The location of limit cycles and the associated mechanism in the state switched nonlinear system

Optik 127 (2016) 2931–2935

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The location of limit cycles and the associated mechanism in the state switched nonlinear system Chun Zhang ∗ School of Mathematical Science, Huaiyin Normal University, Huai’an, Jiangsu 223300, PR China

a r t i c l e

i n f o

Article history: Received 4 September 2015 Accepted 11 December 2015 Keywords: Switched system Bifurcation mechanism Periodic oscillations Period-decreasing sequence Period-adding sequence

a b s t r a c t By introducing switching law associated with the values of the state variables, a switched mathematical model is established. Poincaré map of the whole switched system is defined by suitable local sections and local maps, and the formal expression of its Jacobian matrix is obtained. The location of the fixed point corresponding to the limit cycle of the switched system is calculated by the shooting method. To investigate switching behavior of this system, the equilibrium points and their bifurcations of the subsystems are derived. An interesting switching behavior, i.e., the so-called 4T-focus/focus/focus periodic switching is explored in detail to present the mechanism of the movement. With the increase of the parameter, the turning points on the switching surface may be attracted by different attractors of the subsystem, causing the turning points decrease from four to two. Then the system forms other types of periodic solutions. Furthermore, period-decreasing and period-adding sequences have been obtained, which can be explained by the changes of the duration time in the subsystems. © 2015 Elsevier GmbH. All rights reserved.

1. Introduction Switched dynamical systems are one of the most interesting class of piece-wise smooth systems that arise in scientific problems and engineering applications such as the electrical circuits [1], chemical processing [2], communication networks [3]. Typically, a switched dynamical system is composed of a family of subsystems and rules that governs the switching of them. Generally, switching rules are determined by the values of the state variables or related to the fixed time for the occurrence of the alteration [4,5]. When the switching rules are satisfied, the vector field may alternate from one dynamical subsystem to another, leading to the vector field is not differentiable at turning point [6–8]. Because of the wide existence of switches, the behaviors of switched system have received much interest during the last decades and a lot of results have been reported [9–12]. In [9], some new exponents were defined by which the essential patterns that guarantee the stability of fast switching systems could be figured out and the capacity and efficiency of random switch for stabilizing a switched system were described. In [10], new type of characteristic exponent was introduced to capture

∗ Tel.: +86 15052627056. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijleo.2015.12.075 0030-4026/© 2015 Elsevier GmbH. All rights reserved.

the stability feature of continues-time switched systems and two criteria of asymptotic exponential stability for linear and nonlinear case were obtained. In [11], the stabilization of switched nonlinear systems with passive and non-passive subsystems was studied via the average dwell time method. In [12], some sufficient conditions were obtained to ensure global asymptotical stability and global robust stability of the unique equilibrium of switched neural networks. Up to now, much attention has been paid to the stability, controllability, reachability, observability and design of the switched system [13–17]. However, little work has been done in the dynamical evolution with the variation of the parameters, the difference between the behaviors caused by two types of switches, respectively, and the parameter bifurcations associated with the switches as well as the mechanism of the complexity. Here we consider a switched system alternating between two subsystems described by Rössler system and Lorenz oscillation with switches defined on the conditions related to the state variables. Some interesting phenomena such as periodic switching, period-decreasing, period-adding sequences have been obtained. Based on the equilibrium points analysis of the two subsystems as well as the critical behaviors at the switches, the mechanism related to the periodic orbits observed are presented to account for the evolutions of the trajectories.

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3. The location of limit cycles In this section, we investigate the generation of the period oscillation of the switched system (2.1). The multiple-shooting method present in [18] will be adopted to locate limit cycle. With the definition of the trajectory of the switched system given by (2.2) and (2.3), the following two local maps are defined T1 : S2,3 −→ S1 X2i−1 −→ X2i = (i2 , X2i−1 ),

Fig. 1. (a) Trajectory partition and (b) the switching scheme.

X2i −→ X2i+1 = (i1 , X2i ).

2. The model of switched system Let us consider a state switched system, which alternates between Rössler system and Lorenz oscillator, written in the form dX = f (X), dt



 ∈ {1, 2},

(2.1)

T

vector fields and f2 (X) =

˛(y − x), x(ı − z) − y, xy − ˇz .  is a switching scheme. When  = 1 means the subsystem 1, named SW1, be active, while  = 2 means the subsystem 2, named SW2, be active. In order to investigate the typical dynamics of the switched system, the switch scheme  related to three switching bound1 } and S 2 3 aries S1 = {X ∈ R3 | x = xref 2,3 = {X ∈ R | x = ±xref } are introduced, shown in Fig. 1. For the trajectory of the switched system starting from an initial point X0 , governed by SW1, i.e.,  = 1. Once one of the state vari1 twice at t =  1 , it may able x passes across the reference value xref 0 then be governed by SW2, i.e.,  change from 1 to 2. The trajectory moving according to SW2 may change back to the vector field f1 (X) 2 or −x2 when the trajectory passes across the reference value xref ref

1 four times at t = 01 + 12 , until the state variable x passes across xref twice again and the motion then continues as above. Based on the switching scheme described above, two sequences (Xi , i ∈ N) and (ik , i ∈ N, k ∈ {1, 2}) can be obtained, 2 1

1 1

2 2

2

1

1

3

2

where i = 1, 2, . . .. Assume that the switching surface S2,3 is the poincaré section, the poincaré mapping T from S2,3 to S2,3 can be expressed as T (X2i−1 ) = T2 ◦ T1 (X2i−1 ) = (i1 , (i2 , X2i−1 )).

X = (x, y, z)T .f ,  ∈ {1, 2} are the f1 (X) = (−y − x, x + y, xz − z + )T

where with

3

4

Essentially, the periodic oscillation of the switched system (2.1) is equivalent to the existence of the fixed point of the poincaré mapping T, namely X∗ − T (X∗ ) = 0.

DT = DT 2 × DT 1 =

i=0

⎡ ⎤ i i   1 ⎣ (j1 + j2 ), (j+1 + j2 )⎦ ,(2.2) j=0

j=0

the other is determined by SW2, i.e.,

X(t) = (t, X2i+1 ),

t ∈

∞ i=0

⎡ ⎤ i i   1 1 2 ⎣ (j+1 + j2 ), (j+1 + j+1 )⎦, j=0

∂X2i+1 ∂X2i × ∂X2i ∂X2i−1

(3.4)

Notice that X2i = (i2 , X2i−1 ), X2i+1 = (i1 , X2i ) and the duration time i1 and i2 are dependent on the state variation X2i and X2i−1 , respectively, then the Jacobian matrix

∂X ∂X2i and 2i+1 can ∂X2i−1 ∂X2i

be written as the follow form

∂i2 ∂X2i ∂ ∂ = × + ∂X2i−1 ∂i2 ∂X2i−1 ∂X2i−1

(3.5)

∂i1 ∂X2i+1 ∂ ∂ = × + ∂X2i ∂i1 ∂X2i ∂X2i

(3.6)

matrices

∞

(3.3)

Since the analytic expression of the poincaré map T is unknown, and in order to compute the fixed point, we need to compute its Jacobian matrix

∂i

where the points (Xi , i ∈ N) are the turning points and (i1 , i ∈ N) and (i2 , i ∈ N) are the duration time in the two subsystems, respectively. Therefore, the trajectory of the whole switched system can be divided into two parts, one is governed by SW1,expressed as

t ∈

(3.2)

where ∂2 = f2 (X2i−1 ), ∂1 = f1 (X2i ), while the elements of the

2

X1 →X2 →X3 →X4 →X5 →X6 →X7 →· · ·

X(t) = (t, X2i ),

(3.1)

T2 : S1 −→ S2,3

j=0

(2.3) where 02 = 0. According to the switching scheme, it is easy to know that the points (X2i−1 , i = 1, 2, · · ·) are on the switching surface S1 , while the points (X2i , i = 1, 2, · · ·) are on the switching surface S2,3 .

∂

∂X2i−1

∂ i ∂  and can be computed by the following vari∂X2i

ational equations

⎧ d ⎪ ⎪ ⎨ dt (D) = f2X × D  ⎪ ∂  = I ⎪ ⎩ ∂X2i−1 

(3.7)

t=0

⎧ d ⎪ ⎪ ⎨ dt (D) = f1X × D  ∂  ⎪ ⎪ =I ⎩ ∂X  2 2i

(3.8)

t=

i

from t = 0 to t = i2 and t = i2 to t = i2 + i1 , respectively. Where f1X and f2X are the Jacobian matrix of f1 and f2 . I is an identity matrix. Notice that X2i and X2i+1 are on the switching surfaces S1 and S2,3 ,respectively. That is to say 1 q1 (X2i ) ≡ x − xref =0 2 q2 (X2i+1 ) ≡ x ± xref =0

(3.9) (3.10)

C. Zhang / Optik 127 (2016) 2931–2935 10

10

(a)

0 0

20

5

10

15

20

25

30

35

40

45

x

(b)

0

5

10

15

20

25

30

35

40

45

50

50

5

10

15

20

25

30

35

40

45

50

(d)

0 10

20

30

40

50

60

70

80

90

Fig. 2. Periodic switching oscillations for different parameter ı. (a) ı = 12.0; (b) ı = 21.0; (c) ı = 43.0 and (d) ı = 141.5.

i.e., q1 ((i2 , X2i−1 )) = 0

(3.11)

q2 ((i1 , X2i ))

(3.12)

=0

 ∇ q2

∂i2 ∂ ∂ × + 2 ∂X2i−1 ∂X2i−1 ∂i ∂i1 ∂ ∂ × + ∂i1 ∂X2i ∂X2i

 =0

(3.13)



=0

(3.14)

where ∇ q1 = ∇ q2 = (1, 0, 0). Since the orbit is transversal to the two sections, which implies ∇ q1 · ∂2 = / 0 and ∇ q2 · ∂1 = / 0. Thus, we ∂i

obtain the following result

∂i2 ∂X2i−1 ∂i1 ∂X2i

=−

1

∇ q1 ∂2

· ∇ q1

∂i

=−

1

∇ q2 ∂1 ∂i

· ∇ q2

∂ i

⎛

∂X2i−1

∂ ∂X2i

(3.15)

(3.16)

⎞

⎝I −

1

∇ q1 ∂2

⎛ ×

−70 −4

−20 −50

2

4

6

⎝I −

SF−

8

0

H− δ

50

UF− 100

Fig. 3. (a) Two equilibrium points for Rössler system with  = 0.1,  = 4.0,  = 6.0 and (b) the equilibrium points of Lorenz system for different parameter ı.

attractor may decrease (see in Fig. 2(c)). However, when the parameter increase to a special value, the period of the periodic solution increasing sharply, leading to the period-adding phenomenon occur, as shown in Fig. 2(d).

∂i

∂i



3ı − 3, ±

3ı − 3, ı − 1).

As shown in Fig. 3(b), the solid branches SF± represent stable focuses and the dashed branches UF± and Sa represent unstable focuses and saddles, respectively, the points H± are Hopf bifurcation points of equilibria EL± with ı = 55.0 and the point BP is fold bifurcation of the equilibria with ı = 1.0. As ı increases from 1.0, the two equilibria EL± becomes gradually far from the saddle EL0 .

In the following, based on the results obtained above, we study the dynamical behaviors and the mechanism of the periodic oscillations.

⎞

∇ q2 ∂1

EL0 = (0, 0, 0), EL± = (±

5. Dynamical mechanisms of switching oscillations

∂ ∂ · ∇ q1 ⎠ ∂X2i−1 ∂i2

1

Because the switched system (2.1) may alternate between two vector fields, the stability analysis of equilibrium point related to the two subsystems is very important. Note that Rössler system has a stable focus ER− = (0.067, − 0.674, 0.674) and a saddle ER+ = (5.933, − 59.326, 59.326) for the fixed parameter above (see Fig. 3(a)). By varying the parameter ı, the equilibrium points of Lorenz system is presented in Fig. 3(b). It can be seen that when ı > 1 there are three equilibria in Lorenz system, denoted by



∂

Substituting (3.15), (3.16) into (3.5), (3.6) and considering (3.4) results in DT =

−15 0

Sa

−10

R+

−2

BP

4. Equilibrium points and bifurcations of the subsystems

Thereby

∇ q1

+

100

t



E

−60

x 0

UF

0 −5

−50 0

H+

SF+

5

x

−40

(c)

(b)

10

−20

0

−50

ER−

y −30

50

−50

15

−10

50

0 −20

20

(a)

0 −10

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∂ ∂ · ∇ q2 ⎠ ∂X2i ∂i1

(3.17)

The location of the fixed point in (3.3) which corresponding to the periodic oscillation of the switched system can be approximated by Newton-Raphson method. 1 = 0, x2 = 5.0 Once the switching law assigned above with xref ref is introduced, the different types of periodic switching attractors can be found. Because of the initial conditions may influence the structures of the attractors of the switched system, here, it is assumed that the switched system (2.1) is initialized at X0 = (4.2778, 4.6964, 0.3049)T in the whole paper. Now, we fix the parameters at ˛ = 5.0, ˇ = 3.0,  = 0.1,  = 4.0,  = 6.0, and take ı as the bifurcation parameter to investigate the dynamical evolution of the oscillator. In Fig. 2(a), the periodic switching oscillation of the system (2.1) is synthesized for ı = 12.0. Intriguingly, with the increasing of the parameter ı, the number of the turning points in the orbit of the periodic attractor changes from four to two (see Fig. 2(b)). Further increasing the parameter reveals that the period of the

5.1. 4T-focus/focus/focus periodic switching We consider the oscillation in Fig. 2(a) for ı = 12.0. By the method mentioned in Section 3, the fixed point of the associated Poincaré map is computed as X∗ = (5.0, 3.06049, 11.46105)T and the corresponding phase portrait is shown in Fig. 4(a). By the overlap of the phase portrait with the attractors of the two subsystems in (x, y, z) space (see Fig. 4(b)), we may find a clear understanding that such oscillation alternates among three behaviors associated with the stable focuses ER− and EL± . As shown in Fig. 4(b), assuming F is the initial point, the trajectory of the system, governed by Rössler system, may move along with FA. Because of the trajectory goes through the boundaries S1 twice at the point A, the vector field instantaneously turns to Lorenz oscillation, causing the trajectory settles down to the stable focus EL− along with AB instead of settling down to the stable focus ER− . The boundaries S3 interrupt the procedure, leading to another turning point B, at which the trajectory tends asymptotically to the stable focus ER− again along BC. The differences between FA and BC lie in the different initial points F and B which causing anther

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long period of transients, it also forms the periodic switching oscillation among three focuses with four turning points. When ı ∈ (20.532, 39.882) , qualitative changes may take place on the switching boundary S1 . All the turning points on the S1 are attracted by the focus EL+ (see Fig. 5(c) for ı = 21.0), which implies that when switches to Lorenz system, the trajectory always settles down to the stable focus EL+ until the next switching boundaries is satisfied. It forms the periodic switching oscillation between ER− and EL+ with two turning points, called as 2T-focus/focus periodic switching. 5.3. Period-decreasing and period-adding cascades Fig. 4. 4T-focus/focus/focus periodic switching for ı = 12.0. (a) Phase portrait in (x, y, z) space and (b) overlap of the phase portrait and the attractors of the subsystems, where S1 , S2 and S3 are switching boundaries.

turning point C on the boundaries S1 . The trajectory does not tend asymptotically to the stable focus ER− from Rössler system but to another stable focus EL+ along with CD governed by Lorenz oscillation. When the trajectory goes through the switching boundaries S2 fourth at point D, the vector field instantaneously turns back to Rössler system again, causing the trajectory settles down to the stable focus ER− along with the path DF to the starting point F. Thus the periodic switching oscillation among three focuses with four turning points is creased, called as 4T-focus/focus/focus periodic switching. 5.2. 2T-focus/focus periodic switching With the increase of the parameter ı, the number of turning points in the orbit of the periodic switching attractor reduces from four to two shown in Fig. 2(b). Now we focus on the phase portraits for several typical values of the parameter ı (see Fig. 5(a)–(c)) and overlap the phase portraits with the attraction domains of the stable focus EL− and EL+ on the switching boundaries S1 to reveal the details of dynamical evolution of this phenomenon. With the increase of the parameter ı, the turning points on the boundary S1 attracted by the stable focuses EL± alternately will change to always attracted by the stable focus EL+ , leading the turning point attracted by EL− disappear. Fig. 5(a)–(c) gives the gradual process. It can be seen that the first turning point will change into the attraction domains of EL+ when ı > 18.485 (see Fig. 5(b) for ı = 19.5). At the beginning, the trajectory separately goes through the switching boundaries S2 and S3 twice, and then the vector field turns to Rössler system. However, the turning point A1 is still in the domain of the focus EL− . After neglecting a sufficiently

Further increase of the parameter ı, although the equilibria EL± are still stable focus for ı ∈ (39.882, 55) (seen in Fig. 3(b)), the turning points on the switching boundaries S1 are neither attracted by the stable focus EL− nor EL+ . Note that regular oscillation and chaotic movements may alternate with the increase of the parameter ı in Lorenz oscillators. When the parameter increases from 40.0 to 140.56, the structure of the periodic switching may change a bit (see Fig. 6(a) and (b)). Further investigation reveals that the period of the periodic switching attractor may decrease with the increase of parameter ı, seeing the related time histories in the finite time sub-interval of [0, 20] for different ı in Fig. 6(d) and (e). For the former case, it can be seen that the number of switches is only 12 and the period can be computed at 3.12, but for the latter, the number of switches is increased to 29 and the period can be approximated at 1.36, implying rapid decrease of the period of the switching attractor may occur with the increase of ı. However, when the parameter increase to 140.56, qualitative changes may take place on the phase trajectory, leading the period of the periodic switching increasing sharply. The phase portrait for ı = 141.0 is shown in Fig. 6(c) and its time histories are in Fig. 6(f), from which, the number of switches is 6 in the finite time sub-interval of [0, 100] and the period can be computed at 28.80. The phenomenon can be understood by the analysis of the turning points on the switching boundary S2 associated with the stable focus defined by Rössler system. Note that the eigenvalues of the stable focus in Rössler system are changeless, leading the trajectories converge to EL− with the same rates. However, with the increase of the parameter from 39.882 to 140.56, the location of turning point which is the initial point when the system switches to Rössler system on S2 may change, leading the trajectory becomes shorter and shorter in Rössler system. As seen in Fig. 7, where A1 , A2 are the turning points and A1 B1 , A2 B2 are the trajectory governed by Rössler system. Thus the period-decreasing cascade

Fig. 5. Turning points on the switching boundaries S1 , where the light-grey areas and while areas are attraction domains of EL− and EL+ , respectively, and the charcoal grey lines are the phase portraits after neglecting a sufficiently long period of transients. (a) ı = 12.0; (b) ı = 19.5 and (c) ı = 21.0.

C. Zhang / Optik 127 (2016) 2931–2935 50

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0

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(b)

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(a)

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(c)

100 50

y

0

0 −50

−50 −50 −30 −20 −10

0

10

20

−100

−40

−20

x 20

0

−150

20

−40

−20

x 30

(d)

40

(e)

20

x

x

x

0

−10

−20

20

40

80

100

(f)

20

10

0

0

x

0

−20

−20 −40

−40

−30 0

5

10

15

20

0

5

t

10

15

20

0

t

20

40

60

t

Fig. 6. Phase portraits and its time histories of periodic switching for (a, d) ı = 50.0; (b, e) ı = 70.0 and (e, f) ı = 141.0.

Acknowledgment This work is supported by the National Natural Science Foundation of China (Grant Nos. 11502091 and 11572141). References

Fig. 7. Ai Bi , i = 1, 2, 3, 4 are the trajectories governed by Rössler system initialed at the turning points Ai on the boundary S3 for different parameters ı, where the subfigure gives a clear understanding of the circular region.

appears in the periodic movement of the system. When the parameter ı ∈ (140.56, 148.0), as presented in Fig. 7, where A3 B3 , A4 B4 are the trajectory governed by Rössler system for the parameter ı = 141.0, ı = 145.0, they spend much more time in Rössler system until the next switched condition is satisfied, causing the periodadding phenomenon. 6. Conclusion Switches between different subsystems may exhibit very complex and fascinating behaviors such as 4T-focus/focus/focus periodic switching, 2T-focus/focus periodic switching. It is found that the trajectory of periodic solution can be divided into parts determined by the transient processes of different attractors of the subsystems. Meanwhile, by analyzing behaviors of the turning points on the switching boundaries, the evolution processes and the associated mechanisms of the periodic solutions are investigated. Furthermore, the duration time toward to the attractors of the subsystems may change with the increase of the parameter, which may lead to the period-adding or period-decreasing cascades in the dynamical evolution of switched system.

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