Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices

Zaitao Liang; Shengjun Li; Xin Li

doi:10.1515/anona-2022-0287

Article Open Access

Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices

Zaitao Liang , Shengjun Li and Xin Li

Published/Copyright: February 17, 2023

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From the journal Advances in Nonlinear Analysis Volume 12 Issue 1

Abstract

In this article, we consider a four-dimensional singular differential system that can describe the dynamics of configurations bearing a small number of vortices in atomic Bose-Einstein condensates. On the basis of the topological degree theory and some analysis methods, we prove that such a system has two distinct families of periodic solutions and two distinct families of quasi-periodic solutions. Some results in the literature are generalized and improved.

Keywords: periodic solutions; quasi-periodic solutions; singular differential system; topological degree theory

MSC 2010: 34C25; 34A34; 37N20

1 Introduction and main results

In this article, we consider the following four-dimensional singular differential system:

(1.1) x 1 ′ = − a ( t ) y 1 + f ( t , r ) y 1 − y 2 r , y 1 ′ = a ( t ) x 1 − f ( t , r ) x 1 − x 2 r , x 2 ′ = a ( t ) y 2 − f ( t , r ) y 2 − y 1 r , y 2 ′ = − a ( t ) x 2 + f ( t , r ) x 2 − x 1 r ,

where r = ( x 1 − x 2 ) 2 + ( y 1 − y 2 ) 2 , a ∈ C ( R / T Z , R + ) , T > 0 , and the nonlinearity f ∈ C ( ( R / T Z ) × R + , R + ) may be singular at the origin, and

lim r → 0 + f ( t , r ) = + ∞ , uniformly in t ∈ R .

The system (1.1) can describe the dynamics of configurations bearing a small number of vortices in atomic Bose-Einstein condensates. The research of vortices is an important topic in nonlinear science [41]. Vortices are persistent circulating flow patterns that appear in a lot of different mathematical and physical contexts, such as the fluid mechanics [4], the nonlinear optics [10,11,27], and the physics of superfluids, including the Bose-Einstein condensates [2,12,13,25,26,39,42,43,51]. In the aforementioned equation, the Bose-Einstein condensates were shown to form a pristine setting in which a great deal of the exciting nonlinear dynamical properties of single- and multi-charge vortices, as well as of vortex lattices and vortex crystals, can be theoretically researched and experimentally observed. During the past two decades, the literature of the study of dynamics of vortices in Bose-Einstein condensates is vast, on both the mathematical and physical sides. We just refer the reader to [9,14,28–32,44,52,53] and the references therein for a more detailed introduction.

Meanwhile, numerical studies have shown that the motion of vortices evolving under the Gross-Pitaevskii equation is mimicked by the motion arising in the ordinary differential equation dynamics [33,40,45,46]. Such ordinary differential systems have allowed for the prediction of many different dynamical phenomena in the Gross-Pitaevskii equation. Naturally, the dynamics of such systems have been gaining wide attention from scholars, see [24,48–50] and the references therein. For example, in [48], Torres et al. considered an ordinary differential system describing the vortices’ motion. They mainly considered the dynamics of interacting vortices trapped in the harmonic trap. The vortex dynamics is driven by density and phase gradients of the background, in which the coherent structure is placed. In view of the effect of vortex precession, which induced by the harmonic trap [37,38] and vortex–vortex interactions [26,39], they illustrated that the dynamics of a small cluster of interacting vortices with different charges can be described by the following differential systems:

x i ′ = − a ( t ) L i y i − b ∑ j ≠ i L j y i − y j r i j α , y i ′ = a ( t ) L i x i + b ∑ j ≠ i L j x i − x j r i j α , i = 1 , 2 , … , N ,

where ( x i , y i ) are the coordinates, the trap coefficient a ( t ) is a positive function and may be periodic, b and α are two positive constants, L i is the charge of vortex i , and

r i j = ( x i − x j ) 2 + ( y i − y j ) 2 .

Torres et al. in [48] mainly considered the dynamics of two vortices ( N = 2 ), for both the case of opposite charges and the case of same charges. They found that the case of same charge vortices can be analytically tackled in a closed form, upon consideration of the relevant equations in the center of mass frame. The motion of the vortices may be periodic or quasi-periodic, but is always analytically tractable. For both cases of oppositely and same charged vortices, they also researched a situation in which the trap frequency is time dependent. In the case of opposite charges, the use of the center of mass frame is still relevant; however, the dynamical picture is more complicated: in principle, the relevant evolution may not only be periodic or quasi-periodic, but it can also be chaotic.

In fact, for two vortices with opposite charges, with a simple rescaling, it can be assumed that L 1 = − L 2 = 1 without loss of generality. Thus, the motion of two opposite charge vortices is ruled by the following four-dimensional singular differential system:

(1.2) x 1 ′ = − a ( t ) y 1 + b y 1 − y 2 r α , y 1 ′ = a ( t ) x 1 − b x 1 − x 2 r α , x 2 ′ = a ( t ) y 2 − b y 2 − y 1 r α , y 2 ′ = − a ( t ) x 2 + b x 2 − x 1 r α ,

where r = r 12 . On the basis of the Krasnoselskii fixed point theorem, Torres et al. in [48] obtained the following result.

Theorem 1.1

[48, Theorem 4.4] Let a ( t ) = 1 + ε sin ( p t ) with p > 2 . Then there exists some N 0 > 0 such that the system (1.2) has a mT-periodic solution for every integer m ≥ N 0 . Moreover, the system (1.2) has a family of quasi-periodic solutions.

Notice that the authors in [48] are only concerned about the existence of one family of periodic and quasi-periodic solutions for a special four-dimensional singular differential system (1.2). Motivated by this, in this article, we will continue this topic and study the existence and multiplicity of periodic and quasi-periodic solutions for a more general four-dimensional singular differential system (1.1). By using the topological degree theory and some analysis methods, we obtain the following two results.

Theorem 1.2

Assume that

(1.3) a ¯ = 1 T ∫ 0 T a ( t ) d t < π T ,

there exist positive constants l , q , β , and a continuous function g ( r ) such that

(1.4) f ( t , r ) ≥ l a ( t ) r q , ∀ r ∈ ( 0 , + ∞ )

and

(1.5) 0 < f ( t , r ) ≤ g ( r ) , ∀ ( t , r ) ∈ [ 0 , T ] × [ β , + ∞ ) ,

where g ( r ) > 0 is non-increasing in r ∈ [ β , + ∞ ) . Then there exists m ∗ ≥ 1 such that, for every integer m ≥ m ∗ , the system (1.1) has two distinct periodic solutions ( x 1 m , y 1 m , x 2 m , y 2 m ) with minimal period mT.

Theorem 1.3

Assume that (1.3), (1.4), and (1.5) hold. Then there exists ϑ ¯ ∗ > 0 such that, for every number ϑ with 0 < ϑ < ϑ ¯ ∗ not commensurable with 2 π T , the system (1.1) has two distinct quasi-periodic solutions ( x 1 ϑ , y 1 ϑ , x 2 ϑ , y 2 ϑ ) of the frequencies ϑ 1 = 2 π T , ϑ 2 = ϑ , where ϑ will be defined in Section 2.

As an application, we consider the system (1.2). The system (1.2) can be regarded as a special from of the system (1.1), where

f ( t , r ) = b r α − 1 .

If α > 1 , we choose positive constants l and q with 0 < l ≤ b a ∗ and q = α − 1 , notice that

f ( t , r ) = b r α − 1 ≥ l a ( t ) r q , ∀ r ∈ ( 0 , + ∞ )

and

f ( t , r ) = b r α − 1 ≤ 1 = g ( r ) , ∀ r ∈ b 1 α − 1 , + ∞ ,

where a ∗ = max t ∈ [ 0 , T ] a ( t ) . That is, (1.4) and (1.5) hold for the equation (1.2) with α > 1 . Therefore, Theorems 1.2 and 1.3 can be directly applied to equation (1.2) with α > 1 , as the following corollary shows.

Corollary 1.4

Assume that α > 1 , b > 0 , a ∈ C ( R / T Z , R + ) with (1.3) hold. Then the system (1.2) has two distinct families of periodic solutions and two distinct families of quasi-periodic solutions.

For the case α ∈ ( 0 , 1 ] , we notice that r = 0 is not a singularity of f ( t , r ) = b r α − 1 = b r 1 − α , and (1.4) and (1.5) fail to hold. Therefore, Theorems 1.2 and 1.3 cannot be applied to the system (1.2).

Remark 1

If α ∈ ( 0 , 1 ] , similar to [48, Theorem 4.4], we can obtain that there exists m ∗ ≥ 1 such that, for every integer m ≥ m ∗ , the system (1.2) has a m T -periodic solution. Moreover, the system (1.2) has a family of quasi-periodic solutions. Whether the system (1.2) with α ∈ ( 0 , 1 ] has another distinct family of periodic or quasi-periodic solutions is still unknown. We leave this problem open and as a future topic to be considered.

Compared with Theorem 1.1, we not only consider a more general four-dimensional singular differential system but also obtain some results on the existence of two distinct families of periodic and quasi-periodic solutions. Therefore, the results in this article can be seen as the generalization and improvement of Theorem 1.1.

Theorems 1.2 and 1.3 are direct consequences of Theorems 2.1 and 2.2 (in Section 2), which are the main results of the equivalent system of the system (1.1). Moreover, we also analyze the sign of Green’s function associated with the equivalent system in Section 2. A priori estimates of the solutions are established in Section 3, and some connected sets of solutions are obtained in Section 4. The proof of Theorems 2.1 and 2.2 are given in Section 5. Finally, we give our conclusions.

2 Main results of the equivalent system

By using the following new variables:

v 1 = x 1 − x 2 , v 2 = y 1 − y 2 , v 3 = x 1 + x 2 , v 4 = y 1 + y 2 ,

the system (1.1) is transformed into

v 1 ′ = − a ( t ) v 4 , v 2 ′ = a ( t ) v 3 , v 3 ′ = − a ( t ) v 2 + 2 f ( t , r ) v 2 r , v 4 ′ = a ( t ) v 1 − 2 f ( t , r ) v 1 r .

Then by eliminating v 3 and v 4 in the third and fourth equations of the aforementioned system, we obtain the following singular planar differential system:

v 1 ′ a ( t ) ′ = − a ( t ) v 1 + 2 f ( t , r ) v 1 r , v 2 ′ a ( t ) ′ = − a ( t ) v 2 + 2 f ( t , r ) v 2 r ,

that is,

(2.1) u ′ a ( t ) ′ + a ( t ) u = 2 f ( t , ∣ u ∣ ) u ∣ u ∣ , u ∈ R 2 ⧹ { 0 } ,

where u ( t ) = ( v 1 ( t ) , v 2 ( t ) ) and

∣ u ∣ = r = v 1 2 + v 2 2 .

Obviously, the periodic and quasi-periodic solutions of the system (1.1) correspond to the periodic and quasi-periodic solutions of the system (2.1), respectively.

Notice that the system (2.1) is a radially symmetric system. The main feature of such a system is that its solutions are contained in a plane. During the past few years, the existence of periodic solutions for second order radially symmetric systems has been studied by means of the topological degree approach [5,16–22] and the qualitative analysis of Poincaré map with action angle variables [36]. Moreover, some perturbation results were obtained in [23] by introducing a rotational symmetry. According to the ideas presented in the aforementioned literature, we can split the system into its angular and radial components, that is, we introduce the following polar coordinates:

u ( t ) = ( v 1 ( t ) , v 2 ( t ) ) = ( r ( t ) cos θ ( t ) , r ( t ) sin θ ( t ) ) ,

here, θ ( t ) ∈ R , ∀ t ∈ R , then the system (2.1) is transformed into

(2.2) r ′ a ( t ) ′ − r θ ′ 2 a ( t ) + a ( t ) r = 2 f ( t , r ) , d d t r 2 θ ′ a ( t ) = 0 .

Define

μ ( t ) = r 2 ( t ) θ ′ ( t ) ,

which is the angular momentum of u . Then the system (2.2) can be written as the following singular differential equation:

(2.3) r ′ a ( t ) ′ + a ( t ) r = ω 2 a ( t ) r 3 + 2 f ( t , r ) , ω = r 2 θ ′ a ( t ) = μ ( t ) a ( t ) ,

where ω is a constant in time along any solution u .

A solution u : R → R 2 ⧹ { 0 } of the system (2.1) is said to be radially T -periodic if the radial component r ( t ) is T -periodic. In this case, the number

ϑ = θ ( T ) − θ ( 0 ) T

can be interpreted as the average angular speed of u and will be called the rotation number of u . Then, a radial solution u is T -periodic if and only if ϑ is an integer multiple of 2 π T . If the rotation number belongs instead to ( 2 π / T ) Q , then u will not be T -periodic, but a subharmonic. On the other hand, if the modulus r is not constant and ϑ ∉ ( 2 π / T ) Q , then u will not be periodic of any period; instead, it will be quasi-periodic on the two frequencies ϑ 1 = 2 π T , ϑ 2 = ϑ .

By using the topological degree theory [54] and some analysis methods, we obtain the following two differential results for the system (2.1).

Theorem 2.1

Suppose that (1.3), (1.4), and (1.5) hold. Then the following conclusions hold:

There exists m 0 ≥ 1 such that, for every integer m ≥ m 0 , the system (2.1) has a periodic solution u m with minimal period m T , which makes exactly one revolution around the origin in the period time mT. Moreover,
(2.4) lim m → + ∞ ∣ u m ( t ) ∣ = + ∞ , lim m → + ∞ μ m ( t ) = + ∞ , u n i f o r m l y i n t ,
where μ m ( t ) is the angular momentum of the solution u m ;
There exists ϑ ¯ > 0 such that, for every number 0 < ϑ < ϑ ¯ not commensurable with 2 π T , the system (2.1) has a quasi-periodic solution u ϑ ( t ) = r ϑ ( cos θ ϑ , sin θ ϑ ) of the frequencies ϑ 1 = 2 π T , ϑ 2 = ϑ . Moreover,
(2.5) lim ϑ → 0 ∣ u ϑ ( t ) ∣ = + ∞ , lim ϑ → 0 μ ϑ ( t ) = + ∞ , u n i f o r m l y i n t ,
where μ ϑ ( t ) is the angular momentum of the solution u ϑ .

Theorem 2.2

Assume that (1.3), (1.4), and (1.5) hold. Then the following conclusions hold:

There exists m 1 ≥ 1 such that, for every integer m ≥ m 1 , the system (2.1) has a periodic solution u m with minimal period m T , which makes exactly one revolution around the origin in the period time mT. Moreover, if μ m ( t ) is the angular momentum of the solution u m , then
lim m → + ∞ μ m ( t ) = 0 , ∀ t ∈ R .
There exists ϑ ¯ 0 > 0 such that, for every number 0 < ϑ < ϑ ¯ 0 not commensurable with 2 π T , the system (2.1) has a quasi-periodic solution u ϑ ( t ) = r ϑ ( cos θ ϑ , sin θ ϑ ) of the frequencies ϑ 1 = 2 π T , ϑ 2 = ϑ . Moreover, if μ ϑ ( t ) is the angular momentum of the solution u ϑ , then
lim ϑ → 0 μ ϑ ( t ) = 0 , ∀ t ∈ R .

Notice that the periodic and quasi-periodic solutions obtained in Theorem 2.1 rotate around the origin with large angular momentum and large amplitude, and the periodic and quasi-periodic solutions obtained in Theorem 2.2 rotate around the origin with small angular momentum. Therefore, the system (2.1) has two distinct families of periodic solutions and two distinct families of quasi-periodic solutions.

By Theorems 2.1, 2.2 and the following facts:

x 1 = v 1 + v 3 2 , x 2 = − v 1 + v 3 2 , y 1 = v 2 + v 4 2 , y 2 = − v 2 + v 4 2 ,

we can obtain Theorems 1.2 and 1.3 directly.

Moreover, we also analyze the sign of Green’s function of the following equation:

(2.6) r ′ a ( t ) ′ + a ( t ) r = h ( t )

associated with the boundary conditions:

(2.7) r ( 0 ) = r ( T ) , r ′ ( 0 ) = r ′ ( T ) .

Motivated by the ideas in [8, Theorem 5.1], we obtain the following result.

Theorem 2.3

Assume that (1.3) holds. Then the Green’s function G ( t , s ) of the problems (2.6)–(2.7) is positive for all ( t , s ) ∈ [ 0 , T ] 2 , and r is a solution of the problem (2.6)–(2.7) if and only if

r ( t ) = ∫ 0 T G ( t , s ) h ( s ) d s .

Proof

Under the change of time

(2.8) s = τ ( t ) = ∫ 0 t a ( ξ ) d ξ ,

and the problem (2.6)–(2.7) is transformed into

(2.9) R ″ ( s ) + R ( s ) = h ˜ ( s ) , R ( 0 ) = R ( T ˜ ) , R ′ ( 0 ) = R ′ ( T ˜ ) ,

where

R ( s ) = r ( τ − 1 ( s ) ) , h ˜ ( s ) = h ( τ − 1 ( s ) ) a ( τ − 1 ( s ) ) ,

which are T ˜ -periodic with T ˜ = τ ( T ) = T a ¯ .

By [7, Example 2.9], we notice that if T ˜ = T a ¯ < π , i.e., (1.3) holds, then the Green’s function G ˜ ( s , s ˜ ) of the problem (2.9) is positive, ∀ ( s , s ˜ ) ∈ [ 0 , T ˜ ] 2 , and R is a solution of the problem (2.9) if and only if

R ( s ) = ∫ 0 T ˜ G ˜ ( s , s ˜ ) h ( τ − 1 ( s ˜ ) ) a ( τ − 1 ( s ˜ ) ) d s ˜ .

See also [15,47]. Substituting s = τ ( t ) and s ˜ = τ ( s ) into the aforementioned equality, we obtain that r is a solution of the problem (2.6)–(2.7) if and only if

r ( t ) = R ( τ ( t ) ) = ∫ 0 T G ˜ ( τ ( t ) , τ ( s ) ) h ( s ) d s ,

which implies that the Green’s function of the problem (2.6)–(2.7) is equal to

□ G ( t , s ) = G ˜ ( τ ( t ) , τ ( s ) ) > 0 , ∀ ( t , s ) ∈ [ 0 , T ] × [ 0 , T ] .

3 A priori estimates

In order to prove Theorems 2.1 and 2.2 by the topological degree theory, a priori estimates of possible solutions to the following equation

(3.1) r ′ a ( t ) ′ + a ( t ) r = h n λ ( t , r ) + ω 2 a ( t ) r 3 + a ( t ) n , λ ∈ [ 0 , 1 ]

are established in this section, where n ∈ N + ,

h n λ ( t , r ) = 2 λ f n ( t , r ) + ( 1 − λ ) 2 l a ( t ) r q ,

f n : R 2 → R + , f n ( t , r ) = f t , 1 n , for r ≤ 1 n , f ( t , r ) , for r ≥ 1 n .

By Theorem 2.3, we obtain that r is a T -periodic solution of (3.1) if and only if

r ( t ) = ∫ 0 T G ( t , s ) h n λ ( s , r ( s ) ) + ω 2 a ( s ) r 3 ( s ) + a ( s ) n d s = ∫ 0 T G ( t , s ) h n λ ( s , r ( s ) ) + ω 2 a ( s ) r 3 ( s ) d s + 1 n ,

and here, we have used the fact that

∫ 0 T G ( t , s ) a ( s ) d s = 1 .

Lemma 3.1

Suppose that r is a T-periodic solution of equation (3.1), then

r ( t ) ≥ σ ‖ r ‖ , ∀ t ∈ [ 0 , T ] ,

where the norm ‖ ⋅ ‖ is defined as follows:

‖ r ‖ = max t ∈ [ 0 , T ] ∣ r ( t ) ∣

and

σ = G ∗ G ∗ , G ∗ = max 0 ≤ s , t ≤ T G ( t , s ) , G ∗ = min 0 ≤ s , t ≤ T G ( t , s ) .

Proof

If r is a T -periodic solution of equation (3.1), by some estimates, we obtain

□ r ( t ) = ∫ 0 T G ( t , s ) h n λ ( s , r ( s ) ) + ω 2 a ( s ) r 3 ( s ) + a ( s ) n d s ≥ G ∗ ∫ 0 T h n λ ( s , r ( s ) ) + ω 2 a ( s ) r 3 ( s ) + a ( s ) n d s = σ G ∗ ∫ 0 T h n λ ( s , r ( s ) ) + ω 2 a ( s ) r 3 ( s ) + a ( s ) n d s ≥ σ max 0 ≤ t ≤ T ∫ 0 T G ( t , s ) h n λ ( s , r ( s ) ) + ω 2 a ( s ) r 3 ( s ) + a ( s ) n d s = σ ‖ r ‖ .

Lemma 3.2

For every ι > 0 , there exists ω ( ι ) ≥ 1 such that, if ω ≥ ω ( ι ) and r is a T-periodic solution of the equation (3.1), then ‖ r ‖ ≥ ι .

Proof

Conversely, suppose that there exist ι > 0 , sequences { λ n } ⊂ [ 0 , 1 ] , { ω n } ⊂ R and { r n } such that

lim n → ∞ ω n = + ∞ ,

and r n is a T -periodic solution of the following equation:

r n ′ a ( t ) ′ + a ( t ) r n = h n λ n ( t , r n ) + ω n 2 a ( t ) r n 3 + a ( t ) n ,

with ‖ r n ‖ < ι . Multiplying both sides of the aforementioned equation by r n 3 ( t ) and integrating into the interval [ 0 , T ] , we obtain

∫ 0 T a ( t ) r n 4 ( t ) d t = ∫ 0 T 3 r n 2 ( r n ′ ) 2 a ( t ) d t + ∫ 0 T h n λ n ( t , r n ( t ) ) r n 3 ( t ) d t + ∫ 0 T ω n 2 a ( t ) d t + ∫ 0 T a ( t ) r n 3 ( t ) n d t .

By the aforementioned equality, we conclude that

T a ¯ ι 4 > ∫ 0 T a ( t ) r n 4 ( t ) d t > ω n 2 ∫ 0 T a ( t ) d t = T ω n 2 a ¯ ,

which is contrary to

□ lim n → ∞ ω n = + ∞ .

Lemma 3.3

Given w 1 , w 2 with w 2 ≥ w 1 ≥ ω ¯ = ω ( ι ) , if ω ∈ [ w 1 , w 2 ] and r is a T-periodic solution of the equation (3.1), then there exists a positive constant ρ 0 such that

1 ρ 0 < r ( t ) < ρ 0 , ∣ r ′ ( t ) ∣ < ρ 0 , ∀ t ∈ [ 0 , T ] .

Proof

We first claim that any possible T -periodic solution r of the equation (3.1) satisfies

‖ r ‖ < ρ 1 ,

where ρ 1 is a positive constant with

(3.2) ρ 1 > max β σ , 2 g ( β ) G ˜ ∗ + 2 l β q + ω 2 β 3 + 1

and

G ˜ ∗ = max t ∈ [ 0 , T ] ∫ 0 T G ( t , s ) d s .

Assume by contradiction that

(3.3) ‖ r ‖ ≥ ρ 1 .

Then, by (3.2) and Lemma 3.1, we have

(3.4) r ( t ) ≥ σ ‖ r ‖ ≥ σ ρ 1 > β .

Notice that

r ( t ) = ∫ 0 T G ( t , s ) h n λ ( s , r ( s ) ) + ω 2 a ( s ) r 3 ( s ) d s + 1 n ≥ 1 n ,

which leads to

h n λ ( t , r ) = 2 λ f ( t , r ) + ( 1 − λ ) 2 l a ( t ) r q .

Then, by (1.5), (3.2), and (3.4), we obtain

r ( t ) = ∫ 0 T G ( t , s ) h n λ ( s , r ( s ) ) + ω 2 a ( s ) r 3 ( s ) d s + 1 n = ∫ 0 T G ( t , s ) 2 λ f ( s , r ) + ( 1 − λ ) 2 l a ( s ) r q ( s ) + ω 2 a ( s ) r 3 ( s ) d s + 1 n ≤ ∫ 0 T G ( t , s ) 2 g ( r ) + 2 l a ( s ) r q ( s ) + ω 2 a ( s ) r 3 ( s ) d s + 1 n < 2 g ( β ) ∫ 0 T G ( t , s ) d s + 2 l β q + ω 2 β 3 + 1 n ≤ 2 g ( β ) G ˜ ∗ + 2 l β q + ω 2 β 3 + 1 < ρ 1 ,

which contradicts (3.3). Hence, the claim is proved.

Moreover, by (1.4), we have

r ( t ) = ∫ 0 T G ( t , s ) h n λ ( s , r ( s ) ) + ω 2 a ( s ) r 3 ( s ) d s + 1 n = ∫ 0 T G ( t , s ) 2 λ f ( s , r ( s ) ) + ( 1 − λ ) 2 l a ( s ) r q ( s ) + ω 2 a ( s ) r 3 ( s ) d s + 1 n ≥ ∫ 0 T G ( t , s ) 2 λ l a ( s ) r q ( s ) + ( 1 − λ ) 2 l a ( s ) r q ( s ) + ω 2 a ( s ) r 3 ( s ) d s + 1 n

≥ ∫ 0 T G ( t , s ) 2 l a ( s ) ρ 1 q + ω 2 a ( s ) ρ 1 3 d s + 1 n > 2 l ρ 1 q + ω 1 2 ρ 1 3 ≔ ρ 2 .

Next, we verify that there exists a positive constant ρ 3 such that

∣ r ′ ( t ) ∣ ≤ ρ 3 .

Since r is a T -periodic function with r ( 0 ) = r ( T ) , there exists a t 0 ∈ [ 0 , T ] such that r ′ ( t 0 ) = 0 . Integrating the equation (3.1) into the interval [ 0 , T ] , we have

∫ 0 T a ( s ) r d s = ∫ 0 T [ h n λ ( s , r ) + ω 2 a ( s ) r 3 + a ( s ) n ] d s .

Note that

(3.5) ‖ r ′ ‖ = max t ∈ [ 0 , T ] ∣ r ′ ( t ) ∣ ≤ max t ∈ [ 0 , T ] ∣ a ( t ) ∣ max t ∈ [ 0 , T ] r ′ ( t ) a ( t ) = a ∗ max t ∈ [ 0 , T ] ∫ t 0 t r ′ a ( s ) ′ d s = a ∗ max t ∈ [ 0 , T ] ∫ t 0 t − a ( s ) r + h n λ ( s , r ) + ω 2 a ( s ) r 3 + a ( s ) n d s ≤ a ∗ ∫ 0 T a ( s ) r + h n λ ( s , r ) + ω 2 a ( s ) r 3 + a ( s ) n d s = 2 a ∗ ∫ 0 T a ( s ) r ( s ) d s ≤ 2 a ∗ ρ 1 ∫ 0 T a ( s ) d s = 2 T a ¯ a ∗ ρ 1 ≔ ρ 3 .

Then the proof is done if we choose

□ ρ 0 = max ρ 1 , 1 ρ 2 , ρ 3 .

For equation (3.1) with ω = 0 , repeating the proof of the aforementioned lemma, we can obtain the following results.

Corollary 3.4

If r is a T-periodic solution of the equation (3.1) with ω = 0 , then there exists a positive constant ϱ , such that

1 ϱ < r ( t ) < ϱ , ∣ r ′ ( t ) ∣ < ϱ , ∀ t ∈ [ 0 , T ] .

4 Connected sets of solutions

In this section, we prove that there exist some connected sets, whose elements are the solutions of equation (3.1) with λ = 1 .

By the time rescaling (2.8), equation (3.1) is transformed into

(4.1) R ″ + R = h ˜ n λ ( s , R ) a ˜ ( s ) + ω 2 R 3 + 1 n ,

where

R ( s ) = r ( τ − 1 ( s ) ) , a ˜ ( s ) = a ( τ − 1 ( s ) ) , h ˜ n λ ( s , R ( s ) ) = h n λ ( τ − 1 ( s ) , r ( τ − 1 ( s ) ) ) .

Notice that a ˜ and h ˜ are T ˜ -periodic in s . Moreover, we have

(4.2) max s ∈ [ 0 , T ˜ ] ∣ R ( s ) ∣ = max s ∈ [ 0 , T ˜ ] ∣ r ( τ − 1 ( s ) ) ∣ = max t ∈ [ 0 , T ] ∣ r ( t ) ∣ , min s ∈ [ 0 , T ˜ ] ∣ R ( s ) ∣ = min s ∈ [ 0 , T ˜ ] ∣ r ( τ − 1 ( s ) ) ∣ = min t ∈ [ 0 , T ] ∣ r ( t ) ∣ ,

and

(4.3) ∣ R ′ ( s ) ∣ = ∣ r ′ ( τ − 1 ( s ) ) ( τ − 1 ( s ) ) ′ ∣ ≤ max s ∈ [ 0 , T ˜ ] ∣ r ′ ( τ − 1 ( s ) ) ( τ − 1 ( s ) ) ′ ∣ = max t ∈ [ 0 , T ] r ′ ( t ) a ( t ) .

4.1 The case of large angular momentum

Lemma 4.1

Given w 1 and w 2 with w 2 ≥ w 1 ≥ ω ¯ , there exists a continuum C w 1 , w 2 in [ w 1 , w 2 ] × C T 1 , connecting { ω 1 } × C T 1 with { ω 2 } × C T 1 , whose elements ( ω , r ) are the solutions of equation (3.1) with λ = 1 , where C T 1 = C 1 ( R / T Z ) .

Proof

Define the operator

L : D ( L ) ⊂ C 1 [ 0 , T ˜ ] → L 1 ( 0 , T ˜ ) , L ( R ) = R ″ + R ,

where

D ( L ) = { R ∈ W 2 , 1 ( 0 , T ˜ ) : R ( 0 ) = R ( T ˜ ) , R ′ ( 0 ) = R ′ ( T ˜ ) } ,

and the operator

N : [ w 1 , w 2 ] × C 1 [ 0 , T ˜ ] → L 1 ( 0 , T ˜ ) , N ( ω , R ) = 2 f ˜ n ( s , R ) a ˜ ( s ) + ω 2 R 3 + 1 n ,

where f ˜ n ( s , R ( s ) ) = f n ( τ − 1 ( s ) , r ( τ − 1 ( s ) ) ) . Then equation (4.1) with λ = 1 is equivalent to the operator equation:

L ( R ) = N ( ω , R ) ,

that is,

(4.4) R = L − 1 N ( ω , R ) ,

because L is invertible.

By Lemma 3.3, (4.2), and (4.3), if R is a T ˜ -periodic solution of equation (4.1), then we have

1 ρ 0 < R ( s ) < ρ 0 , ∣ R ′ ( s ) ∣ ≤ ρ 3 a ∗ ,

where a ∗ = min t ∈ [ 0 , T ] ∣ a ( t ) ∣ . Choose

ρ ˜ 0 = max ρ 0 , ρ 3 a ∗ ,

and set

Ω = R ∈ C 1 [ 0 , T ˜ ] : 1 ρ ˜ 0 < R ( s ) < ρ ˜ 0 , ∣ R ′ ( s ) ∣ < ρ ˜ 0 , ∀ s ∈ [ 0 , T ˜ ] ,

which is an open and bounded subset of C 1 [ 0 , T ˜ ] . Obviously, equation (4.4) has no solution on [ w 1 , w 2 ] × ∂ Ω .

Next, we verify that the degree is nonzero for some ω ∈ [ w 1 , w 2 ] . Since the degree has to be the same for each λ ∈ [ 0 , 1 ] , we just need to consider equation (4.1) with λ = 0 :

R ″ + R = 2 l R q + ω 2 R 3 + 1 n ,

which is equivalent to the following system:

U ′ = H ( U ) = V , − R + 2 l R q + ω 2 R 3 + 1 n ,

where U = ( R , V ) . It is easy to verify that H has a unique zero ( R 0 , 0 ) and the determinant of Jacobian matrix

∣ J H ( R 0 , 0 ) ∣ > 0 .

Then, by [3, Theorem 1], the Leray-Schauder degree of I − L − 1 N ( ω , ⋅ ) is equal to the Brouwer degree of H ,

deg L ( I − L − 1 N ( ω , ⋅ ) , Ω , 0 ) = deg B H , 1 ρ ˜ 0 , ρ ˜ 0 × ( − ρ ˜ 0 , ρ ˜ 0 ) ≠ 0 .

Then by the global continuation principle of Leray-Schauder [54, Theorem 14.C], we prove that there exists a continuum C w 1 , w 2 in [ w 1 , w 2 ] × C T ˜ 1 , connecting { ω 1 } × C T ˜ 1 with { ω 2 } × C T ˜ 1 , whose elements ( ω , R ) are solutions of equation (4.1) with λ = 1 .

Going back to the original variables, the proof is finished.□

4.2 The case of small angular momentum

Let Y be a Banach space of functions such that

C 1 ( [ 0 , T ˜ ] ) ⊆ Y ⊆ C ( [ 0 , T ˜ ] ) ,

with continuous immersions. Define the operators

L : D ( L ) ⊂ Y → L 1 ( 0 , T ˜ ) , L ( R ) = R ″ + R ,

and

N ω : Ω + → L 1 ( 0 , T ˜ ) , N ω ( R ) = 2 f ˜ n ( s , R ) a ˜ ( s ) + ω 2 R 3 + 1 n ,

where

D ( L ) = { R ∈ W 2 , 1 ( 0 , T ˜ ) : R ( 0 ) = R ( T ˜ ) , R ′ ( 0 ) = R ′ ( T ˜ ) } ,

and

Ω + = { R ∈ Y : min s R ( s ) > 0 } .

Then equation (4.1) with λ = 1 can be converted to the abstract equation:

L ( R ) = N ω ( R ) ,

that is,

(4.5) R = L − 1 N ω ( R ) ,

because L is invertible.

By Corollary 3.4, (4.2), and (4.3), we obtain that if R is a T ˜ -periodic solution of equation (4.5) with ω = 0 , then it satisfies

(4.6) 1 ϱ < R ( s ) < ϱ , ∣ R ′ ( s ) ∣ < ϱ a ∗ , ∀ s ∈ [ 0 , T ˜ ] .

Set

Ω 1 = R ∈ C 1 [ 0 , T ˜ ] : 1 ϱ < R ( s ) < ϱ , ∣ R ′ ( s ) ∣ < ϱ a ∗ , ∀ s ∈ [ 0 , T ˜ ] ,

which is an open and bounded subset of C 1 [ 0 , T ˜ ] .

Lemma 4.2

There exists a continuum C in [ 0 , w 0 ] × Ω ˜ 1 connecting { 0 } × Ω ˜ 1 with { w 0 } × Ω ˜ 1 , whose elements ( ω , r ) satisfy equation (3.1) with λ = 1 , where

Ω ˜ 1 = r ∈ C 1 [ 0 , T ] : 1 ϱ < r ( t ) < ϱ , ∣ r ′ ( t ) ∣ < ϱ , ∀ t ∈ [ 0 , T ] .

Proof

By (4.6), we note that equation (4.5) with ω = 0 has no solution on ∂ Ω 1 . Now, we show that there exists a constant w 0 such that for every ω ∈ [ 0 , w 0 ] , equation (4.5) has no solution on ∂ Ω 1 . Conversely, suppose that there exist sequences { ω n } and { R n } such that

ω n → 0 , n → + ∞ , R n ∈ ∂ Ω 1 , R n = L − 1 N ω n ( R n ) .

By the fact { R n } is a uniformly bounded sequence that { ( ω n , R n ) } is bounded in L 1 ( 0 , T ˜ ) . Moreover, owing to L − 1 is a compact operator, we can affirm that there exists a subsequence { R n j } ⊂ { R n } such that

L − 1 N ω n j ( R n j ) → R ˜ , j → + ∞ ,

for some R ˜ . Therefore,

R n j → R ˜ , j → + ∞ .

Since ∂ Ω 1 is closed, we obtain R ˜ ∈ ∂ Ω 1 , and

R ˜ = L − 1 N 0 ( R ˜ ) ,

that is, R ˜ ∈ ∂ Ω 1 is a solution of equation (4.5) with ω = 0 , which is a contradiction.

Since the degree deg ( I − L − 1 N ω , Ω 1 , 0 ) is the same for every ω ∈ [ 0 , ω 0 ] , we consider equation (4.5) with ω = 0 , which is equivalent to equation (4.1) with λ = 1 and ω = 0 . Moreover, the degree is also the same for equation (4.1) for every λ ∈ [ 0 , 1 ] ; therefore, we just need to consider equation (4.1) with λ = 0 and ω = 0

R ″ + R = 2 l R q + 1 n ,

which is equivalent to the following system:

U ′ = H ( U ) = V , − R + 2 l R q + 1 n ,

where U = ( R , V ) . It is easy to verify that H has a unique zero ( R 0 , 0 ) and the determinant of Jacobian matrix

∣ J H ( R 0 , 0 ) ∣ > 0 .

Then, by [3, Theorem 1], the Leray-Schauder degree of I − L − 1 N ω is equal to the Brouwer degree of H ,

deg ( I − L − 1 N ω , Ω 1 , 0 ) = deg B H , 1 ϱ , ϱ × − ϱ a ∗ , ϱ a ∗ ≠ 0 .

Then, by the global continuation principle of Leray-Schauder [54, Theorem 14.C], we prove that there exists a continuum C in [ 0 , w 0 ] × Ω 1 connecting { 0 } × Ω 1 with { w 0 } × Ω 1 , whose elements ( ω , R ) satisfy the system (4.5) with λ = 1 .

Going back to the original variables, we know that there exists a continuum C in [ 0 , w 0 ] × Ω ˜ 1 connecting { 0 } × Ω ˜ 1 with { w 0 } × Ω ˜ 1 , whose elements ( ω , r ) satisfy equation (3.1) with λ = 1 .□

5 Proof of Theorems 2.1 and 2.2

5.1 Proof of Theorem 2.1

By Lemma 4.1, we have shown that equation (3.1) with λ = 1 has a T -periodic solution r n for any fixed n ∈ N with ω n . Moreover, { r n } is a bounded and equi-continuous sequence. Then by the Arzelà-Ascoli Theorem, we obtain that { r n } has a subsequence { r n i } , converging uniformly to a function r ∈ C ( [ 0 , T ] ) . And we can suppose that ω n i → ω as i → ∞ . Obviously, r n i satisfies

r n i ( t ) = ∫ 0 T G ( t , s ) 2 f n i ( s , r n i ) + ω n i 2 a ( s ) r n i 3 ( s ) d s + 1 n i .

Then, as i → ∞ , we have

r ( t ) = ∫ 0 T G ( t , s ) 2 f ( s , r ) + ω 2 a ( s ) r 3 ( s ) d s ,

which means that r is a periodic solution of the first equation of the system (2.3).

Moreover, we can deduce from Lemma 4.1 that there exists a connected set C , contained in [ ω ¯ , + ∞ ] × C T 1 , connecting { ω ¯ } × C T 1 with { ω ∗ } × C T 1 for every ω ∗ > ω ¯ , and its elements ( ω , r ) are T -periodic solutions of the first equation of the system (2.3).

For every ε > 0 , similar to [16, Lemma 5], we can prove that there exists ω ε ≥ ω ¯ , if ( ω , r ) ∈ C with ω ≥ ω ε , then

1 T ∫ 0 T ω a ( t ) r 2 ( t ) d t ≤ ε .

Therefore, the function

φ ( ω , r ) = 1 T ∫ 0 T ω a ( t ) r 2 ( t ) d t ,

is continuous from C to R , and its image is an interval of the type ( 0 , ϑ ¯ ] , for some ϑ ¯ > 0 . Then, for ϑ ∈ ( 0 , ϑ ¯ ] , there exists ( ω , r ) ∈ C such that

φ ( ω , r ) = ϑ ,

and ( ω , r ) satisfies the first equation in (2.3) and r is T -periodic.

Define

θ ( t ) = ∫ 0 t ω a ( s ) r 2 ( s ) d s ,

which satisfies the second equation of (2.3). Moreover, we have

θ ( t + T ) − θ ( t ) T = 1 T ∫ t t + T ω a ( t ) r 2 ( t ) d t = 1 T ∫ 0 T ω a ( t ) r 2 ( t ) d t = θ ( T ) − θ ( 0 ) T = ϑ .

For every ϑ ∈ ( 0 , ϑ ¯ ] , it is obvious that the solution of the system (2.3) in the aforementioned equation provides a solution of the system (2.1), such that

u ( t + T ) = u ( t ) e i ϑ T , ∀ t ∈ R .

On the one hand, if we choose ϑ = 2 π m T , where m is an integer with m ≥ 2 π T ϑ ¯ , then ϑ ∈ ( 0 , ϑ ¯ ] and

u ( t + m T ) = u ( t ) , ∀ t ∈ R ,

that is, we obtain a m T -periodic solution u m ( t ) of the system (2.1), which rotates exactly once around the origin in the period time m T . Let ( r m , θ m ) be its polar coordinates and ω m be the constant which relates to its angular momentum. By the aforementioned arguments, ( ω m , r m , θ m ) satisfy the system (2.3), ( ω m , r m ) ∈ C , and

1 T ∫ 0 T ω m a ( t ) r m 2 ( t ) d t = ϑ = 2 π m T .

Obviously,

(5.1) lim m → ∞ ∫ 0 T ω m a ( t ) r m 2 ( t ) d t = 0 .

Now, we prove that

(5.2) lim m → ∞ ω m = + ∞ .

Assume by contradiction that { ω m i } is a bounded subsequence, with { ω m i } ∈ [ ω ¯ , ω 1 ] for some ω 1 ≥ ω ¯ = ω ( ι ) > 1 . By Lemma 3.3 with λ = 1 , there exists a constant ρ 0 > 0 such that

‖ r m i ‖ < ρ 0 ,

and therefore,

∫ 0 T ω m i a ( t ) r m i 2 ( t ) d t > ω ¯ T a ¯ ρ 0 2 , i ∈ N ,

contrary to (5.1). Therefore, (5.2) holds. Furthermore, by (5.2), Lemma 3.1–3.2 with λ = 1 , we have

lim m → ∞ ‖ r m ‖ = + ∞ .

Then, by the aforementioned limit, (5.2) and the fact μ m ( t ) = ω m a ( t ) , we obtain that (2.4) holds.

On the other hand, when ϑ ∉ ( 2 π / T ) Q with ϑ ∈ ( 0 , ϑ ¯ ] , we obtain that the system (2.1) has a quasi-periodic solution u ϑ of the frequencies ϑ 1 = 2 π T , ϑ 2 = ϑ . This is easy to check as, in complex notation, u ϑ = r ϑ ( t ) e i θ ϑ ( t ) may be decomposed as the product of the T -periodic function r ϑ ( t ) e i θ ϑ ( t ) − i ϑ t and the 2 π ϑ -periodic function e i ϑ t , where ( r ϑ , θ ϑ ) be its polar coordinates and ω ϑ be the constant which relates to its angular momentum.

By the aforementioned arguments, ( ω ϑ , r ϑ , θ ϑ ) satisfy the system (2.3), ( ω ϑ , r ϑ ) ∈ C , and

1 T ∫ 0 T ω ϑ a ( t ) r ϑ 2 ( t ) d t = ϑ .

Obviously,

(5.3) lim ϑ → 0 ∫ 0 T ω ϑ a ( t ) r ϑ 2 ( t ) d t = 0 .

Finally, we prove that

(5.4) lim ϑ → 0 ω ϑ = + ∞ .

Assume by contradiction that { ω ϑ i } is a bounded subsequence, with { ω ϑ i } ∈ [ ω ¯ , ω 1 ] for some ω 1 ≥ ω ¯ = ω ( ι ) > 1 . By Lemma 3.3 with λ = 1 , there exists a constant ρ 0 > 0 such that

‖ r ϑ i ‖ < ρ 0 ,

and therefore,

∫ 0 T ω ϑ i a ( t ) r ϑ i 2 ( t ) d t > ω ¯ T a ¯ ρ 0 2 , i ∈ N ,

contrary to (5.3). Therefore, (5.4) holds. Furthermore, by (5.4), Lemma 3.1–3.2 with λ = 1 , we have

lim ϑ → 0 ‖ r ϑ ‖ = + ∞ .

Then, by the aforementioned limit, (5.4) and the fact μ ϑ ( t ) = ω ϑ a ( t ) , we obtain that (2.5) holds.

The proof of Theorem 2.1 is completed.

5.2 Proof of Theorem 2.2

By Lemma 4.2, we have shown that equation (3.1) with λ = 1 has a T -periodic solution r n for any fixed n ∈ N with ω n . Moreover, { r n } is a bounded and equi-continuous sequence. Then by the Arzelà-Ascoli Theorem, we obtain that { r n } has a subsequence { r n i } , converging uniformly to a function r ∈ C ( [ 0 , T ] ) . And we can suppose that ω n i → ω as i → ∞ . Obviously, r n i satisfies

r n i ( t ) = ∫ 0 T G ( t , s ) 2 f n i ( s , r n i ) + ω n i 2 a ( s ) r n i 3 ( s ) d s + 1 n i .

Then, as i → ∞ , we have

r ( t ) = ∫ 0 T G ( t , s ) 2 f ( s , r ) + ω 2 a ( s ) r 3 ( s ) d s ,

which means that r is a periodic solution of the first equation of the system (2.3). Moreover, we can deduce from Lemma 4.2 that there exists a continuum C in [ 0 , w 0 ] × Ω ˜ 1 connecting { 0 } × Ω ˜ 1 with { w 0 } × Ω ˜ 1 , whose elements ( ω , r ) satisfy the first equation of the system (2.3).

Let us consider the function φ : C → R , defined by

φ ( ω , r ) = 1 T ∫ 0 T ω a ( t ) r 2 ( t ) d t .

It is continuous and defined on a compact and connected domain, so its image is a compact interval. Since φ ( 0 , r ) = 0 and φ is not identically zero, this interval is of the type [ 0 , ϑ ¯ 0 ] , for some ϑ ¯ 0 > 0 . Furthermore, for each ϑ ∈ ( 0 , ϑ ¯ 0 ] , there exists ( ω , r , θ ) verifying the system (2.3), for which ( ω , r ) ∈ C and

r ( t + T ) = r ( t ) , θ ( t + T ) = θ ( t ) + T ϑ , ∀ t ∈ R ,

where

θ ( t ) = ∫ 0 t ω a ( s ) r 2 ( s ) d s .

Hence, these elements provide solutions of the system (2.1) through polar coordinates and

u ( t + T ) = u ( t ) e i T ϑ , ∀ t ∈ R .

On the one hand, if we choose ϑ = 2 π m T , where m is an integer with m ≥ 2 π T ϑ ¯ 0 , then ϑ ∈ ( 0 , ϑ ¯ 0 ] and

u ( t + m T ) = u ( t ) , ∀ t ∈ R ,

that is, we obtain a m T -periodic solution u m ( t ) of the system (2.1), which rotates exactly once around the origin in the period time m T . Let ( r m , θ m ) be its polar coordinates and ω m be the constant, which relate to its angular momentum. By the aforementioned arguments, ( ω m , r m , θ m ) satisfy the system (2.3), ( ω m , r m ) ∈ C , and

∫ 0 T ω m a ( t ) r m 2 ( t ) d t = 2 π m .

Moreover, by the fact r m ∈ Ω ˜ 1 that

ω m T a ¯ ϱ 2 < ∫ 0 T ω m a ( t ) r m 2 ( t ) d t = 2 π m ,

which implies that

lim m → ∞ ω m = 0 .

Then by the fact μ m ( t ) = ω m a ( t ) , we obtain that

lim m → + ∞ μ m ( t ) = 0 .

On the other hand, when ϑ ∉ ( 2 π / T ) Q with ϑ ∈ ( 0 , ϑ ¯ 0 ] , we obtain that the system (2.1) has a quasi-periodic solution u ϑ of the frequencies ϑ 1 = 2 π T , ϑ 2 = ϑ . Let ( r ϑ , θ ϑ ) be its polar coordinates, and ω ϑ be the constants, which relate to its angular momentum. By the aforementioned arguments, ( ω ϑ , r ϑ , θ ϑ ) satisfy the system (2.3), ( ω ϑ , r ϑ ) ∈ C , and

1 T ∫ 0 T ω ϑ a ( t ) r ϑ 2 ( t ) d t = ϑ .

Moreover, by the fact r ϑ ∈ Ω ˜ 1 that

ω ϑ T a ¯ ϱ 2 < ∫ 0 T ω ϑ a ( t ) r ϑ 2 ( t ) d t = T ϑ ,

which implies that

lim ϑ → 0 ω ϑ = 0 .

Then by the fact μ ϑ ( t ) = ω ϑ a ( t ) , we obtain that

lim ϑ → 0 μ ϑ ( t ) = 0 .

The proof of Theorem 2.2 is completed.

6 Conclusion

In this article, we consider a four-dimensional singular differential system (1.1), which can describe the dynamics of configurations bearing a small number of vortices in atomic Bose-Einstein condensates. By introducing some new variables, the system (1.1) is equivalent to the radially symmetric system (2.1). Based on topological degree theory and some analysis methods, we obtained some results on the existence and multiplicity of periodic and quasi-periodic solutions of the system (2.1). Besides the existence, we also obtain some dynamics properties of these solutions. Detailed conclusions are listed as follows:

The system (2.1) has two distinct families of periodic solutions with the following distinct dynamical behaviors: one rotates around the origin with large angular momentum and large amplitude, and the other one rotates around the origin with small angular momentum;
The system (2.1) also has two distinct families of quasi-periodic solutions with the following distinct dynamical behaviors: one rotates around the origin with large angular momentum and the other one rotates around the origin with small angular momentum.

When we reinterpret the aforementioned results in the original vortex dynamical system, we obtain that the system (1.1) has two distinct families of periodic solutions and two distinct families of quasi-periodic solutions with distinct dynamical behaviors. Some results in the literature are generalized and improved.

Finally, for the radially symmetric systems, with the exception of the existence, the stability of periodic solutions has also been studied, and see [1,6,34,35] and the references therein. Naturally, how is the stability of periodic solutions of the system (2.1)? How to find sufficient conditions for the Lyapunov stability of periodic solutions of the system (2.1)? These will be the topics of our future research.

ztliang@aust.edu.cn

Acknowledgments

We would like to express our great thanks to the referees for their valuable suggestions. We also would like to show our thanks to Professor Jifeng Chu for his constant supervision and support.

Funding information: Zaitao Liang was supported by the Major Program of University Natural Science Research Fund of Anhui Province (No. 2022AH040112). Shengjun Li was supported by the Hainan Provincial Natural Science Foundation of China (No. 120RC450) and National Natural Science Foundation of China (No. 11861028).
Conflict of interest: The authors declare that they have no competing interest.

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Received: 2022-06-27

Revised: 2023-01-07

Accepted: 2023-01-08

Published Online: 2023-02-17

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https://doi.org/10.1515/anona-2022-0287

Keywords for this article

periodic solutions; quasi-periodic solutions; singular differential system; topological degree theory

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