ORIGINAL PAPER

Study on stability and bifurcation of electromagnet-trackbeam coupling system for EMS maglev vehicle

Xiaohao Chen . Weihua Ma . Shihui Luo

Received: 20 October 2019 /Accepted: 25 August 2020 / Published online: 7 September 2020

� Springer Nature B.V. 2020

Abstract The stability and bifurcation behavior of

the electromagnet-track beam coupling system of the

electromagnetic suspension maglev vehicle are stud-

ied by the theoretical and numerical analyses. The

stability domain of three key dynamics parameters of

the track beam as well as the Lyapunov coefficient at

the degenerated equilibrium point is calculated. The

topological structure of the solution to the coupling

system near the Bautin bifurcation point is determined.

The results show that in the engineering practice, the

intermediate frequency should be avoided in the

natural frequency of the track beam; when a low

frequency is taken, it should be reduced as much as

possible; when a high frequency is taken, it should be

increased as much as possible and the damping ratio

should also be increased, so that the system can remain

stable while a lighter-weight track beam is used,

thereby reducing the engineering costs. Besides, when

the parameters are near the Bautin bifurcation point,

the system will have complex dynamics behaviors.

Within a certain parameter range, multiple stable and

unstable limit cycles exist simultaneously, so that the

system tends to have different stable solutions under

different initial disturbances.

Keywords Maglev vehicle � Stability analysis �Hopf bifurcation � Bautin bifurcation � Limit cycle

1 Introduction

EMS (electromagnetic suspension) maglev vehicles

rely on electromagnetic levitation to avoid mechanical

contact in traditional wheel-rail trains, thus enjoying

the advantages of being smooth and comfortable, with

low noise, small turning radius and strong climbing

ability [6, 14]. It also has broad application and

development prospects in the intercity highway and

low-speed urban traffic system. With electromagnet

suspended under the track beam by electromagnetic

attraction, the stable levitation of the vehicle is

realized by the active control of the electromagnetic

coil current [21]. However, the variation of the

electromagnetic force acting simultaneously on the

flexible track beam will cause the vibration, which

then changes the size of the air gap between the

electromagnet and the track, thus in turn affecting the

variation of the electromagnetic force, thereby form-

ing the electromagnet-track beam coupled vibration

system. If the parameters of the vehicle system do not

match the track beam system, coupled self-excited

vibrations may occur between the electromagnet and

the track beam, affecting the stable levitation of the

vehicle. The EMS maglev transportation system often

X. Chen � W. Ma (&) � S. LuoTraction Power State Key Laboratory, Southwest Jiaotong

University, Chengdu, People’s Republic of China

e-mail: mwh@swjtu.edu.cn

X. Chen

e-mail: xiaohao_chen@126.com

123

Nonlinear Dyn (2020) 101:2181–2193

https://doi.org/10.1007/s11071-020-05917-8(0123456789().,-volV)( 0123456789().,-volV)

uses an elevated mode for the track. The flexibility of

the track beam, which has a significant impact on the

levitation stability [13, 18], is the main cause of the

self-excited vibration of the electromagnet-track beam

coupling, especially during stationary or low-speed

operation [1, 8].

In view of the vibration problem of the electro-

magnet-track beam coupling system of maglev vehi-

cles, some scholars studied from the perspective of

levitation controller. Liang [9] expounded the charac-

teristics of electromagnetic interaction between the

electromagnet and the track beam, and studied the

influence of the control system parameters on the

performance of levitation system. Zou [23] concluded

that the system would have the same homoclinic

bifurcation, Hopf bifurcation and chaos under differ-

ent parameters, which was the root cause of the self-

excited vibration of the coupling system. Wang

[11, 12] focused on the time delay of the control

system, pointing out that the time delay parameter can

not only suppress the resonance response, but also

control the generation of chaos. Zhang [17] studied the

amplitude variation of the periodic solution to the

coupling system when self-excited vibration occurred,

using the perturbation method to calculate the limit

cycle caused by Hopf bifurcation, and verified its

effectiveness by numerical simulation. Li [7] used the

Nyquist stability criterion, the Routh table and the root

locus diagram to obtain the stability conditions of the

electromagnet-track beam coupling system under

static levitation. Other scholars have concentrated on

the impact of the mechanical structure of the vehicle

on the coupled vibration. Zhao [19, 20] equated the

electromagnetic force with linear spring-damping

force, established a high-speed maglev vehicle-track

beam coupled vibration model, and simulated its

vibration characteristics in high-speed operation. Kim,

Han et al. [3, 4] established a three-dimensional

vehicle-track beam model for coupling numerical

calculation, and compared the calculation results with

the experimental results to prove the validity of the

model.

As the coupled self-excited vibration problem has a

great impact on engineering applications, it is of great

practical significance to study the stability of electro-

magnet-track beam coupling system of EMS maglev

vehicles. In the field of the traditional railway, many

researchers used the Hurwitz determinant to determine

the Hopf bifurcation point of the railway wheelset

[10, 22] and the central manifold reduction method to

reduce the dimensionality of the system, then giving

the stability or bifurcation form of the system [16], but

few researchers used this method to study the maglev

system.

Due to the diversified structural modes of the EMS

maglev vehicle, the parameters of the track beam and

the vehicle vary a lot in different modes. In order to

obtain general conclusions, this paper establishes the

minimum coupling system model by only considering

the flexible track beam and the vertical motion of the

single electromagnet levitated underneath. The non-

linearity of the model is mainly derived from the

inverse square characteristics of the electromagnetic

interaction. This paper is aimed to figure out the

calculation method of the stability domain of the

electromagnet-track beam coupling system, and deter-

mine the stability of the degenerated equilibrium point

and the form of Hopf bifurcation by using the

Lyapunov coefficient when it has valid critical

parameters in the system.

2 System model and differential equation

of motion

The electromagnet-track beam coupling system is the

basic unit of the maglev vehicle levitation system. The

dynamics model is shown in Fig. 1. The electromag-

net of mass m is levitated below the track beam.

It is proved by engineering practice that almost all

the coupled vibration problems are caused by the first-

order vibration instability of the track beam [15].

Therefore, when examining the coupled vibration

problem between the electromagnet and the track

beam, it is reasonable to consider that the track beam is

Fig. 1 System dynamics model

123

2182 X. Chen et al.

equivalent to a single-degree-of-freedom vibrating

body with mass M, support stiffness k and damping c.

The coupling system has two mechanical degrees of

freedom, namely the vertical displacement of the

electromagnet z and the deflection of the track beam at

the levitation position w. The electromagnetic force

between the electromagnet and the track beam fm is a

function of the electromagnet coil current i and the

levitation gap d. The coil current i is driven by a

voltage u acting across the coil, which has a total

resistance of R. The levitation gap is d ¼ d0 þ z� w,

where d0 is the rated levitation gap. The electromag-

netic force between the electromagnet and the track

beam fm can be expressed as follows

fm i; dð Þ ¼ l0N2A

4

i

d

� �2

; ð2:1Þ

where l0 is the permeability in vacuum, N is the

number of turns of the coil, and A is the area of the

magnetic pole of the electromagnet.

According to Kirchhoff’s law, the relationship

between coil current and control voltage can be

written as

u ¼ iRþ L0 _i� kL _d; ð2:2Þ

where coefficients L0 ¼ l0N2A

2d and kL ¼ l0N2Ai

2d2.

The dynamics equation of the mechanical system

can be written in the following form by Newton’s law

of motion

m z:: ¼ mg� fm i; dð Þ;

€w ¼ r fm i; dð Þ � mg½ � � x2w� 2nx _w;

�ð2:3Þ

where g is the acceleration of gravity, r ¼ 1M is the

dynamic exchange coefficient of the electromagnetic

force on the track beam, x ¼ffiffiffikM

qis the natural

frequency of the track beam, n ¼ c2ffiffiffiffiffiMk

p is the damping

ratio of the track beam, and the zero points of z and

w are located in the static equilibrium position.

Due to the inverse square characteristic of the

electromagnetic force, the open-loop electromagnetic

levitation system is unstable, thus feedback control of

the control voltage u is required. The levitation system

of EMS maglev vehicle mostly uses double-loop

feedback control to adjust the levitation gap. The

levitation electromagnet is equipped with gap sensor,

acceleration sensor and current sensor, which can

monitor the levitation gap d, the vertical acceleration

of the electromagnet €z and the coil current I, respec-

tively. By the feedback on the levitation gap d and theintegrated vertical acceleration of the electromagnet €z,

that is, the vertical velocity of the electromagnet _z, thetarget levitation current is obtained

ie ¼ kP d� d0ð Þ þ kD _zþ i0; ð2:4Þ

where kP and kD are the state feedback coefficients, i0

is the stable levitation current, and i0 ¼ 2d0N

ffiffiffiffiffiffimgl0A

q.

To speed up the current tracking speed, current loop

is used to have feedback control on the voltage across

the coil

u ¼ ke ie � ið Þ þ iR; ð2:5Þ

where ke is the current feedback coefficient.

Taking Eqs. (2.4) and (2.5) into Eq. (2.2), the

differential equation of the circuit can be obtained

L0 _i� kL _d ¼ ke kP d� d0ð Þ þ kD _zþ i0 � i½ �: ð2:6Þ

In conclusion, the motions of the electromagnet and

the track beam are represented by Eq. (2.3). Current in

the coil is represented by Eq. (2.6). The ordinary

differential equations describing electromagnet-track

beam coupling system are obtained. To convert

equations into standard forms, by transforming

x1; x2; x3; x4; x5½ � ¼ z; _z;w; _w; i� i0½ � and expanding

d, L0 and kL, the system equations can be given by

_x1 ¼ x2;

_x2 ¼ � l0N2A

4m

x5 þ i0x1 � x3 þ d0

� �2

þg;

_x3 ¼ x4;

_x4 ¼ rl0N

2A

4

x5 þ i0x1 � x3 þ d0

� �2

�mg

" #� x2x3 � 2nxx4;

_x5 ¼2keðx1 � x3 þ d0Þ

l0N2AkP x1 � x3ð Þ þ kDx2 � x5½ � þ x5 þ i0

x1 � x3 þ d0x2 � x4ð Þ:

8>>>>>>>>>>><>>>>>>>>>>>:

ð2:7Þ

3 Analysis on stability and bifurcation

To study the influence of the track beam parameters r,x and n on the stability of the coupling system, let a ¼,

then (2.6) can be written into the following state space

for

_x ¼ J að Þxþ f x; að Þ; ð3:1Þ

where J að Þ is the Jacobian matrix of the system,

_x ¼ _x1; _x2; _x3; _x4; _x5½ �T, x ¼ x1; x2; x3; x4; x5½ �T and

123

Study on stability and bifurcation of electromagnet-track 2183

f x; að Þ ¼ 0; f2 x; að Þ; 0; f4 x; að Þ; f5 x; að Þ½ �T.Then the stability of the system at the equilibrium

point is examined. Obviously, x ¼ 0 is a steady-state

solution of the system, and the Jacobian matrix at the

equilibrium point is given by

J að Þ ¼

0 1 0 0 02g

d00 �2g

d00 �N

d0

ffiffiffiffiffiffiffiffiffiffil0Agm

r

0 0 0 1 0

�2mgrd0

02mgrd0

�x2 �2nxNrd0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffil0Amg

p2d0kekPl0N2A

2d0kekDl0N2A

þ 2

N

ffiffiffiffiffiffiffiffimg

l0A

r�2d0kekP

l0N2A� 2

N

ffiffiffiffiffiffiffiffimg

l0A

r� 2d0kel0N2A

26666666664

37777777775:

ð3:2Þ

The characteristic equation of (3.2) can be written

in the following alternative form

k5 þ a1 að Þk4 þ a2 að Þk3 þ a3 að Þk2

þ a4 að Þkþ a5 að Þ ¼ 0;ð3:3Þ

Where

a1 að Þ ¼ 2d0kel0N2A

þ 2nx;

a2 að Þ ¼ 2kekDN

ffiffiffiffiffiffiffiffiffiffiffig

ml0A

rþ 4ked0nx

l0N2Aþ x2;

a3 að Þ ¼ 2kekPN

ffiffiffiffiffiffiffiffiffiffiffig

ml0A

r� 4gkel0N2A

þ 2kekPN

ffiffiffiffiffiffiffiffimg

l0A

r� 4kemg

l0N2A

� �r

þ 4kekDnxN

ffiffiffiffiffiffiffiffiffiffiffig

ml0A

rþ 2ked0x2

l0N2A

a4 að Þ ¼ 4kekPN

ffiffiffiffiffiffiffiffiffiffiffig

ml0A

r� 8gkel0N2A

� �nxþ 4kekDx2

N

ffiffiffiffiffiffiffiffiffiffiffig

ml0A

r

a5 að Þ ¼ 2kekPN

ffiffiffiffiffiffiffiffiffiffiffig

ml0A

r� 4gkel0N2A

� �x2

Let Ta ¼ span r;x; nf g be the space of the track

beam parameters r;x and n, and set S has all the entireelements in the parameter space Ta that allow all the

eigenvalues of Eq. (3.3) to have negative real parts.

According to the first method of Lyapunov, if all the

eigenvalues of Eq. (3.3) have negative real parts, the

equilibrium point of the system (3.1) is stable. The

elements in the set S make the system stable at the

equilibrium point.

In order to study the Hopf bifurcation of the system

(3.1), it is necessary to find a condition that the

Eq. (3.3) has a pair of pure imaginary eigenvalue and

the other eigenvalues have negative real parts. If the

transversality condition is not zero at the same time,

the Hopf bifurcation occurs at this point. Since there is

not only one parameter being discussed, whether the

transversality condition is satisfied is determined by

the changing directions of the parameters, which will

be discussed by the numerical analysis in Sect. 4.

Here, the following theorem is applied to Eq. (3.3).

Theorem 3.1 [15]. Let the real coefficient algebraic

equation

kn þ a1kn�1 þ � � � þ an�1kþ an ¼ 0; ð3:4Þ

the necessary and sufficient condition for having a

pair of pure imaginary eigenvalue and the remaining

n - 2 eigenvalues have negative real parts is

ai [ 0 i ¼ 1; 2; . . .nð Þ;Di [ 0 i ¼ n� 3; n� 5; . . .ð Þ;Dn�1 ¼ 0;

where Di is the ith order principal minor of the

Hurwitz determinant of Eq. (3.4).

By applying Theorem (3.1) to Eq. (3.3), the con-

dition that there is a pair of pure imaginary eigenvalue

and the remaining eigenvalues have negative real parts

is

ai [ 0 i ¼ 1; 2; 3; 4; 5ð Þ;

D2 ¼a1 1

a3 a2

�������� ¼ a1a2 � a3 [ 0;

D4 ¼

a1 1

a3 a1

0 0

a1 1

a5 a4

0 0

a3 a2

a5 a4

���������

���������¼ a1a2a3a4 � a23a4 � a21a

24

� a1a22a5 þ a2a3a5 þ 2a1a4a5 � a25 ¼ 0:

ð3:5Þ

Let the set C be the entire elements satisfying the

condition (3.5) in the parameter space Ta. The

elements in set C put the system in a critical state

with a degenerated equilibrium point. In order to

determine the stability of this point, it is necessary to

analyze the nonlinear part of the system. System (3.1)

is a 5-dimensional system. In order to examine its

stability and bifurcation, the central flow reduction

method is usually used to simplify complex high-

dimensional system into corresponding planar system,

and then the paradigm theory and Poincare canonical

theory are applied to determine the form of bifurcation

that may occur in a degenerated system [2]. However,

these methods require multiple transformations of the

base of the space, which are very complicated and

difficult to program. Therefore, the method based on

Fredholm’s theorem is used to ‘‘project’’ the system

123

2184 X. Chen et al.

into the critical feature space and its complement

space to obtain its restricted version of the general

normalization equation. According to this method, the

simplification and normalization of the central man-

ifold are performed simultaneously. The obtained

formula contains only the Jacobian matrix and its

transposed critical eigenvectors, as well as the Taylor

series expansion according to the original base at the

degenerated equilibrium point, greatly reducing the

complexity of symbol calculation and numerical

calculation [5].

Let 8ac 2 C and J acð Þ have a pair of pure

imaginary eigenvalues �ixc while the remaining

eigenvalues have negative real parts, then the nonlin-

ear part of the system fc xð Þ ¼ f x; acð Þ ¼ Oðjjxjj2Þ is

smooth enough near the equilibrium point. Taylor

series expansion at the equilibrium point is

f c xð Þ ¼ 1

2B x; xð Þ þ 1

6C x; x; xð Þ þ Oðjjxjj4Þ; ð3:6Þ

where B n; gð Þ and C w; g; fð Þ are symmetric multi-

linear vector functions of w; g; f 2 C5, which can be

given by

Bi w; gð Þ ¼X5j;k¼1

o2fi x; acð Þoxjoxk

jx¼0wjgk; i ¼ 1; . . .; 5;

ð3:7Þ

Ci w; g; fð Þ ¼X5j;k;l¼1

o3fi x; acð Þoxjoxkoxl

jx¼0wjgkfl; i ¼ 1; . . .; 5:

ð3:8Þ

Let q 2 C5 be the complex eigenvector correspond-

ing to the pure imaginary eigenvalue �ixc,

J acð Þq ¼ ixcq; J acð Þ�q ¼ �ixc�q; ð3:9Þ

and introduce adjoint eigenvector p 2 C5,which not

only has the following properties

J acð ÞTp ¼ �ixcp; J acð ÞT �p ¼ ixc�p; ð3:10Þ

but also satisfy the normalization condition p; q ¼ 1,

where p; q ¼P5i¼1

�piqi 2 C is the standard inner product

in C5.

The eigenvalue subspace of J acð Þ corresponding toq and �q is Tc ¼ span Re qcð Þ; Im qcð Þf g. Ts is the

characteristic subspace corresponding to other

eigenvalues of J acð Þ, then Ts ¼ fyjy 2 R5; p; y ¼ 0g,and the state space x 2 R5 can be decomposed into

x ¼ zqþ �z�qþ y; zqþ �z�q 2 Tc; y 2 Ts; ð3:11Þ

and

z ¼ p; x;y ¼ x� p; xq� �p; xq:

�ð3:12Þ

In the (z, y) coordinates, system (3.1) becomes

_z ¼ ixczþ hp; f c zqþ �z�qþ yð Þi;_y ¼ J acð Þyþ f c zqþ �z�qþ yð Þ � hp; f c zqþ �z�qþ yð Þiq��p; f c zqþ �z�qþ yð Þ�q:

8<:

ð3:13Þ

Integrating (3.6) into (3.13), the system’s restricted

version of the general normalization equation is obtained

based on the central manifold reduction method and

Poincare canonical theory, and then the expression of the

first Lyapunov coefficient of the system is obtained. There

is a detailed derivation process in Ref. [5]. The first

Lyapunov coefficient is expressed as

l1 acð Þ ¼ 1

xcRec1; ð3:14Þ

where

c1 ¼1

2p;C q; q; �qð Þ þ B �q; 2ixcI � J acð Þð Þ�1B q; qð Þ

� �D

�2B q; J acð Þ�1B q; �qð Þ� �E

If l1(ac)\ 0, the degenerated equilibrium point is

stable. If the changes of the system parameters satisfy

the transversality condition at the same time, super-

critical Hopf bifurcation occurs.When the equilibrium

point loses stability as the parameter value changes, a

stable limit cycle is split from the equilibrium point. If

l1(ac)[ 0, the degenerated equilibrium point is

unstable, and if the transversality condition is satisfied,

subcritical Hopf bifurcation occurs in the system.

When the equilibrium point loses stability as the

parameter value changes, an unstable limit cycle

merges with the equilibrium point. However, if

l1(ac) = 0, Bautin bifurcation may occur. To investi-

gate the stability of the equilibrium point and the

topological structure near the bifurcation point, further

analysis on the higher-order parts of the nonlinear part

of the system is required.

Extend Eq. (3.6) into

123

Study on stability and bifurcation of electromagnet-track 2185

f c xð Þ ¼ 1

2!B x; xð Þ þ 1

3!C x; x; xð Þ þ 1

4!D x; x; x; xð Þ

þ 1

5!E x; x; x; x; xð Þ þ Oðjjxjj6Þ

;

ð3:15Þ

where

Di w; g; f; mð Þ ¼X5

j;k;l;s¼1

o4fi x; acð Þoxjoxkoxloxs

jx¼0wjgkflms;

i ¼ 1; . . .; 5;

ð3:16Þ

Ei w; g; f; m; tð Þ ¼X5

j;k;l;s;r¼1

o5f i x; acð Þoxjoxkoxloxsoxr

jx¼0wjgkflmstr

i ¼ 1; . . .; 5:

ð3:17Þ

According to the derivation process of the first

Lyapunov coefficient l1 acð Þ, the expression of the

second Lyapunov coefficient l2 acð Þ can be obtained.

Similarly, the detailed derivation process is given in

Ref. [5].

l2 acð Þ ¼ 1

12xcRe½ p;E q; q; q; �q; �qð Þ þ D q; q; q; �h20

�þ 3D q; �q; �q; h20ð Þ þ 6D q; q; �q; h11ð Þ þ C �q; �q; h30ð Þþ 3C q; q; �h21

þ 6C q; �q; h21ð Þ þ 3C q; �h20; h20

þ 6C q; h11; h11ð Þ þ 6C �q; h20; h11ð Þ þ 2B �q; h31ð Þþ 3B q; h22ð Þ þ B �h20; h30

þ 3B �h21; h20

þ 6B h11; h21ð Þi�;

ð3:18Þ

where

h20 ¼ 2ixcI � J acð Þð Þ�1B q; qð Þ;h11 ¼ �J acð Þ�1B q; �qð Þ;h30 ¼ 3ixcI � J acð Þð Þ�1 C q; q; qð Þ þ 3B q; h20ð Þ½ �;h31 ¼ 2ixcI � J acð Þð Þ�1½D q; q; q; �qð Þ þ 3C q; q; h11ð Þ

þ 3C q; �q; h20ð Þ þ 3B h20; h11ð Þ þ B �q; h30ð Þþ 3B q; h21ð Þ � 6c1h20�;

h22 ¼ �J acð Þ�1½D q; q; �q; �qð Þ þ 4C q; �q; h11ð Þþ C �q; �q; h20ð Þ þ C q; q; �h20

� þ 2B h11; h11ð Þ

þ 2B q; �h21

þ 2B �q; h21ð Þ þ B �h20;h20

�:

At this time, the stability of the system at the

equilibrium point and the local topological structure

near the bifurcation point can be determined by the

plus/minus sign of the second Lyapunov coefficient

l2 acð Þ.Let the set U ¼ faja 2 Ta&a 62 S

SCg and let it be

the complement of the parameter space Ta for the

union of sets S and C. Every element in the set

U makes at least one eigenvalue of Eq. (3.3) with a

positive real part, making the system unstable at the

equilibrium point.

The parameter space Ta is divided into three

subsets. If the track beam parameter a 2 S, the system

is stable at the equilibrium point. If the track beam

parameter a 2 U, the system is unstable at the

equilibrium point. However, if a 2 C, it is compli-

cated: if the first Lyapunov coefficient l1 að Þ\0, the

system is stable at the equilibrium point and super-

critical Hopf bifurcation occurs if the transversality

condition is satisfied; if l1 að Þ[ 0, the system is

unstable at the equilibrium point and subcritical Hopf

bifurcation occurs if the transversality condition is

satisfied; if l1 að Þ ¼ 0, Bautin bifurcation occurs, and

the stability at the equilibrium point and the topolog-

ical structure near the bifurcation point are determined

by the plus/minus sign of the second Lyapunov

coefficient l2 að Þ.

4 Numerical analysis

In this section, analysis on an electromagnet-track

beam coupling system is taken as an example to

demonstrate the conclusions obtained in Sect. 3, and

the stability domain and the bifurcation diagrams of

the equilibrium point under different track beam

parameters are drawn by using the parameters

described in the appendix.

4.1 Stability domain of the equilibrium point

Since the values of all parameters are known, it is easy

to calculate the distribution of the sets S,C andU in the

parameter space Ta according to Theorem (3.1), as

shown in Fig. 2. In this figure, the blue area is the set S,

the uncolored area is the set U, and the curved yellow

surface is the set C.

Figure 3 shows the curves composed of the

elements of the set C in the space span r;xf g when

n takes different values. The elements on the curve all

belong to the set C, so that Eq. (3.3) has a pair of pure

123

2186 X. Chen et al.

imaginary eigenvalue and the remaining eigenvalues

have negative real parts. At this time, the system has a

degenerated equilibrium point, and the first Lyapunov

coefficient corresponding to the elements on the curve

calculated according to Eq. (3.14) is indicated by the

solid line when l1\0 and indicated by dotted lines

when l1 [ 0, respectively. The elements of the solid

line make the degenerated equilibrium point stable,

and the supercritical Hopf bifurcation occurs when the

parameters cross the curve. The elements of the dotted

line make the degenerated equilibrium point unstable,

and the subcritical Hopf bifurcation occurs when the

parameters cross the curve. If the parameters does not

cross the curve but is tangent to the curve, the Hopf

bifurcation does not occur at this time because the

transversality condition is not satisfied.

The elements below the curve belong to the set S, so

that all eigenvalues of Eq. (3.3) have negative real

parts and the system is stable at the equilibrium point.

The elements above the curve belong to the set U, so

that there is at least one eigenvalue of the Eq. (3.3)

with a positive real part, and the system is unstable at

the equilibrium point.

It can be seen from Fig. 3 that the curve formed by

the elements of the set C has a significant inflection

point around x ¼ 150 rad/s. In the low frequency part

before the inflection point, the critical value rdecreases with the increase of x, that is, before the

inflection point, as the natural frequency of the track

beam increases, a larger equivalent mass of track beam

is required to stabilize the system. When the natural

frequency is smaller than that of the inflection point,

the stability domain varies little with the damping ratio

of the track beam n. In the high frequency part after theinflection point, the critical value r increases with the

increase of x, that is, after the inflection point, as the

natural frequency increases, a lower equivalent mass

of track beam can also stabilize the system. When the

natural frequency is larger than that of the inflection

point, the stability domain changes significantly with

the change of n. Therefore, increasing n can signifi-

cantly increase the stability domain of the system.

When the natural frequency x of the track beam is

close to that of the inflection point, the value range of rfor the system to be stabilized is very small, so that a

considerably large equivalent mass of the track beam

is required for the stability.

It can be seen from the analysis above that the range

of the natural frequency of the track beam can be

roughly divided into three parts: low frequency

(before the inflection point), intermediate frequency

(near the inflection point), and high frequency (after

the inflection point). In engineering practice, the

intermediate frequency (near the inflection point)

should be avoided in the natural frequency of the

track beam; because in this range, the stability of the

equilibrium point can be ensured only when the track

beam is considerably heavy, which greatly increases

the engineering cost. When a low frequency (before

the inflection point) is taken as the natural frequency of

the track beam, it should be reduced as much as

possible, which can help keep the system stable while

a lighter-weight track beam is chosen. Similarly, when

a high frequency (after the inflection point) is taken,

both the damping ratio of the track beam and the

Fig. 2 Distribution of the sets S,C andU in the parameter space

Ta

Fig. 3 Curves composed of the elements of the set C in the

space span r;xf g when n takes different values

123

Study on stability and bifurcation of electromagnet-track 2187

natural frequency should be increased as much as

possible, since the damping ratio of the track beam nwill greatly affect the performance of the system. In

this case, choosing a lighter-weight track beam can

also keep the system stable, which reduces engineer-

ing costs.

4.2 Bifurcation analysis

The bifurcation analysis of the system is conducted by

selecting the comparatively more typical track beam

damping ratio: let n ¼ 0:5%. Figure 4 is the stability

domain diagram of the system in the parameter space

span r;xf g, and the first Lyapunov coefficient at the

edge of the stability domain.

The points in the shaded part of the figure are the

elements in the set S. When the parameters are in the

shaded part, the system is stable at the equilibrium

point. The points of the blue curve are the elements in

the set C. When the parameters are on the blue curve,

the equilibrium point is degenerated. The first Lya-

punov coefficient indicated by the red curve shows

that when x\64:3 rad=s or x[ 147:7 rad=s, the first

Lyapunov coefficient of the degenerated equilibrium

point is l1\0, the equilibrium point is stable, and

supercritical Hopf bifurcation occurs when the param-

eters cross the curve; when 64:3 rad=s\x\147:7 rad=s, the first Lyapunov coefficient l1 [ 0,

the equilibrium point is unstable, and the subcritical

Hopf bifurcation occurs when the parameters cross the

curve; when x ¼ 64:3 rad=s or x ¼ 147:7 rad=s, the

first Lyapunov coefficient l1 ¼ 0, and Bautin bifurca-

tion occurs when the parameters cross the curve from

this degenerated equilibrium point.

In order to better understand the topological

structure near the bifurcation point, the dynamics

behaviors of the system near the Bautin bifurcation

point at x ¼ 64:3 rad=s are investigated. Four differ-

ent groups of parameters around the point are used for

numerical simulation and shown in Fig. 5, where

a1 � a4 are the parameter combinations: x1; r1½ � ¼62 rad/s; 5:5� 10�3 kg�1�

, x2; r2½ � ¼ 62 rad/s;½6:3� 10�3 kg�1�, x3; r3½ � ¼ 66 rad/s; 5:5�½10�3 kg�1� and x4; r4½ � ¼ 66 rad/s; 6:1� 10�3½kg�1�. A non-stiff differential equation solver (the

standard NDSolve function in MATHEMATICA) is

used for the numerical integration. Under the initial

disturbance x ¼ 0; 0; 0:003 m; 0; 0½ �T, the projections

of the phase diagram on the plane d� _d under the fourparameters combination are presented. The blue curve

is the phase trajectory, with the black point and the

gray point representing the stable and unstable equi-

librium points, respectively, and the black solid curve

and the imaginary curve representing the stable and

unstable limit cycles, respectively. What needs to be

pointed out is that the system dynamics behavior is

complicated when the parameter combination a3 is

taken, therefore, an additional set of initial distur-

bances x ¼ 0; 0; 0:0033 m; 0; 0½ �T is added for simu-

lation, with its phase trajectory represented by a red

curve.

It is worth noting that due to the physical limitations

of electromagnets, the completely smooth Eq. (2.6)

does not describe the global behavior of the system. It

is necessary to modify Eq. (2.6) as follows:

Fig. 4 Stability domain diagram of the system in the parameter

space span r;xf g and the first Lyapunov coefficient at its edge

when n ¼ 0:5%Fig. 5 Simulation parameter points selected near the Bautin

bifurcation point

123

2188 X. Chen et al.

1. Since the coil current is not reversible, the

minimum value is 0A, that is, when x5 þ i0\0,

the value of x5 in the numerical integration is �i0;

2. Since the levitation gap cannot be negative, when

x1 � x5 þ c0\0, it is considered to have a colli-

sion and the calculation is stopped accordingly.

As can be seen from Fig. 6, the results of numerical

analysis and theoretical analysis are consistent. When

the parameter takes a1, the system can return to the

equilibrium position under the initial disturbance.

When the parameters take a2 and a4, under the initialdisturbance, the system will tend to maintain a

constant vibration amplitude and finally be stabilized

in the periodic solution, and at that time, the equilib-

rium point is unstable. When the parameters take a3,the situation is much more complicated. The

stable equilibrium point, the stable limit cycle and

the unstable limit cycle coexist in the figure. When the

initial disturbance is small, the system can return to the

equilibrium position, yet when the initial disturbance

is larger, the system is inclined to the stable limit cycle

for periodic motion.

Figure 5 can be used to determine the direction of

the Hopf bifurcation transition. Obviously, as the

bifurcation parameter moves from left to right relative

to the Bautin bifurcation point, the supercritical Hopf

(a) (c)

(b) (d)

Fig. 6 The phase diagram of the system with different

parameters a The phase diagram when the parameter takes a1b The phase diagram when the parameter takes a2 c The phase

diagramwhen the parameter takes a3 d The phase diagramwhen

the parameter takes a4

123

Study on stability and bifurcation of electromagnet-track 2189

bifurcation point becomes a subcritical Hopf bifurca-

tion point. To observe this transition process, the

Newton–Raphson iterative method is applied to track

the evolution of periodic solutions. At the same time,

the Floqurt multiplier of the periodic solution and the

eigenvalues of the Jacobian matrix of the static

solution are monitored, as they provide the stability

information of the limit cycle and the equilibrium

point, respectively. Figure 7 is the two Hopf bifurca-

tion diagrams near the Bautin bifurcation point. In

Fig. 7, the dotted line indicates the unstable equilib-

rium point or the limit cycle, while the solid line

indicates the stable equilibrium point or the limit

cycle, with the symbols * and o indicating the Hopf

bifurcation point and the fold bifurcation point,

respectively. Figure 8 is a blowup of Fig. 7 near the

Hopf bifurcation point. It can be clearly seen that as xincreases and passes the Bautin bifurcation point, a

supercritical Hopf bifurcation transforms into a sub-

critical Hopf bifurcation.

As shown in Fig. 7a, when damping ratio of the

track beam n ¼ 0:5% and the natural frequency

x ¼ 62 rad=s, if r is smaller than that of the fold

bifurcation point C1, the system can always return to

the equilibrium position under the initial disturbance;

if r is larger than that of the fold bifurcation point C1

and smaller than that of the Hopf bifurcation point A1,

the system will return to the equilibrium position

under a small initial disturbance, yet tend to have a

periodic motion with a large amplitude under a large

initial disturbance; if r is larger than that of the Hopf

bifurcation point A1 and less than that of the fold

bifurcation point B1, the system tends to have a

periodic motion with a smaller amplitude under a

smaller initial disturbance, and have a periodic motion

with a larger amplitude under a larger initial distur-

bance, with equilibrium position being unstable; if r is

larger than that of the fold bifurcation point B1, the

system will always tend to have a periodic motion with

a large amplitude under the initial disturbance, and the

equilibrium position is unstable. As shown in Fig. 7b,

when damping ratio of the track beam n ¼ 0:5% and

the natural frequency x ¼ 65 rad=s, if r is smaller

than that of the fold bifurcation point B2, the system

can always return to the equilibrium position under the

initial disturbance; if r is greater than that of the fold

bifurcation point B2 and less than that of the Hopf

bifurcation point A2, the system will return to the

equilibrium position under a small initial disturbance

and tend to have a periodic motion with a large

amplitude under a large initial disturbance; if r is

greater than that of the Hopf bifurcation point A2, the

system will always tend to have a periodic motion with

a large amplitude under the initial disturbance, and the

equilibrium position is unstable.

5 Conclusion

In this paper, stability and bifurcation analyses are

studying based on the mathematical model of the

*

(a)

(b)

*

Fig. 7 Hopf bifurcation diagram near Bautin bifurcation point

a Bifurcation diagram of the system as r changes when x ¼62 rad=s bBifurcation diagram of the system as r changes whenx ¼ 65 rad=s

123

2190 X. Chen et al.

electromagnet-track beam coupling system of EMS

maglev vehicles. A simplified dynamics model of a

single electromagnet levitated below the flexible track

beam is established under some certain assumptions,

which reduces the degree of freedom of the whole

system without losing the basic characteristics, and

some general conclusions applicable to different

structural modes of EMS maglev vehicles are drawn.

Based on the Lyapunov’s first method, the space

formed by the three key parameters of the track beam

r, x and n is decomposed into a stable set S, an

unstable set U and a degenerated set C. In order to

examine the stability of the degradation condition and

study the bifurcation behavior of the system near the

degenerated equilibrium point, the first Lyapunov

coefficient at the degenerated equilibrium point is

calculated by using codimension one and codimen-

sion two bifurcation theory of the nonlinear dynamics

system. The first Lyapunov coefficient is used to

determine which kind of Hopf bifurcation occurs in

the system under some certain bifurcation parameters.

When the first Lyapunov coefficient is 0, the second

Lyapunov coefficient obtained by the higher-order

Taylor expansion of the nonlinear part of the model is

needed to understand the local topology of the

coupling system better.

The distribution of the set S, U and C in the

parameter space is obtained by numerical analysis.

The shape of the stable domain set S in the parameter

space indicates that the range of the natural frequency

of the track beam x can be roughly divided into three

parts: low frequency (before the inflection point),

intermediate frequency (near the inflection point), and

high frequency (after the inflection point). In engi-

neering practice, the intermediate frequency (near the

inflection point) should be avoided in the natural

frequencyx of the track beam. If x takes a value from

the low frequency part (before the inflection point), it

should be reduced as much as possible, which can help

keep the system stable while a lighter-weight track

beam is used. At this time, the damping ratio of the

track beam n has an insignificant influence on the

stability domain. If x takes a value from the high

frequency (after the inflection point) part, the damping

ratio n will greatly affect the performance of the

system. In this case, both n and x should be increased

as much as possible, so as to ensure that the system can

keep stable with a lighter-weight track beam, thus

reducing the engineering costs.

When the parameter n has a fixed value, the Bautinbifurcation point of the system is determined by

calculating the first Lyapunov coefficient correspond-

ing to the elements in the set C. Numerical simulations

are then carried out by selecting the combination of

parameters near the Bautin bifurcation point, and it is

found that the system has complex dynamics behav-

iors near the Bautin bifurcation point.

In order to better observe the Hopf bifurcation

transition near the Bautin bifurcation point, the

Newton–Raphson iterative method is used to track

the evolution of the periodic solution. From the change

of the Hopf bifurcation diagram, it can be found that

the supercritical Hopf bifurcation is transformed into

subcritical Hopf bifurcation as the bifurcation

(a)

(b)

Fig. 8 Blowup of Fig. 7 near the Hopf Bifurcation points

123

Study on stability and bifurcation of electromagnet-track 2191

parameter moves from left to right relative to the

Bautin bifurcation point. In the case of supercritical

Hopf bifurcation, there are two fold bifurcations,

while in the case of subcritical Hopf bifurcation, there

is one fold bifurcation. As the bifurcation parameters

change, there may be multiple limit cycles coexisting,

so that the system tends to have different stable solu-

tions under different initial disturbances.

Acknowledgements This study was funded by the National

Natural Science Foundation of China (Grant Number

51875483), the National Key R&D Program of China (Grant

Number 2016YFB1200601-A03) and Independent Topics of

National Key Laboratories (Grant Number 2020TPL-T04).

Funding This study was funded by the National Natural

Science Foundation of China (Grant Number 51875483).

Compliance with ethical standards

Conflict of interest The authors declare that they have no

conflict of interest.

Appendix: Parameters of electromagnet-track

beam coupling system in Sect. 4

Parameter Description Value

m Mass of electromagnet 550 kg

g Gravity acceleration 9.8 m/s2

l0 Permeability in vacuum 4p� 10�7 T m/A

N Coil turns of electromagnet

core

800

A Area of the magnetic pole of

the electromagnet

0.014 m2

d0 Rated levitation gap 8 mm

i0 Stable levitation current 11.07 A

kP Gap feedback coefficient 3500

kD Velocity feedback coefficient 300

ke Current feedback coefficient 25

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Study on stability and bifurcation of electromagnet-track 2193