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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.

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 [

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 [

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 [

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 m e = ℏ = e = a 0 = 1 ) can be written as:

H = 1 2 p 2 − 1 r + α ( x 2 + y 2 + λ 2 z 2 ) + γ ( x 2 + y 2 ) + β z (1)

where r 2 = x 2 + y 2 + z 2 , p 2 = p x 2 + p y 2 + p z 2 , and 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 B 0 = 2.35 × 10 5 T , controls the quadratic Zeeman effect; β, measured in units of electric field F 0 = 5.14 × 10 11 V / cm , controls the Stark effect; and the a-dimensional number λ controls the anharmonicity associated to the van der Waals interaction [

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 common-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.

The regularized Hamiltonian is obtained by transforming Equation (1) to semiparabolic coordinates and the corresponding momenta, namely,

x = u v cos ( φ ) , y = u v sin ( φ ) , z = u 2 − v 2 2 , p x = v cos ( φ ) u 2 + v 2 v cos ( φ ) u 2 + v 2 p u + u cos ( φ ) u 2 + v 2 p v − sin ( φ ) u v p φ , p y = v sin ( φ ) u 2 + v 2 v sin ( φ ) u 2 + v 2 p u + u sin ( φ ) u 2 + v 2 p v + cos ( φ ) u v p φ , p z = u u 2 + v 2 u u 2 + v 2 p u − v u 2 + v 2 p v . (2)

Note that the equations of motion associated with the Hamiltonian (1) have a collision singularity at r = 0 , which can be removed with the following change in the independent variable

d t d τ = r = ( u 2 + v 2 )

After such transformation, the regularized Hamiltonian reads

H ˜ ( u , v ) = 2 = 1 2 ( p u 2 + p v 2 ) + p φ 2 2 ( 1 u 2 + 1 v 2 ) − H ( u 2 + v 2 ) + α λ 2 4 ( u 6 + v 6 ) + β 2 ( u 4 − v 4 ) + ( α + γ − α λ 2 4 ) ( u 4 v 2 + u 2 v 4 ) (3)

In this equation, p φ is a conserved quantity, represents the value of the cyclic integral associated to the cyclic coordinate φ, and we put p φ = l , 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. [

The equations of motion associated with H ˜ ( u , v ) in the new time τ are, therefore,

u ˙ = p u , p ˙ u = 2 h u − 2 β u 3 − 6 α λ 2 4 u 5 + l 2 u 3 − ( α + γ − α λ 2 4 ) ( 4 u 3 v 2 + 2 u v 4 ) , v ˙ = p v , p ˙ v = 2 h u + 2 β v 3 − 6 α λ 2 4 v 5 + l 2 v 3 − ( α + γ − α λ 2 4 ) ( 4 u 2 v 3 + 2 u 4 v ) . (4)

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, p φ = l = 0 .

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 two dimensional regularized Hamiltonian (3) is separable if the relation (5) is verified

α + γ − α λ 2 4 = 0 ⇔ λ = ± 2 1 + γ α (5)

in this case the second integral of motion reads

F = f = ( u 2 p v 2 − v 2 p u 2 ) 2 ( u 2 + v 2 ) + ( α + γ ) ( v 6 u 2 − u 6 v 2 ) + 2 v 2 u 2 + v 2 − β ( v 4 u 2 + u 4 v 2 ) 2 ( u 2 + v 2 ) (6)

F = f = z p x 2 − x p x p z + 1 − z x 2 + z 2 − 2 ( α + γ ) x 2 z − β 2 x 2 (7)

It is easy to verify that Equation (2) becomes

x = u v , z = u 2 − v 2 2 , p x = v u 2 + v 2 v u 2 + v 2 g 1 ( u ) + u u 2 + v 2 g 2 ( v ) , p z = u u 2 + v 2 u u 2 + v 2 g 1 ( u ) − v u 2 + v 2 g 2 ( v ) . (8)

where

g 1 ( u ) = p u 2 = 2 ( 2 − f + h u 2 − β 2 u 4 − ( α + γ ) u 6 ) (9)

g 2 ( v ) = p v 2 = 2 ( f + h v 2 + β 2 v 4 − ( α + γ ) v 6 ) (10)

in these conditions, the equations of motion are

u ˙ = p u = ± g 1 ( u ) , v ˙ = p v = ± g 2 ( v ) . (11)

H = h and F = f are the first integrals of motion, functions of ( u , v , p u , p v ) which are constant along the solutions of Equation (11). The system is called (meromorphically) Liouville integrable or completely integrable.

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:

A ℝ = { ( u , v , p u , p v ) ∈ R 4 : H = h , F = f } ⊂ R 4

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

( u , v , p u , p v ) → ( H , F )

It appears, according to several works carried out in this sense (for details, see Ref. [

B = B 1 ∪ B 2

B = { ( h , f , α , β , γ ) ∈ R 5 : d i s c r ( g 1 ( u ) ) = 0 } ∪ { ( h , f , α , β , γ ) ∈ R 5 : d i s c r ( g 2 ( v ) ) = 0 }

It is convenient to consider separately, for some values of the constants α, β and γ, two cases | λ | ≥ 2 and | λ | < 2 , because they generate different structures of the topology of the real level sets A ℝ and the corresponding Poincaré map.

B 1 = { ( h , f ) ∈ R 2 : f = 2 , f = 1348 675 − h 15 ± 2 675 1 + 45 h + 675 h 2 + 3375 h 3 } ⊂ R 2 B 2 = { ( h , f ) ∈ R 2 : f = 0 , f = − 2 675 − h 15 ± 2 675 1 + 45 h + 675 h 2 + 3375 h 3 } ⊂ R 2

(14)

The set { ℝ 3 \ B } ∩ { | λ | ≥ 2 } consists of 10 connected components (as it is shown in

Theorem 1. The set { ℝ 3 \ B } ∩ { | λ | ≥ 2 } consists of ten connected components. The sections of these components with the plane { λ = c o n s t . } are shown on

Proof. Consider the complexified system

A ⊄ = { ( x , z , p x , p z , r ) ∈ ⊄ 5 : H = h = c o n s t . , F = f = c o n s t . , r 2 = x 2 + z 2 , r ≠ 0 }

consider also the hyperelliptic curves Γ 1 : { ω 1 2 = g 1 ( u ) } and Γ 2 : { ω 2 2 = g 2 ( v ) }

and the corresponding Riemann surfaces R 1 and R 2 of the same genus j 1 = j 2 = 2 . We obtain the explicit solutions of the initial problem (11) by solving the Jacobi inversion problem [

Domain | Roots of g 1 ( u ) | Roots of g 2 ( v ) | Accessible region | A ℝ |
---|---|---|---|---|

1 | 0 | v 1 < 0 < v 2 | ∅ × [ v 1 , v 2 ] | ∅ |

2 | u 1 < u 2 < 0 < u 3 < u 4 | v 1 < 0 < v 2 | [ u 1 , u 2 ] ∪ [ u 3 , u 4 ] × [ v 1 , v 2 ] | 2T |

3 | u 1 < u 2 < 0 < u 3 < u 4 | v 1 < 0 < v 2 | [ u 1 , u 2 ] ∪ [ u 3 , u 4 ] × [ v 1 , v 2 ] | 2T |

4 | u 1 < 0 < u 2 | v 1 < 0 < v 2 | [ u 1 , u 2 ] × [ v 1 , v 2 ] | T |

5 | u 1 < 0 < u 2 | v 1 < 0 < v 2 | [ u 1 , u 2 ] × [ v 1 , v 2 ] | T |

6 | u 1 < 0 < u 2 | v 1 < 0 < v 2 | [ u 1 , u 2 ] × [ v 1 , v 2 ] | T |

7 | u 1 < 0 < u 2 | v 1 < 0 < v 2 | [ u 1 , u 2 ] × [ v 1 , v 2 ] | T |

8 | u 1 < 0 < u 2 | v 1 < v 2 < 0 < v 3 < v 4 | [ u 1 , u 2 ] × [ v 1 , v 2 ] ∪ [ v 3 , v 4 ] | 2T |

9 | u 1 < 0 < u 2 | v 1 < v 2 < 0 < v 3 < v 4 | [ u 1 , u 2 ] × [ v 1 , v 2 ] ∪ [ v 3 , v 4 ] | 2T |

10 | u 1 < 0 < u 2 | 0 | [ u 1 , u 2 ] × ∅ | ∅ |

Thus x , z , p x , p z can be expressed in terms of hyperelliptic functions living in the Jacobi variety Γ = Γ 1 ⊗ Γ 2 (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 Γ 1 ⊗ Γ 2 there correspond two values of ( x , z , p x , p z ) . Thus we define the natural projection

π : A ⊄ → Γ 1 ⊗ Γ 2

corresponding to the involution i

i : ( x , z , p x , p z ) → ( x , z , − p x , − p z )

the real level sets A ℝ = R e ( A ⊄ ) is the set of fixed points of the complex conjugation on A ⊄ :

η : ( x , z , p x , p z ) → ( x ¯ , z ¯ , p ¯ x , p ¯ z ) (15)

Consider also the natural projection ξ on the Riemann surface R = R 1 ⊗ R 2 given in u, v coordinates by:

ξ : ( u , v ) → ( u ¯ , v ¯ )

It induces an involution on the Jacobi variety and hence on A ⊄ by the natural 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 ℝ it is enough to study the projection:

π : A ⊄ → J a c ( R ) = Γ 1 ⊗ Γ 2

Definition. A connected component of the set of fixed points of η on the curves Γ 1 and Γ 2 is called an oval.

To determine the ovals of Γ 1 and Γ 2 it suffices to study the real roots of the polynomials g 1 ( u ) and g 2 ( v ) for different values of h and f as shown in

B ′ 1 = { ( h , f ) ∈ R 2 : f = 2 , f = 484 243 + h 9 ± 2 243 1 − 27 h + 243 h 2 − 729 h 3 } ⊂ R 2

B ′ 2 = { ( h , f ) ∈ R 2 : f = 0 , f = − 2 243 + h 9 ± 2 243 1 − 27 h + 243 h 2 − 729 h 3 } ⊂ R 2

Theorem 2. The set { ℝ 3 \ B ′ } ∩ { | λ | < 2 } consists of ten connected components. The sections of these components with the plane { λ = c o n s t . } are shown

on

Suppose now that the constants h, f are changed in such a way that ( h , f ) passes through the bifurcation diagram B. Then the topological type of A ℝ may change and the bifurcation of Liouville tori takes place.

For the first case | λ | ≥ 2 , According to Fomenko surgery on Liouville tori [

Domain | Accessible region | A ℝ |
---|---|---|

1’ | ] − ∞ , u 1 ] ∪ [ u 2 , + ∞ [ × ] − ∞ , + ∞ [ | 2 R 2 |

2’ | ] − ∞ , u 1 ] ∪ [ u 2 , + ∞ [ × ] − ∞ , + ∞ [ | 2 R 2 |

3’ | ] − ∞ , u 1 ] ∪ [ u 2 , + ∞ [ × ] − ∞ , v 1 ] ∪ [ v 2 , v 3 ] ∪ [ v 4 , + ∞ [ | 2 C + 4 R 2 |

4’ | ] − ∞ , u 1 ] ∪ [ u 2 , u 3 ] ∪ [ u 4 , + ∞ [ × ] − ∞ , v 1 ] ∪ [ v 2 , v 3 ] ∪ [ v 4 , + ∞ [ | T + 4 C + 4 R 2 |

5’ | ] − ∞ , u 1 ] ∪ [ u 2 , u 3 ] ∪ [ u 4 , + ∞ [ × ] − ∞ , + ∞ [ | C + 2 R 2 |
---|---|---|

6’ | ] − ∞ , + ∞ [ × ] − ∞ , + ∞ [ | R 2 |

7’ | ] − ∞ , + ∞ [ × ] − ∞ , v 1 ] ∪ [ v 2 , v 3 ] ∪ [ v 4 , + ∞ [ | C + 2 R 2 |

8’ | ] − ∞ , u 1 ] ∪ [ u 2 , u 3 ] ∪ [ u 4 , + ∞ [ × ] − ∞ , v 1 ] ∪ [ v 2 , + ∞ [ | 2 C + 4 R 2 |

9’ | ] − ∞ , + ∞ [ × ] − ∞ , v 1 ] ∪ [ v 2 , + ∞ [ | 2 R 2 |

10’ | ] − ∞ , + ∞ [ × ] − ∞ , v 1 ] ∪ [ v 2 , + ∞ [ | 2 R 2 |

6 → 1 6 → 10 | 2 → 1 9 → 10 | 2 → 5 , 9 → 10 3 → 4 , 8 → 4 |
---|---|---|

T → ϕ | 2 T → ϕ | 2 T → T |

As shown in

1) Bifurcation T → S → ∅ : The two dimensional tori T is contracted to the circle S corresponding to the periodic solution, and then vanishes.

2) Bifurcation 2 T → S × ( S ∧ S ) → T : The two dimensional two tori 2T merge into two dimensional tori T by passing through the complex S × ( S ∧ S ) where ( S ∧ S ) is a union of two circles having exactly one common point.

3) Bifurcation 2 T → 2 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 | λ | < 2 , The Fomenko classification of bifurcation of Liouville tori [

To prove that, it suffices to look at the bifurcation of roots of the polynomials g 1 ( u ) and g 2 ( v ) as shown in

When the bifurcation of Liouville tori takes place, the level set A ℝ becomes completely degenerate. Then we can have exceptional families of periodic solutions. It is seen from

[ v 1 , v 2 ] and u = u 1 = u 2 is equal to the double root of the polynomial g 1 ( u ) (see

Then we obtain from Equation (8) the following parameterization of fixed periodic solution:

4 ′ → 3 ′ 4 ′ → 8 ′ | 4 ′ → 5 ′ 4 ′ → 7 ′ | 3 ′ → 2 ′ 8 ′ → 9 ′ | 5 ′ → 6 ′ 7 ′ → 6 ′ |
---|---|---|---|

T + 4 C + 4 R 2 → 2 C + 4 R 2 | T + 4 C + 4 R 2 → C + 2 R 2 | 2 C + 4 R 2 → 2 R 2 | C + 2 R 2 → 2 R 2 |

Curves | Accessible region | A ℝ |
---|---|---|

C 0 | { u 1 = u 2 } × [ v 1 , v 2 ] | S |

C 1 | { u 1 = u 2 } ∪ { u 3 = u 4 } × [ v 1 , v 2 ] | 2S |

C ′ 0 | [ u 1 , u 2 ] × { v 1 = v 2 } | S |

C ′ 1 | [ u 1 , u 2 ] × { v 1 = v 2 } ∪ { v 3 = v 4 } | 2S |

{ x = 0 z = − v 2 2 and { p x = 0 p z = ± g 2 ( v ) v (16)

to derive the differential equation satisfied by v, we use

d v d τ = p v

we obtain from (10)

d v g 2 ( v ) = ± d τ

thus v ( τ ) and hence z ( τ ) can be expressed in terms of hyperelliptic functions.

On the curve C 0 , the second integral of motion f is equal to 2, as well as the characteristic polynomial g 2 ( v ) = p v 2 = 2 ( 2 + h v 2 + v 4 − 5 v 6 ) depends only on h. Taking this variable change:

R = v 2 ⇒ d R 2 R = d v

d τ = ± d R 2 2 R ( 2 + h R − 5 R 3 + R 2 ) = ± d R 2 2 P 2 ( R ) (17)

the polynomial P 2 ( R ) has four distinct roots

R 0 = 0 , R 1 = D 1 3 30 + 2 ( h + 1 15 ) D 1 3 + 1 15 , R 2 = a 0 + i b 0 et R 3 = a 0 − i b 0

where

a 0 = − D 1 3 60 − ( h + 1 15 ) D 1 3 + 1 15 , b 0 = 3 2 [ − D 1 3 60 − ( h + 1 15 ) D 1 3 ]

D = 180 h + 5408 + 60 − 60 h 3 − 3 h 2 + 540 h + 8124

By an inversion of the elliptic integral Equation (17), one explicitly obtains the expression the periodic solution v ( τ ) :

∮ 0 τ d τ = 1 2 2 ∫ R 0 R d R ( R − R 0 ) ( R − R 1 ) ( R − R 2 ) ( R − R 3 ) (18)

τ = n 0 C n − 1 ( cos φ , k 0 )

where

n 0 = 1 2 2 A 0 B 0 , k 0 = R 1 2 − ( A 0 − B 0 ) 2 4 A 0 B 0 , A 0 2 = R 1 2 + a 0 2 , B 0 2 = b 0 2 + a 0 2

C n − 1 ( cos φ , k 0 ) = F ( φ , k 0 ) being the incomplete elliptic integral of first kind. The expression of periodic solution v ( τ ) is given by solving the Jacobi inversion problem:

τ = ∫ τ d t = ∫ d v g 2 ( v ) ⇒ v ( τ ) = ± R 1 B 0 { C n ( 2 t 2 A 0 B 0 ) − 1 } ( A 0 + B 0 ) + ( A 0 − B 0 ) C n ( 2 t 2 A 0 B 0 )

The period T v associated with the solution v ( t ) is obtained by calculating the elliptic integral Equation (17) over the totality of the admissible oval for v:

T v = ∮ d τ = 1 2 2 ∮ d R P 2 ( R ) = 2 2 2 ∫ R 0 R 1 d p y R ( 2 + h R − 5 R 3 + R 2 )

It’s sufficient to replace in Equation (18) the upper bound in the integral by R 1 and multiply by 2, in these conditions cos φ = − 1 , we obtain a period dependent only on the energy h:

T v ( h ) = 2 n 0 C n − 1 ( − 1 , k 0 ) = 2 n 0 F ( π , k 0 )

where C n − 1 ( − 1 , k 0 ) = F ( π , k 0 ) is the complete elliptic integral of the first kind.

In the same way, for the periodic solution on the curve C 1 the parameter v

takes values in the admissible oval [ v 1 , v 2 ] , the double roots of the polynomial g 1 ( u ) are equal to u 1 = u 2 = − m and u 3 = u 4 = m (see

The values of the first integrals H and F on the curve C 1 are related by

f = 1348 675 − h 15 + 2 675 1 + 45 h + 675 h 2 + 3375 h 3

From Equation (8) we obtain the following parameterization of fixed periodic solution

{ x = ± m v z = m 2 − v 2 2 and { p x = ± m m 2 + v 2 g 2 ( v ) p z = ± v m 2 + v 2 g 2 ( v ) (19)

v ( t ) = ± R 1 B 1 { C n ( 2 t 2 A 1 B 1 ) − 1 } ( A 1 + B 1 ) + ( A 1 − B 1 ) C n ( 2 t 2 A 1 B 1 )

The period of v ( t ) is

T v ( h ) = 2 n 1 C n − 1 ( − 1 , k 1 ) = 2 n 1 F ( π , k 1 )

where

n 1 = 1 2 2 A 1 B 1 , k 1 = R 1 2 − ( A 1 − B 1 ) 2 4 A 1 B 1 , A 1 2 = R 1 2 + a 1 2 , B 1 2 = b 1 2 + a 1 2 ,

R 0 = 0 , R 1 = 1 15 ( C 1 3 + 15 h + 1 C 1 3 + 1 ) , a 1 = − 1 30 ( C 1 3 + 15 h + 1 C 1 3 − 2 ) ,

b 1 = 3 30 ( C 1 3 − 15 h + 1 C 1 3 ) ,

C = 675 ( 1 + ( 15 h + 1 ) 3 ) + 911249 + 20503080 h + 307546200 h 2 + 1537731000 h 3 + 911250 ( 15 h + 1 ) 3

From Equation (19), the Cartesian equation for this periodic solution in the ( x , z ) plan is

z ( x ) = m 2 2 − x 2 2 m 2

In this case, it is seen from

v ( t ) = ± R 1 B ′ 0 { C n ( 2 t 2 A ′ 0 B ′ 0 ) − 1 } ( A ′ 0 + B ′ 0 ) + ( A ′ 0 − B ′ 0 ) C n ( 2 t 2 A ′ 0 B ′ 0 )

The associated period is

T v ( h ) = 2 n ′ 0 C n − 1 ( − 1 , k ′ 0 ) = 2 n ′ 0 F ( π , k ′ 0 )

where

n ′ 0 = 1 2 2 A ′ 0 B ′ 0 , k ′ 0 = R 1 2 − ( A ′ 0 − B ′ 0 ) 2 4 A ′ 0 B ′ 0 , A ′ 0 2 = R 1 2 + a ′ 0 2 , B ′ 0 2 = b ′ 0 2 + a ′ 0 2 , R 0 = 0 ,

R 1 = D 0 1 3 18 − 2 ( h − 1 9 ) D 0 1 3 − 1 9 , a 0 = − D 0 1 3 36 + ( h − 1 9 ) D 0 1 3 − 1 9 , b 0 = 3 2 ( D 0 1 3 18 + 2 ( h − 1 9 ) D 0 1 3 ) ,

D 0 = 108 h − 1952 + 36 36 h 3 − 3 h 2 − 324 h + 2940

Curves | Accessible region | A ℝ |
---|---|---|

L 0 | ] − ∞ , u 1 ] ∪ { u 2 = u 3 } ∪ [ u 4 , + ∞ [ × ] − ∞ , v 1 ] ∪ [ v 2 , v 3 ] ∪ [ v 4 , + ∞ [ | S + 2 R v + 2 C + 4 R 2 |

L ′ 0 | ] − ∞ , u 1 ] ∪ [ u 2 , u 3 ] ∪ [ u 4 , + ∞ [ × ] − ∞ , v 1 ] ∪ { v 2 = v 3 } ∪ [ v 4 , + ∞ [ | S + 2 R u + 2 C + 4 R 2 |

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 [

The Poincaré surfaces of section are plotted in the plane ( q 1 , p 1 ) = ( u , p u ) .

Figures 7(a)-(h) represent Poincaré surfaces of section for β = 2 (electric field) and values of γ (magnetic field) near the integrable case γ = 4 (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 ℝ is diffeomorphic to a two-dimensional tori T, or a disjoint union of two-dimensional two-tori 2T. When the magnetic field increases for γ = 4 to γ = 4.5 , the corresponding Poincaré surfaces of section show that the system go through a transition from regularity to chaotical dynamics. At γ = 5 (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 β = 0 (no electric field) and γ = 4 , 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.

domain 8 to domain 4 where h = 7.155 and f = − 0.738 (black), 0 (blue), 1.112 (orange). Figures 9(e) represents the bifurcation 2 T → S × ( S ∧ S ) → T , from domain 9 to domain 7 where h = 3.021 and f = − 0.569 (black), (red), 0.305 (blue). Figures 9(f) represents the bifurcation 2 T → S × ( S ∧ S ) → T , from domain 2 to domain 5 where h = 3.88 and f = 1.616 (black), 2 (red), 2.625 (blue). Figures 9(g) represents the bifurcation T → S → ϕ , from domain 6 to domain 1 where h = − 4.067 and f = 1 (black), 1.437 (blue), 1.891 (red), 2 (black dot). Figures 9(h) represents the bifurcation T → S → ϕ , from domain 6 to domain 10 where the values of integrals of motion are h = − 4.067 and f = 1 (black), 0.5 (blue), 0.2 (red), 0 (black dot). The black dots correspond to the periodic solutions in the form of a isolated circles.

For the case | λ | < 2 , the invariant level sets contain a non-compact component (cylinder C or plane R 2 ), only in the domain 4’ on the bifurcation diagram B ′ we find a single tori.

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

different results obtained show the capacity of the method used to provide precise information on this Hamiltonian system. The very important question that we have studied is the topological analysis of the real invariant manifolds A ℝ = { H = h , F = f } of the system. Fomenko’s theory on surgery and bifurcations of the Liouville tori has been combined with that of the algebraic structure to give a rigorous and detailed description of the topology of invariant manifolds A ℝ . For non-critical values of H and F, and for some values of system parameters, we have distinguished two different cases | λ | ≥ 2 where A ℝ contains torus or is empty. On the other hand, for the case | λ | < 2 , it consists of a torus, or a cylinder or a real plane. Indeed, for the first case, all the components of A ℝ are compact, while for the second case, they are not compact. Similarly, we have found that Fomenko’s theory of the bifurcations of the Liouville tori is applicable to the first case, whereas for the second case, the bifurcations of the tori that appear cannot be described by this theory. It needs to be due to the fact that the Fomenko classification theorem on the bifurcations of the Liouville tori is only valid in the case where the invariant manifold A ℝ is formed only of compact components. We have also shown how the periodic orbits can be found, how the period of the solutions is determined, and in what ways explicit formulas can be established. Finally, we have also numerically illustrated the generic bifurcations of the Liouville tori and the regularity-chaos transition when one of the control parameters varies. The numerical results show that only the magnetic interaction has affected the dynamic behavior of the Hydrogen atom in parallel magnetic and electric fields. When the magnetic field is weak, the system is totally regular. When the magnetic field is increasing, we observe a transition from regularity to chaotical dynamics. For high magnetic field, the dynamic is totally chaotic.

Kharbach, J., Chatar, W., Benkhali, M., Rezzouk, A. and Ouazzani-Jamil, M. (2018) On the Regularity and Chaos of the Hydrogen Atom Subjected to External Fields. International Journal of Modern Nonlinear Theory and Application, 7, 56-76. https://doi.org/10.4236/ijmnta.2018.72005