≥ . To consider the case where /n ≤ x ≤ /n we note that the function W ( u ) = − n + 1) "(cid:18) u − /n (cid:19) n +1 + 1Ω /n + 2 n + 1 u satisfies W u ( u ) = − n + 1 (cid:18) u − /n (cid:19) n + 2 n + 1 . We then see that W u ( u ) ≥ x ∈ (cid:2) /n , /n (cid:3) , and since W (cid:0) /n (cid:1) = n +1 (cid:16) − n +1 (cid:17) >
0, weconclude that W ( u ) ≥ (cid:2) /n , /n (cid:3) which in turn implies that V ( x ) | B r (0) ≥ V is distance-invariant it follows that V is a locally positive definite function, moreover it isevident that V is also a decrescent function. Therefore using Theorem II.7 together with V (0) = 0, and V ζ ≤ Theorem II.9.
The equilibrium point 0 of system (16) is asymptotically stable when n is even. Proof . To study the asymptotic stability of the origin, we note that on the domain D = (cid:8) x ≤ /n (cid:9) we have that V | D is positive definite and V ζ | D ≤
0. Unlike in the autonomous case where LaSalle’sinvariance theorem can be used to find a maximally invariant set that the trajectories converge to, in thenonautonomous case it is less clear how to define such a set. In fact since − V ζ is not a locally positivedefinite function, Theorem 8.4 in [9] does not directly apply. We can still use the ideas from Theorem 8.4to construct an invariant set in D . If we consider the closed ball of radius r = /n centered at the origin,then B r (0) ⊂ D . We introduce the following distance invariant set B α = { x ∈ B r (0) | V ( x ) ≤ α } , where α < min k x k = r V ( x ). To calculate min k x k = r V ( x ) we proceed as follows: on B r (0) we can write V ( x ) = − n +1) h(cid:0) x − /n (cid:1) n +1 + /n i + n +1 x + r − x with the derivative V x = − n +1 (cid:0) − Ω /n x (cid:1) n + n +1 − x . We then see that V x (0) = 0, and note that V x x = /n /n (cid:0) − Ω /n x (cid:1) n − −
2. If − /n ≤ x ≤ /n (cid:20) − (cid:16) /n Ω /n (cid:17) n − (cid:21) then V x x ≥
0, and for x ≥ /n (cid:20) − (cid:16) /n Ω /n (cid:17) n − (cid:21) then V x x ≤ x = 0 is the only critical point. Since V ( /n ,
0) is less than both V (0 ,
0) and V ( − /n ,
0) thenmin k x k = r V ( x ) = V ( /n , B α , we have that B α ⊂ B r (0) ⊂ D for all ζ ≥ ζ which implies that B α is bounded.Since V ζ ≤ D we have that for ζ > x , ζ ) stays in B α for all ζ ≥ ζ ,hence the solution is bounded for all ζ ≥ ζ . Moreover, from (30) we have V ( x ( ζ )) − V ( x ( ζ )) = − Z ζζ x ( s ) s ds. (31)Since V is bounded below on D , and V ζ ≤ ζ →∞ V ( x ( ζ ) , ζ ) = V ∞ exists, and V ∞ ≤ V ( x ( ζ ) , ζ ). Taking ζ → ∞ in (31) yields V ( x ( ζ )) − V ∞ = 4 lim ζ →∞ Z ζζ x ( s ) s ds. (32)Next, we claim that lim ζ →∞ x ( ζ ) = 0. By contradiction, suppose that lim ζ →∞ x ( ζ ) = L where L = 0 (note that lim ζ →∞ x ( ζ ) exists since V ∞ exists). Then for ǫ = L there exists ζ > x ( ζ ) > L/ ζ ≥ ζ . This implies thatlim ζ →∞ Z ζζ x ( s ) s ds ≥ L ζ →∞ Z ζζ s ds = ∞ . (33)Eq. (33) implies that V ( x ( ζ )) − V ∞ is divergent which is a contradiction, therefore lim ζ →∞ x ( ζ ) = 0.Moreover, x ζ is uniformly continuous since x ζζ = Ω1 + 1 /n (cid:18) x − /n (cid:19) n − x − ζ ( n + 1) (cid:20) Ω (cid:18) x − /n (cid:19) n − (cid:21) + 6 x ζ is bounded because x and x are bounded functions. Thus, from Barbalat’s lemma [15] we see thatlim ζ →∞ x ζ = 0 which implies from (16) thatlim ζ →∞ Ω (cid:0) x − /n (cid:1) n − ζ →∞ x ( ζ ) = 0.At this point, we have established that the origin is asymptotically stable for (16). With this inmind, it is of interest to estimate the basin of attraction for the origin by using the ideas from theproof of Theorem II.9. We start with noting that the constant α from the above proof satisfies α
The equilibrium point 0 of (16) is unstable for odd n . Proof . We apply the above instability Theorem II.10. We define the following scalar valued Lyapunovfunction V ( x, ζ ) = (cid:18) Ω (cid:18) x + 1Ω /n (cid:19) n − (cid:19) x + ( n + 1) x ζ , (34)noting that V (0 , ζ ) = 0, and also that V ( x, ζ ) is a decrescent function. Furthermore, for x = 0 and x = 0, V ( x, ζ ) is strictly positive, hence condition (ii) in Theorem II.10 is satisfied. For the derivativewe obtain V ζ = 1 n + 1 (cid:18) Ω (cid:18) x + 1Ω /n (cid:19) n − (cid:19) + n Ω x (cid:18) x + 1Ω /n (cid:19) n − − n + 1) x ζ = 1 n + 1 (cid:18) Ω (cid:18) x + 1Ω /n (cid:19) n − (cid:19) + x n Ω (cid:18) x + 1Ω /n (cid:19) n − − n + 1) ζ ! . Since (cid:0) x + /n (cid:1) n − ≥ (cid:0) − /n + /n (cid:1) n − = n − Ω − /n , then by taking ζ = 1 + /n )2 n − Ω − /n Ω we have that V ζ ≥ n + 1 (cid:18) Ω (cid:18) x + 1Ω /n (cid:19) n − (cid:19) + x n Ω (cid:18) x + 1Ω /n (cid:19) n − − n + 1) ζ ! = V ( x ) ≥ , (35)provided k x k < r = /n . So V ( x ) is a distance-invariant locally positive definite function, therefore (iii)is satisfied, and Theorem II.10 implies that 0 is unstable at ζ . Thus, we can conclude that 0 is unstablefor all ζ ≥ ζ . III. ORBITAL MODES
In this section, we discuss the effects of n and Ω on the solutions of (1). The formation of stable orbitalmodes in the density function was observed in rotary clouds of atoms [1, 2, 14, 18], and was previouslybelieved to be a consequence of rotation in the cloud. The analysis of the previous section reveals boundedsolutions when n is even which are periodic as long as the starting initial conditions are to the left ofthe unstable positive fixed point ¯ y T , and are inside the basin of attraction B α . Since we use the initialcondition θ ( ζ ) = 1, and to be inside of the basin we require that 1 < /n . This yields to 0 < Ω < n is odd since there is only one unstable equilibrium given by ¯ y T , or if the initial conditions are outsideof the basin of attraction B α . In [13] and [16], the authors discuss orbital modes for the cases γ = 1 , , γ = 1 is discussed in Section 1, and γ = 2 follows from the conclusion of TheoremII.11. It is relevant to note that the case γ = is studied in [12] of which the stability follows fromTheorem II.9.The left panel of Fig. 1 depicts a sublevel set for the function V ( x + ¯ y ), which are the translationsof the level sets of the Lyapunov function V ( x ). The right panel corresponds to an asymptotically stablesolution to (15) with n = 2, Ω = 0 .
5, and left stable fixed point ¯ y T = (cid:0) − /n , (cid:1) . We start frominitial condition x T = (cid:0) /n − ǫ, (cid:1) located to the left of the right unstable fixed point ¯ y T = (cid:0) /n , (cid:1) .The shaded region in the right panel represents the sublevel set (cid:8) x ∈ R | V ( x + ¯ y ) ≤ α (cid:9) , we take thebounded component of this set, then translate it to the right by ¯ y to obtain B α which is used to estimatethe basin of attraction for the origin of system (16). FIG. 1: Left panel: translations of the level sets of the Lyapunov function V ( x ) given by (29). Right panel: basinof attraction for periodic solutions of (16). The fixed points are indicated by the black and red dots, left fixedpoint is always stable while the right fixed point is always unstable. Simulations must start inside of the basin ofattraction which require that 0 < Ω <
1. Also, the necessary conditions to have two fixed points is that n is even.A typical trajectory starts close and to the left of the unstable fixed point and converges to the stable fixed point. The left panel of Fig. 2 shows the solutions z ( ζ ), z ζ ( ζ ) of (15) with the red dashed line representing theright (unstable) equilibrium, and the black dashed line the left (stable) equilibrium point. The right figuredepicts the density function θ ( ζ ) of (1), with γ = , Ω = 0 .
5, and initial conditions θ ( ζ ) = 1, θ ζ ( ζ ) = 0.Since ζ = 0 is a singular point of (1), for numerical simulations, we used ζ = 0 . n = 2 , , γ = , , , and starting with the same initial conditions, we obtain periodic solutions thatare depicted in Fig. 3. The boundary ζ ∗ is given by θ ( ζ ) = 0, is increasing as γ is increasing, andrepresents the nondimensional radius for which the density of atoms is zero.For a fixed adiabatic index, and with the number of atoms that increases then Ω increases, thus thecentral core density increases. This was observed experimentally when a single ring turns into a ring witha central core, causing the cloud to rotate faster [2], as we can see in Fig. 4. IV. CONCLUSION
A detailed stability analysis of a generalized Lane-Emden equation recently elaborated in the physicsof trapped atomic clouds, and previously in astrophysics has been presented. The linearized form of thecorresponding nonautonomous first order system has been shown to be locally asymptotically stable, andthe Lyapunov indirect approach has been used to construct a Lyapunov function for the nonlinear system.When n is even there are two equilibrium points ¯ y < ¯ y , and we showed that the left equilibrium point ¯ y is asymptotically stable. Using the Lyapunov function we provide an estimate for the basin of attraction0
20 40 60 80 100 ζ - (cid:0) (cid:1) (cid:2) z ζ
20 40 60 80 100 ζ θ ( ζ ) FIG. 2: Left panel: solutions z ( ζ ), z ζ ( ζ ) of (15), with the red dashed line representing the right (unstable) fixedpoint, and the black dashed line the left (stable) fixed point. The red dashed function z ( ζ ) and the magenta dottedfunction z ζ ( ζ ) represent the trajectory from the right panel of Fig. 1. Right panel: density function θ ( ζ ) = z of(1), with γ = , Ω = 0 . z ( ζ ) from the left panel.
20 40 60 80 100 ζ θ ( ζ ) γ γ γ FIG. 3: Stable periodic solutions θ ( ζ ) of (1) with Ω = 0 .
5, and three values of the parameter γ = , , and .The top curve is the solution from the right panel of Fig. 2.
20 40 60 80 100 ζ θ ( ζ ) Ω Ω Ω FIG. 4: Stable periodic solutions θ ( ζ ) of (1) with γ = , and three values of the parameter Ω = 0 . , .
5, and 0 . in which initial conditions should be picked that yield periodic solutions. For the other equilibrium point¯ y , we provide an instability result showing that in the case where n is odd this equilibrium is unstable,1and when n is even the same equilibrium ¯ y = ¯ y is also unstable.This analysis yields a special set of solutions for the density of atoms of the generalized Lane-Emdenequation obtained when n is even, and initial conditions are inside a region of attraction to the leftof the stable equilibrium. These periodic solutions (orbital modes) have been observed experimentallypreviously in rotating ultracold atomic clouds, and are demonstrated analytically to be stable.is even, and initial conditions are inside a region of attraction to the leftof the stable equilibrium. These periodic solutions (orbital modes) have been observed experimentallypreviously in rotating ultracold atomic clouds, and are demonstrated analytically to be stable.