On the Regularity and Chaos of the Hydrogen Atom Subjected to External Fields

In this paper, the integrable classical case of the Hydrogen atom subjected to three static external fields is investigated. The structuring and evolution of the real phase space are explored. The bifurcation diagram is found and the bifurcations of solutions are discussed. The periodic solutions and their associated periods for singular common-level sets of the first integrals of motion are explicitly described. Numerical investigations are performed for the integrable case by means of Poincaré surfaces of section and comparing them with nearby living nonintegrable solutions, all generic bifurcations that change the structure of the phase space are illustrated; the problem can exhibit regularity-chaos transition over a range of control parameters of system.


Introduction
Most natural phenomena are generally governed by nonlinear differential equations, for these problems, we must use different mathematical approaches and computational methods to simplify these systems and to study their integrable like lie algebra, the Painlevé criterion, the Ziglin criterion, the Liouville theorem and the Poincaré sections, because physically integrable systems are rare.
The study of the Hydrogen atom has been recently the focus of many works [1]- [10].Its great importance lies in its fields of implications in different branches of physics: solid state physics [11], spectroscopy [12], plasmas physics [13], molecular physics [14], astrophysics [15].
The Hydrogen atom is a simple but very complicated system when introducing external fields.The Hydrogen atom in a static magnetic field is a simple system that can be studied experimentally and theoretically, its dynamics in such a field has been a source of many interesting results in atomic physics over the past two decades [16] [17].It proves to be a prototype of quantum systems whose classical behavior can show chaotic dynamics [18] [19].Concrete systems are often modeled using systems with natural parameters.The regularity of the behavior of the system is observed for certain values of the parameters, but usually the behavior of the dynamical system becomes unpredictable.
The Hydrogen atom under the effect of external fields is a complicated, non-separable problem and generally displays chaotic behavior for some combinations of the appropriate parameters.The chaotic appearance results from the fact that, for certain initial conditions, the behavior of the dynamic system becomes unpredictable [20].The chaos is identified by a high sensitivity to the initial conditions and by the variation of the control parameters of the system.Indeed, a system with n degrees of freedom is regular if there are n independent constants of motion [21].Most problems involving atoms in external fields are very difficult to study analytically and solutions are difficult to obtain.However, for combinations of appropriate parameters, one can find exact solutions.
The isotropic nonrelativistic Hydrogen atom subjected to three static external fields: a magnetic field, an electric field, and a van der Waals interaction.The Hamiltonien (in units such that ) can be written as: α λ γ and β are control parameters representing the magnitude of the applied external fields: α is a constant; γ, measured in units of magnetic field .This combined potential is very interesting because of its similarity to the fields seen by an ion confined in a Paul trap [13].
The paper is organized as follows: In Section 2 we briefly review the regularized equations of motion and we bring out that the regular 2D Hamiltonian is equivalent to the motion of two coupled anharmonic oscillators.Section 3 presents a detailed description of the real phase space topology on the bifurcation diagram.For noncritical values of the constants of motion h and f, the regular level sets of the first integrals of motion of a completely integrable Hamiltonian system are composed of tori, according to the Liouville theorem.All generic bifurcations of Liouville tori, corresponding to critical values of the constants of motion h and f will be described, by using Fomenko's surgery.In Section 4 we give explicit periodic solutions and their associated periods for singular com-mon-level sets of the constants of motion.In section 5 we also investigate numerically via Poincaré surfaces of section, the structure and evolution of the phase space for the integrable case, comparing them with nearby living nonintegrable solutions, the problem can exhibit regularity-chaos transition over a range of system control parameters.Finally, Section 6 contains our conclusions.

Equations of Motion
The regularized Hamiltonian is obtained by transforming Equation (1) to semiparabolic coordinates and the corresponding momenta, namely, ( ) ( ) Note that the equations of motion associated with the Hamiltonian (1) have a collision singularity at 0 r = , which can be removed with the following change in the independent variable ( ) After such transformation, the regularized Hamiltonian reads In this equation, p ϕ is a conserved quantity, represents the value of the cy- clic integral associated to the cyclic coordinate φ, and we put p l ϕ = , with l a constant.
The difference between the Hamiltonian (3) and the Hamiltonian of an ion in a Paul trap is that the total pseudo-energy H  is equal to 2, that of a trapped ion is −2 (for details, see Ref. [11]).
The equations of motion associated with ( ) the dot denotes derivative with respect to τ.
The Hamiltonian (3) characterizes a two dimensional regularized version of the real three dimensional system after ignoring the cyclic integral associated to the cyclic coordinate φ, 0 p l ϕ = = .The study of the dynamics for Hydrogen atom subjected to external fields is easier to process in two-dimensional standardized version than a three-dimensional; and by removing the Coulomb singularity from the original dynamic system, the regularization procedure facilitates the numerical work.Another practical aspect is that the regular 2D Hamiltonian is equivalent to the motion of two coupled anharmonic oscillators with pseudo-energy is equal to 2, a system for which there are effective tools to make the problem easily exploitable.

The Phase Space Topology
The two dimensional regularized Hamiltonian (3) is separable if the relation ( 5) in this case the second integral of motion reads ( ) ( ) It is easy to verify that Equation (2) becomes where ( ) ( ) ( ) ( ) in these conditions, the equations of motion are ( ) .
H h = and F f = are the first integrals of motion, functions of ( ) , , , u v u v p p which are constant along the solutions of Equation (11).The sys- tem is called (meromorphically) Liouville integrable or completely integrable.

The Bifurcation Diagrams and the Admissible Region
In this section we shall give the admissible region on the bifurcation diagrams and a detailed description of the real phase space topology i.e. the topology of the real level sets for all generic constants h and f: , , , : , For doing that we find first the bifurcation diagram B of the problem (12), i.e.
the set of critical values of the energy-momentum mapping It appears, according to several works carried out in this sense (for details, see Ref. [22]- [27]), that the bifurcation diagram B is exactly the discriminant locus of the polynomials ( ) and the parameters of the system ( ) , , , , : 0 , , , , : 0 The set consists of 10 connected components (as it is shown in Figure 1).Thus, in each connected component of the set \ A B  the level set A  has the same topological type, and this latter may be changed only if ( ) The sections of these components with the plane { } . const or it is the empty set ϕ as it is shown in Table 1.
and the numbering of domains for 2 λ ≥ .Thus , , , x z x z p p can be expressed in terms of hyperelliptic functions living in the Jacobi variety Γ = Γ ⊗ Γ (where ⊗ is the symmetric product).These functions however are not single valued as can be seen from formulae ( 8) and (11).
Indeed, to each point on the symmetric product 1 2 Γ ⊗ Γ there correspond two values of ( ) , , , x z x z p p .Thus we define the natural projection is the set of fixed points of the complex conju-gation on A ⊄ :

( ) ( )
: , , , , , , Consider also the natural projection ξ on the Riemann surface given in u, v coordinates by: It induces an involution on the Jacobi variety and hence on A ⊄ by the natu- ral projection π.Formulae ( 6) and (7) imply that this involution ξ coincides with the complex conjugation (15) on A ⊄ the upshot is that in order to describe A  it is enough to study the projection: A connected component of the set of fixed points of η on the curves 1 Γ and 2 Γ is called an oval.
To determine the ovals of 1 Γ and 2 Γ it suffices to study the real roots of the polynomials ( ) , , , x z x z p p ∈  , we find exactly two admissible ovals whose projections on the u-plane and the v-plane are given by 1 ∆ and 2 ∆ .The product of the admissible ovals in Γ ⊗ Γ and the projection π of A  such as, ( ) The sections of these components with the plane { } . const The topological type of A  is (diffeomorphic to) a two-dimensional tori T, to a disjoint union of cylinders C, or it is the real planes 2  as it is shown in Table 2.

Topology of Singular Level Sets and Bifurcations of Liouville Tori
Suppose now that the constants h, f are changed in such a way that ( ) For the first case 2 λ ≥ , According to Fomenko surgery on Liouville tori [29], we can have in our case three types of bifurcation of the level set A  (see Table 3).To prove that, it suffices to look at the bifurcations of roots of the polynomials ( ) Modern Nonlinear Theory and Application Table 3. Generic bifurcations of the level set A  .
As shown in Figure 3, three types of bifurcations of (surgery on) Liouville tori take place: 1) Bifurcation T S → → ∅ : The two dimensional tori T is contracted to the circle S corresponding to the periodic solution, and then vanishes.

S S S T
→ × ∧ → : The two dimensional two tori 2T merge into two dimensional tori T by passing through the complex ( )

S S ∧
is a union of two circles having exactly one common point.
3) Bifurcation 2 2 T S → → ∅ : The two dimensional two tori 2T are contracted to two circles 2S corresponding to two periodic solutions, and then vanishes.
For the second case 2 λ < , The Fomenko classification of bifurcation of Liouville tori [29] cannot be applied as its invariant level sets contain a non-compact component (cylinder C or plane 2  ).Thus, we can have the type of bifurcation of the level set A  passing from domain i to domain j in Table 4.
To prove that, it suffices to look at the bifurcation of roots of the polynomials ( )

For the Case 2 λ ≥
When the bifurcation of Liouville tori takes place, the level set A  becomes completely degenerate.Then we can have exceptional families of periodic solutions.It is seen from Table 5, that if ( ) C C′ (see Figure 5), A  contains a isolated circles S and 2S, respectively which are periodic solutions.Consider now a fixed periodic solution belonging to the curve 0

C . The parameter v takes values in the admissible interval
, v v and u u u = = is equal to the double root of the polynomial ( ) g u (see Table 5).
Then we obtain from Equation ( 8) the following parameterization of fixed periodic solution:   4. Generic bifurcations of the level set A  .
( ) On the curve 0 C , the second integral of motion f is equal to 2, as well as the characteristic polynomial ( ) ( ) depends only on h.
Taking this variable change: the polynomial ( ) 30 15 where By an inversion of the elliptic integral Equation ( 17), one explicitly obtains the expression the periodic solution ( ) ( ) The expression of periodic solution ( ) v τ is given by solving the Jacobi inver- sion problem: ( ) ( ) The period v T associated with the solution ( ) v t is obtained by calculating the elliptic integral Equation ( 17) over the totality of the admissible oval for v: ( ) ( ) It's sufficient to replace in Equation ( 18) the upper bound in the integral by 1 R and multiply by 2, in these conditions cos 1 ϕ = − , we obtain a period de- pendent only on the energy h: u u m = = (see Table 5).
The values of the first integrals H and F on the curve 1 C are related by From Equation ( 8) we obtain the following parameterization of fixed periodic solution ( ) and The period of ( )

For the Case 2 λ <
In this case, it is seen from Table 6, that on the smooth curves ( ) L L′ the values of the second integral of motion f is equal to 2 (f is equal to 2) (see Figure 6), A  contains a circles S which correspond to a periodic solutions.On the curve ( ) ) ( ) The associated period is where ( ) , , , , , , ,

Numerical Investigation
By making use a set of software routines, implemented in Maple, for plotting 2D projections of Poincaré surfaces of section, method introduced by Poincaré and extended by Hénon [30], we give numerical illustrations of the real phase space topology studied in Section 3.For fixed values of constants , , , h f α λ and , β γ vary, the Liouville tori contained in the level set { } , H h F f = = change their topological type, while preserving the integrable behavior of the problem.However, with increasing control parameter associated with magnetic field, the random scattering of points in the sections shows that the system has carried out a transition from regularity through a quasi-regularity to chaotical dynamics.
The Poincaré surfaces of section are plotted in the plane ( ) ( ) , , u q p u p = .4 γ = , the system still has a regular motion, so we can say that the electric field is not responsible for the chaotic behavior of the system, on the other hand only the variation of the intensity of the magnetic field changes the behavior of the system from regularity to chaotically dynamics.and 1 f = (black), 0.5 (blue), 0.2 (red), 0 (black dot).The black dots corres- pond to the periodic solutions in the form of a isolated circles.

Bifurcations of Liouville Tori
For the case 2 λ < , the invariant level sets contain a non-compact compo- nent (cylinder C or plane 2  ), only in the domain 4' on the bifurcation dia- gram B′ we find a single tori.

Conclusion
In this paper, we have studied the classical dynamics of the Hydrogen atom in the generalized van der Waals potential subjected to external parallel magnetic and electric fields.By making use of some reductions, the regularized Hamiltonian of the system is described by a two-degree of freedom dependent on certain control parameters.The converted system is equivalent to the motion of two coupled anharmonic oscillators with pseudo-energy easily exploitable.The effect; and the a-dimensional number λ controls the anharmonicity associated to the van der Waals interaction[10] [11] [12]

h f passes through the
bifurcation diagram B. Then the topological type of A  may change and the bifurcation of Liouville tori takes place.

1 g u and ( ) 2 g
v .The correspondence between bifurcation of roots and Liouville tori is shown in Figure3.

Figure 3 .
Figure 3. Correspondence between bifurcation sets for ( ) , h f B ∈ and

Figure 4 . 2 g
Figure 4. Correspondence between bifurcation of A  and bifurcation of roots of the polynomials
v τ and hence ( ) z τ can be expressed in terms of hyperelliptic func- tions.

1 C
the complete elliptic integral of the first kind.In the same way, for the periodic solution on the curve the parameter v takes values in the admissible oval [ ]

Figures 7 (
Figures 7(a)-(h) represent Poincaré surfaces of section for 2 β = (electric field) and values of γ (magnetic field) near the integrable case 4 γ = (from left to right), where for different values of h and f on the bifurcation diagram B the system is totally regular, the topological type of A  is diffeomorphic to a two-dimensional tori T, or a disjoint union of two-dimensional two-tori 2T.When the magnetic field increases for 4 γ = to 4.5 γ = , the corresponding Poincaré surfaces of section show that the system go through a transition from regularity to chaotical dynamics.At 5 γ = (high magnetic field), all Poincaré surfaces of section show a strong irregular motion, the dynamic is totally chaotic.Figures 8(a)-(h) show the Poincaré surfaces of section for 0 β = (no electric field) and 4 γ = , the system still has a regular motion, so we can say that the

Figure 9 (
Figure 9(a) represents the Poincaré surfaces of section for the bifurcation 2 2 T S φ → → , when the system evolves from domain 2 to domain 1 on the bi- furcation diagram where the values of the integrals of motion are 4.685 h = and 2.221 f = (blue), 2.524 (black), 2.793 (red), 3.363 (orange), 3.467 (black dot).Figure 9(b) represents the bifurcation 2 2 T S φ

Figure 9
, −1.847 (blue), −2.4 (red), −2.587 (black dot).The last values which correspond to black dots are located for ( ) , h f B ∈ , exactly on the curve of the bifurcation diagram, these black dots present periodic solutions in the form of two isolated circles, because the intersection of a circle with the cutting surface gives a dots.Figures 9(c) represents the bifurcation ( ) 2T S S S T → × ∧ → , from domain 3 to domain 4 where 8.873 h = and 1.7 f = (black), 2 (blue), 3.096 (red).Figures 9(d) represent the bifurcation ( ) 2T S S S T → × ∧ → , from Int.J. Modern Nonlinear Theory and Application

Figure 9 .
Figure 9. Poincaré surfaces of section on the bifurcation diagram for ( ) , h f B ∈

Figure 10 .
Figure 10.Poincaré surfaces of section for different values of, h, f on the bifurcation diagram B′ and 2 λ < .
to consider separately, for some values of the constants α, β and γ, two cases 2 λ ≥ and 2 λ < , because they generate different structures of the topology of the real level sets A  and the corresponding Poincaré map.3.1.1.For the Case

Table 1 .
The Topological type of A  , and the admissible set for

Table 1 .
Using the formulae (8) and the condition that ( )4

Table 2 .
The topological type of A  , and the admissible set for