Dynamics Of A Relay Oscillator With Hysteresis

Dynamics of a relay oscillator with hysteresis Tam´as Kalm´ar-Nagy and Pankaj Wahi

Abstract— The dynamics of a hysteretic relay oscillator with sinusoidal forcing is studied in this paper. Periodic excitation gives rise to periodic, quasiperiodic and chaotic responses. A Poincar´e map is introduced to facilitate mathematical analysis. Conditions on the amplitude and frequency of the forcing for the existence of periodic solutions have been obtained. Families of one-period solutions determined as fixed points of the Poincar´e. These families of one-period solutions represent coexisting subharmonic responses. Stability analysis reveals that these solutions can be classified as center or saddle.

I. INTRODUCTION In this paper, the dynamics of a hysteretic relay operator under periodic excitation is studied. We find conditions for the existence of periodic solutions and show that infinitely many subharmonic responses coexist. Relay systems with hysteresis have been attracting increasing attention due to applications in a wide range of engineering problems including voltage regulators, DC motors, and servomechanisms [1], [2], [3], [4], [5], [6], [7]. Relays, in general, have two output branches and the output switches between the branches at certain critical values of the input. Hence, they belong to a general class of piecewise smooth dynamical systems which also include systems with play or backlash [8], systems with friction [9], systems with impacts [9], [10], and other hybrid systems [11], [12], [13], [14]. For an ideal relay, switching between the branches occurs at the same critical input value and the output is a single-valued function of the input. Feedback systems with ideal relays have been studied in greater detail by Johansson et al. and di Bernardo et al. (see [17], [18] and references therein). For a relay with hysteresis the output switches between the branches at different critical values of the input and the output is no longer single-valued for inputs between these two critical values [1]. ˚ om [7] studied the existence and Andronov [2] and Astr¨ stability of one-period solutions (i.e., solutions having exactly two relay switchings per period). Gonc¸alves et al. [15], [16] presented a global analysis of relay systems using Lyapunov functions. Periodic solutions in a relay system with square-wave excitation was considered by Varigonda and Georgiou [3], while Fleishman [19], and Shaw and Holmes [20] focused on periodic response of piecewiselinear systems under sinusoidal forcing. There is also a similarity between relay oscillators and the much studied T. Kalm´ar-Nagy is with the Department of Aerospace Engineering, Texas A&M University, College Station, Texas, 77843, USA.

[email protected] P. Wahi is with the Department of Aerospace Engineering, Texas A&M University, College Station, Texas, 77843, USA.

[email protected]

repeated impact of a ball with a sinusoidally vibrating table [10], [21], [22] and its Hamiltonian analog [23]. A related class of nonlinear systems involves relay operators with delays in the input. Barton et al. [24], Fridman et al. [25], and Norbury and Wilson [26] considered first order delayed relay systems while Bayer and Heiden [27], Sieber [28], Barton et al. [29] and Colombo et al. [30] focused on second order systems. These studies investigate periodic solutions, their bifurcations, as well as chaotic solutions. In this paper, we focus on the rich dynamics of an individual relay hysteretic operator in the presence of sinusoidal forcing. We obtain conditions on the amplitude and frequency of the forcing for which bounded solutions exist. To facilitate the analysis, we introduce a 2D Poincar´e map. Fixed points of the Poincar´e map correspond to periodic solutions of the system. There are two families of one-period solutions representing coexisting subharmonic responses of the system. On the Poincar´e plane, one fixed point is a center while the other is a saddle. As the parameters are varied, the centers and the saddles merge in a saddlecenter bifurcation [31], [32] leaving a single family of nonhyperbolic one-period solutions. Chaotic tangles have been observed numerically implying the presence of infinitely many higher-periodic and aperiodic solutions. II. MATHEMATICAL MODEL OF THE RELAY OSCILLATOR The equation studied in this work is x ¨(t)+F [x(t)] = A cos(ωt+φ) , A ≥ 0 , ω > 0, φ ∈ (−π, π] . (1) Where A, ω, and φ are the amplitude, frequency, and phase of the forcing, respectively. The hysteretic relay operator F [x(t)] (shown in Fig. 1) is defined as   −1, x(t) ≤ 0 e, 0 < x(t) < 1 F [x(t)] =  1, x(t) ≥ 1 where e is −1 or 1 depending on the initial conditions and the time history of the solution. When F [x(t)] = ∓1, the evolution of the dynamical system is described by (I) x ¨I (t) − 1 = A cos(ωt + φI ) (II) x ¨II (t) + 1 = A cos(ωt + φII ) ,

(2) (3)

where the subscripts are used to differentiate between the two subsystems. The complete description of the system also requires initial conditions. Without loss of generality, these initial conditions can be specified as xI (0) = 0,

x˙ I (0) = vI .

(4)

F[x(t)] = 1

(b)

(a)

=0

x(t) II Hysteretic region

II

I

x(t)

^ ^ ^ ^

I

-1

1

0

x(t)

x(t)

1

x(t) =1

^ ^ 0

F[x(t)]

II

I

x(t )

+1

(a)

Hysteretic region

x(t )

F[x(t)]

F[x(t)] = -1

Fig. 1. a) The relay operator with hysteresis. b) Phase portrait of (1) for A=0. Initial conditions are x(0) = 0 and x(0) ˙ = 0. (b) SΙ

Figure 1 (b) shows the phase portrait in x(t) and x(t) ˙ for the free response of the system (i.e., A = 0, see [33]). The dynamics switches between the subsystems when the solution trajectories intersect xI (t) = 1 from the left or xII (t) = 0 from the right. To make the analysis simpler, time is reset when transition occurs between subsystems. To account for this artificial time-shift, the phase of the forcing is ’updated’ at the switchings. Therefore, the evolution of the dynamics is completely specified by x ¨I (t) − 1 = A cos(ωt + φI ) , xI (0) = 0, x˙ I (0) = vI , t ∈ [0, tI ]

(5)

t ∈ [0, tII ].

(6)

x ¨II (t) + 1 = A cos(ωt + φII ) , xII (0) = 1, x˙ II (0) = vII , Here tI and tII are the switching times defined implicitly by xI (tI ) = 1 and xII (tII ) = 0, respectively. III. THE PHASE SPACE AND SOLUTIONS Figure 2 (a) depicts the evolution of solution trajectories in x(t), x(t) ˙ and F [x(t)]. To preserve the uniqueness of solution trajectories in the state space, time t is introduced as another state variable resulting in an extended phase space ([8], [10]). Clearly xI (t) ∈ XI = {xI (t) | xI (t) ≤ 1} and xII (t) ∈ XII = {xII (t) | xII (t) ≥ 0}. Also, x˙ I (t), x˙ II (t) ∈ + R and t ∈ phase space is therefore XI × S R . The extended + R×R XII × R × R+ . This space is a proper subset of R3 × {−1, 1}, where the discrete set {−1, 1} is simply the range of F [x(t)] It is again emphasized in Fig. 2 that the system consists of two distinct subsystems, viz. subsystem I and subsystem II. The dynamics of the system switches from subsystem I to subsystem II when the solution trajectories intersect the surface SI = {(x˙ I (t), t) | xI (t) = 1} and from subsystem II to subsystem I when they intersect the plane SII = {(x˙ II (t), t) | xII (t) = 0}) as demonstrated in Fig. 2(b). Having described the structure of the phase space, we now turn our attention to the solutions. The solution of subsystem I can be written in closed form as   A A 1 2 xI (t) = t + vI − sin(φI ) t + 2 cos(φI ) 2 ω ω

SΙ

SΙΙ

t

SΙΙ

t x ΙΙ (t)

x Ι (t) =0

1

t

0

= (t)

(t)

,x =0

t

= (t)

,x =0

t

x Ι (t)

,x =0

0, t=

1 t)= x(

x ΙΙ (t)

Fig. 2. (a) Phase space of S (1) in x(t), x(t) ˙ and F [x(t)].(b) Extended phase space (xI (t), x˙ I (t), t) (xII (t), x˙ II (t), t)

A cos(ωt + φI ), (7) ω2 A A x˙ I (t) = t + vI − sin(φI ) + sin(ωt + φI ) . (8) ω ω Similarly, the solution of subsystem II is   A A 1 xII (t) = 1 − t2 + vII − sin(φII ) t + 2 cos(φII ) 2 ω ω A − 2 cos(ωt + φII ), (9) ω A A x˙ II (t) = −t + vII − sin(φII ) + sin(ωt + φII ) . (10) ω ω Note that the transformation (vI , φI ) → (−vII , φII + π) in (7) and (8) is equivalent to −

(xI (t), x˙ I (t)) → (1 − xII (t), −x˙ II (t)) ,

(11)

and the substitution (vII , φII ) → (−vI , φI + π) in (9) and (10) leads to (xII (t), x˙ II (t)) → (1 − xI (t), −x˙ I (t)) .

(12)

Therefore a solution of one subsystem with an initial velocity v and initial phase of the forcing φ also represents solution trajectories of the other subsystem (with the corresponding initial velocity −v and initial phase φ+π). As a consequence, solutions appear in pairs, i.e. if (x(t), x(t)) ˙ is a solution of (1), then so is (1 − x(t), −x(t)). ˙ In the next section we compute the switching times that will be needed for obtaining the Poincar´e map.

IV. SWITCHING TIMES

Rearranging (18) and (20) yields

In order to find the switching time tI at which transition takes place between subsystem I and II, the switching criterion xI (tI ) = 1 is substituted into (7) resulting in   1 2 A A t + vI − sin(φI ) tI + 2 cos(φI ) 2 I ω ω A − 2 cos(ωtI + φI ) − 1 = 0 . (13) ω The switching time tI is the first positive root of (13) and is a function of vI and φI for fixed A and ω. Due to the transcendental nature of the equation numerical solution is required. The function tI (vI , φI ) can have discontinuities depending on the parameters A and ω. The velocity at the transition (from (8)) is A A sin(φI ) + sin(ωtI + φI ) . (14) ω ω The phase of the forcing at the transition is simply x˙ I (tI ) = tI + vI −

φI + ω tI , mod 2 π .

(15)

Similarly, to find the time tII of the transition from subsystem II to subsystem I, the switching criterion xII (tII ) = 0 is substituted into equation (9) to yield   1 A A − t2II + vII − sin(φII ) tII + 2 cos(φII ) 2 ω ω A cos(ωtII +φII ) +1 = 0 . (16) ω2 The switching time tII is the smallest positive root of this equation. The velocity at the transition is (from (10)) −

A A sin(φII ) + sin(ωtII + φII ) ω ω (17) + ω tII , mod 2 π.

x˙ II (tII ) = −tII + vII − and the phase is φII

V. POINCAR E´ MAP With the knowledge of the switching times, we are now in the position to construct a map to effectively study the behavior of solutions of (1). First, consider the mapping (x˙ I (0) = vI , φI ) → (x˙ I (tI ), φI + ω tI ) of the initial velocity and phase to the velocity and phase at the time of the transition from subsystem I to subsystem II. Recall that the initial and final positions are uniquely specified by xI (0) = 0 and xI (tI ) = 1. The final velocity and phase at the transition ((14) and (15)) will serve as initial velocity and phase for the solution of subsystem II, i.e. A A sin(φI ) + sin(φII ) ω ω = φI + ω tI , mod 2 π .

vII = x˙ I (tI ) = tI + vI − φII

(18) (19)

Similarly, the final values of velocity and phase of the solution of subsystem II will provide the initial conditions for the solution of subsystem I as A A sin(φII )+ sin(φI ) (20) ω ω + ω tII , mod 2 π. (21)

vI = x˙ II (tII ) = −tII +vII − φI = φII

A A sin(φII ) = tI + vI − sin(φI ) (22) ω ω A A (23) vI − sin(φI ) = −tII + vII − sin(φII ) . ω ω The form of these expressions motivates the introduction of a new variable z = v − A ω sin(φ). Equations (22) and (23) can now be rewritten as vII −

zII = tI + zI ,

(24)

zI = −tII + zII

(25)

A where zI = vI − A ω sin(φI ) and zII = vII − ω sin(φII ). We can now relate initial values of the variables z, φ to their values at the switchings by the two maps ΠI , ΠII as       zII zI zI + t I = ΠI = . (26) φII φI φI + ωtI , mod 2 π       zI zII zII − tII = ΠII = . φI φII φII + ωtII , mod 2 π (27) To specify the range and domain of these maps, we introduce the Poincar´e surfaces ΣI = {(zI , φI )|x(t) = 0} and ΣII = {(zII , φII )|x(t) = 1}. Clearly, ΠI and ΠII are maps from ΣI onto ΣII and from ΣII onto ΣI , respectively. The Poincar´e map (a.k.a. return map) Π is now defined as the map of the plane ΣI onto itself after a pair of switchings. The map Π is therefore obtained by composing the two maps ΠII and ΠI as     zI zI Π = ΠII ◦ ΠI , (28) φI φI

Substituting (26) and (27) in the above results in     zI + tI (zI , φI ) − tII (zI , φI ) zI = . Π φI + ωtI (zI , φI ) + ωtII (zI , φI ), 2π φI (29) The implicit dependence of the switching times tI and tII on (zI , φI ) is also emphasized. With the new variables zI and zII introduced in (13) and (16), the switching times tI and tII are determined as the first positive roots of A A 1 2 t +zI tI + 2 cos(φI )− 2 cos(ωtI +φI ) −1 = 0 , (30) 2 I ω ω 1 A A − t2II +zII tII + 2 cos(φII )− 2 cos(ωtII +φII ) +1 = 0 , 2 ω ω (31) respectively. Substituting for zII and φII from (24) and (19) into (31) results in A 1 − t2II + (zI + tI )tII + 2 cos(φI + ωtI ) 2 ω A − 2 cos(φI +ωtI +ωtII ) +1 = 0 . (32) ω Equations (30) and (32) now define the implicit dependence of tI and tII on (zI , φI ). As mentioned previously, the switching times are not necessarily continuous functions of the parameters. However, in this study we consider parameter

We first locate one-period solutions of (1), i.e., solutions which involve a single pair of switchings between ΣI and ΣII . Equivalently, we are looking for fixed points of the Poincar´e map Π. A. One-period Solutions Fixed points of the Poincar´e map are given by  ∗   ∗  zI zI =Π . φ∗I φ∗I

n2 π 2 nπ + 2 zI∗ =0 2 ω ω which can be solved for zI∗ to give nπ zI∗ = − , n = 1, 3, 5, · · · . (43) 2ω The countably many values of n in (43) define families of fixed points for a given set of parameters A and ω. Also there are two values of φ∗I for a given set of parameters A and ω given by (42). These together define two families of fixed points of the Poincar´e map as  2   ω nπ ∗ ∗ , ± arccos , n = 1, 3, 5, · · · . (zI , φI )1,2 = − 2ω 2A (44) Each of these fixed points corresponds to a one-period solution of (1). The x(t)-x(t) ˙ phase planes of the system corresponding to n = 1, 3, 5 and 7 are shown in Fig. 3. These

Accordingly, (29) yields the conditions

0

−4 0.8

(35)

which after substitution into (34) results in nπ tI = tII = , n ∈ Z+ ast > 0 . (36) ω Having obtained the switching times tI and tII for a fixed point of the Poincar´e map, we still need to determine the values of zI∗ and φ∗I . Using (30), (32) and (36), we obtain

−10 −5

0 5 x(t)

−8 −30

10

0

−5 −10

x(t)

10

30

−10 −60

n π nπ + zI∗ +1+ ((−1)n − 1) = 0 . (38) 2 2ω ω Subtracting (37) from (38) yields A cos(φ∗I ) ((−1)n − 1) = 0 . (39) ω2 Equation (39) can only be solved for odd n. Next, we solve (39) for cos(φ∗I ) as 2+2

ω2 (40) 2A Recall that A > 0. Since | cos(φ∗I )| ≤ 1, the condition for existence of a one-period solution is given by cos(φ∗I ) =

(41)

From (40), the initial phase φ∗I corresponding to the oneperiod solution is obtained as  2 ω ∗ . (42) φI = ± arccos 2A

−20

x(t)

20

60

20

60

0.4 . 0 −0.4 −0.8 −0.5 0

0.5 x(t)

1

1.5

(−π/2, π/3) (−3π/2, π/3)

(−5π/2, π/3) (−7π/2, π/3)

φI

(−π/2, −π/3) (−3π/2, −π/3) (−5π/2, −π/3) (−7π/2, −π/3)

10 5 . 0 −5 −10

zI

−60

(37)

A cos(φ∗I ) ω2

ω 2 ≤ 2A .

5 .

0

−4

2 1 . 0 −1 −2

−20

x(t)

8

4

4

x(t)

n2 π 2 nπ A cos(φ∗I ) + zI∗ −1+ (1 − (−1)n ) = 0 2 2ω ω ω2 and

10

x(t)

tI = tII

.

−2

(34)

Equation (33) gives

2 2

.

4

.

x(t)

+ ω tI + ω tII , mod 2 π .

(33)

8

2

x(t)

=

φ∗I

+ tI − tII

4

x(t)

φ∗I

=

zI∗

x(t)

zI∗

x(t)

VI. PERIODIC SOLUTIONS

Adding (37) and (38), we have

x(t)

values A and ω for which the switching times and consequently the Poincar´e map are continuous. With the introduction of the Poincar´e map we have reduced the study of the original hybrid system (1) to that of the discrete map Π : ΣI → ΣI . In the following, we derive conditions for the existence of fixed points of this map which correspond to one-period solutions of the original system.

0

.

−4 −1

0

x(t)

1

2

0

−4 −10 −5

0 5 x(t)

10

−8 −30

−10

x(t)

10

30

Fig. 3. Phase portraits x(t)-x(t) ˙ for A = 1 and ω = 1 corresponding to the fixed points of the Poincar´e map foe n = 1, 3, 5 and 7.

different one-period motions represent 1 : n subharmonic resonances of the system and they coexist for a given set of parameter values. The one-period solutions shown in Fig. 3 corresponding to (−π/2, π/3), (−π/2, −π/3), (−3π/2, π/3) and (−3π/2, −π/3) are plotted together in Fig. 4 to emphasize their coexistence. The initial conditions for each phase portrait has been chosen to be consistent with the fixed points given in (44), e.g., for A = 1, ω = 1 and n = 1, the initial conditions corresponding to zI = √ −π/2 and φI = −π/3 are x(0) = 0 and x(0) ˙ = −π/2 − 3/2. Having obtained the conditions for a one-period solution of system (1), or equivalently the fixed points of the Poincar´e map (29), we perform a stability analysis of the fixed points

2 A sin(φ∗I ) ¯ nπ z¯I + φI = 0 . (49) ω ω2 Equation (49) can be solved for t¯II (after substitution of (46) for t¯I ) as t¯II = r z¯I + s φ¯I (50)

6

+

4

2

x(t)

where 0

.

r= ω

−2

and s=

−4

−6



−10

−5

0 x(t)

5

10

Fig. 4. Phase portraits x(t)-x(t) ˙ for A = 1 and ω = 1 corresponding to the fixed points (−π/2, π/3) (solid line), (−π/2, −π/3) (dashed line), (−3π/2, π/3) (dotted line) and (−3π/2, −π/3) (dash-dotted line).

of the Poincar´e map in the next subsection. This is equivalent to the stability analysis of the one-period solutions. B. Stability Analysis of One-period Solutions For the purpose of stability analysis of the fixed point of the Poincar´e map (29), we approximate the linearized Poincar´e map about the fixed point (zI∗ , φ∗I ). We first substitute zI = zI∗ + ǫ¯ zI and φI = φ∗I + ǫφ¯I in (30), where ǫ ≪ 1 is a small parameter. For an infinitesimal perturbation in the parameters zI and φI , the switching times tI and tII will be perturbed by infinitesimal amounts only since we are considering parameter values for which there are no discontinuities in the switching times. Accordingly, we also nπ + ǫ t¯I in (30), expand in a Taylor series substitute tI = ω about ǫ = 0 and retain terms of O(ǫ) to get   nπ A sin(φ∗I ) ¯ n π 2 A sin(φ∗I ) ¯ tI + + zI∗ − z¯I − φI = 0 . ω ω ω ω2 (45) Equation (45) can be solved for t¯I in terms of z¯I and φ¯I as t¯I = p z¯I − q φ¯I , where

and q=

2 A2 sin(φ∗I )2 + 2 A sin(φ∗I ) (2 n π + ω zI∗ ) 
 .
2 nπ ∗ 2 ω A sin(φ∗I ) − nπ + ω z − I 2 2

(51)

(52)

Now substituting zI = zI∗ + ǫ¯ zI , φI = φ∗I + ǫφ¯I , tI = n π nπ + ǫ t¯I and tII = + ǫ t¯II in (29), we have ω ω  ∗    zI + ǫ¯ zI zI∗ + ǫ(¯ zI + t¯I − t¯II ) Π = . φ∗I + ǫφ¯I φ∗I + ǫ(φ¯I + ω t¯I + ω t¯II ), 2π (53) Equation (53) after substitution of (46) and (50) gives the linearized Poincar´e map (DΠ) governing the evolution of the perturbations around the fixed point (zI∗ , φ∗I ) as   1+p−r −(s + q) DΠ = . (54) ω(p + r) 1 + ω(s − q) The eigenvalues of the matrix DΠ determine the stability of the fixed point of the Poincar´e map (29) or equivalently the one-period solution of the system (1). The eigenvalues of DΠ are given by p tr (DΠ)2 − 4 det(DΠ) tr (DΠ) ± , (55) λ1,2 = − 2 2 where tr (DΠ) = 2 + p − r + ω(s − q) and det(DΠ) = 1 + p − r + ω (2ps + 2qr − q + s) . Substituting for p, q, r and s from (47)-(52) in the above, it is easy to verify that det(DΠ) = 1 and 2

(46)

nπ ∗ A sin(φI ) − ω zI∗ − n π

(47)

2 A sin(φ∗I ) . ω (A sin(φ∗I ) − ω zI∗ − n π)

(48)

p=

n π ω 2 zI∗ − 3 n π A ω sin(φ∗I )

2 nπ ∗ 2 A sin(φ∗I ) − nπ + ω z − I 2 2

We next substitute zI = zI∗ + ǫ¯ zI , φI = φ∗I + ǫφ¯I , tI = n π nπ + ǫ t¯I and tII = + ǫ t¯II in (32), expand in a Taylor ω ω series about ǫ = 0 and retain terms of O(ǫ) to yield     n π 2 A sin(φ∗I ) ¯ A sin(φ∗I ) ¯ ∗ tII + tI zI + + ω ω ω

tr (DΠ) =

2 (2 A sin(φ∗I ) + n π) + 16 A n π sin(φ∗I ) (2 A sin(φ∗I ) − n π)

2

.

(56) Since λ1 λ2 = det(DΠ) = 1, there are three possibilities for the eigenvalues: 1) Both λ1 and λ2 are real and distinct. In this case, one has a modulus greater than one (eigenvalue outside the unit circle) and the other smaller than one (eigenvalue inside the unit circle). This fixed point is a saddle. 2) λ1 and λ2 are complex conjugate with |λ1 | = |λ2 | = 1 (eigenvalues on the unit circle). The fixed point is a center. 3) Either λ1 = λ2 = 1 or λ1 = λ2 = −1. The fixed point is a non-hyperbolic fixed point and nonlinear analysis is required to determine the behavior of the fixed point.

From (55), we note that the eigenvalues λ1,2 are real and distinct if tr (Dπ) > 2, and they are complex  2  conjugate if ω ∗ tr (DΠ) < 2. Substituting φI = arccos in (56) gives 2A √
2 4 A2 − ω 4 + n2 π 2 + 6 n π 4 A2 − ω 4 √
. tr (DΠ) = 4 A2 − ω 4 + n2 π 2 − 2 n π 4 A2 − ω 4

√ Clearly tr (Dπ) > 2 for ω < 2 A and hence, the eigenvalues are real and distinct. Therefore,  2 the  family of fixed points ω corresponding to φ∗I = arccos are saddles. Similarly, 2A   2 ω a substitution of φ∗I = − arccos results in 2A √
2 4 A2 − ω 4 + n2 π 2 − 6 n π 4 A2 − ω 4 √
<2 tr (DΠ) = 4 A2 − ω 4 + n2 π 2 + 2 n π 4 A2 − ω 4

1.5 Saddle 1

Phase space in zI -φI plane for A = 1 and ω = 1.

C. Global Dynamics The dynamics of the system in the zI -φI variables for A = 1 and ω = 1 is shown in Fig. 6. While it seems from Fig. 6 that there is a homoclinic orbit around the center, in reality the unstable and the stable manifolds of the saddles intersect transversally giving rise to chaotic tangles. These can be clearly seen in Fig. 7 where the results of forward and backward iterations of 250 × 250 points in a small neighborhood of the saddle for A = 1 and ω = 1 are plotted. This numerical evidence shows the existence of a Smale horseshoe which implies an infinite number of higherperiodic and bounded aperiodic solutions. More details about

0 φ

I

φ

I

0.5 *

Fig. 6.

√

2 A. Hence,the family of fixed points correspond ω2 ∗ are centers. These two branches ing to φI = − arccos 2A of one-period solutions are shown in Fig. 5. for ω <

−0.5

−1 Center

0

0.2

0.4

0.6

0.8 1/2

ω/A

Fig. 5.

1

1.2 z

1.4

I

−1.5

I

The two branches of fixed points of the Poincar´e map (29).

In the limiting case of ω 2 = 2 A, the saddle and the center merge in a reverse saddle-center bifurcation [31], [32] leaving a single family of fixed points   −n π ∗ ∗ √ ,0 n = 1, 3, 5, · · · . (zI , φI ) = 2 2A At these points both the eigenvalues are equal to 1. Therefore, this is a non-hyperbolic fixed point and nonlinear analysis is needed to determine stability.

Fig. 7. Phase space in zI -φI plane for A = 1 and ω = 1 showing transverse intersection of the stable and unstable manifolds of the saddle.

the complicated dynamics in this system including higherperiodic, quasiperiodic and chaotic responses can be found in [33].

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