HOPF BIFURCATION ANALYSIS FOR SIMPLE THIRD-ORDER QUADRATIC SYSTEMS

HOPF BIFURCATION ANALYSIS FOR SIMPLE THIRD-ORDER QUADRATIC SYSTEMS G. Innocenti ∗ , R. Genesio ∗ , A. Tesi ∗

∗

Dipartimento di Sistemi e Informatica, Universit` a di Firenze, via S. Marta 3, 50139, Firenze, Italy {ginnocen,genesio,tesi}@dsi.unifi.it

Abstract: The paper deals with the characterization of Hopf bifurcations in families of simple nonlinear systems, i.e., third-order autonomous systems with few nonlinear terms. By employing Harmonic Balance (HB) tools, the complete set of system parameters corresponding to supercritical and subcritical bifurcations is determined. In addition, it is shown how parameters defining supercritical bifurcations can be used as starting points in bifurcation analysis, in order to locate quite simple system able to display complex behaviours. Keywords: Hopf bifurcation, periodic solutions, harmonic balance, bifurcation analysis, chaos.

1. INTRODUCTION In the last few years a growing interest in discovering simple nonlinear systems, i.e., thirdorder autonomous systems with very few (possibly one) nonlinear terms which are able to display rich complex dynamical behaviours, has been observed (see, e.g., (Eichhorn et al., 1998; Sprott and Linz, 2000; Sprott, 2003)). The interest in these systems stems from the need to provide a deeper understanding of the elementary mechanisms which are at the basis of the emergence of complex behaviour. The standard way to determine simple systems consists in first choosing a family of nonlinear systems with a fixed structure and described by few free parameters, and then employing numerical tools for parameter bifurcation analysis. Unfortunately, it is well known that these tools are difficult to use for more general classes of simple nonlinear systems which may depend on many free parameters. In these cases the knowledge of suitable starting points in the parameter space is of fundamental importance. Obviously, these starting points should correspond to behaviours

which are close to complex ones and, most importantly, should be determined in an analytic way. The aim of the present paper is to provide analytical tools to determine suitable starting points for a class of simple nonlinear systems. More specifically, families of third-order autonomous systems, described by a differential equation involving only a quadratic term in addition to the linear ones and containing four free parameters, are considered. Since these systems admit a feedback representation composed by a linear block in the forward path and an explicit nonlinearity in the feedback path, the Harmonic Balance (HB) tools can be fruitfully used to investigate their dynamical behaviour (Mees, 1981; Moiola and Chen, 1996). In particular, the location in the four-dimensional parameter space of all supecritical Hopf bifurcations of the family is analytically characterized. Such bifurcations parameters can be efficiently employed as starting points to determine simple nonlinear systems displaying rich complex dynamics, as shown in the illustrative examples. The paper is organized as follows. The considered family of simple nonlinear systems is introduced in

Section 2. Section 3 provides the complete characterization of supercritical/subcritical Hopf bifurcations via HB. Some illustrative examples are discussed in Section 4, while some brief comments are drawn in Section 5. Notation R: real space; C: complex space; : imaginary unit; ℜ [x]: real part of x ∈ C; ℑ [x]: imaginary part of x ∈ C, D: d/ dt operator.

2. PROBLEM SETUP Consider the family of simple nonlinear systems which obey to the following third-order differential equation ... x + a1 x ¨ + a2 x˙ + a3 x + a4 g(x, x, ˙ x ¨) = 0, (1) where x ∈ R is the system output signal and a = (a1 , a2 , a3 , a4 ) ∈ R4 is the vector describing the system parameter space. The function g(x, x, ˙ x ¨), which provides the pure nonlinear part of the system, is assumed to be given by a unique quadratic term, i.e.,  g(x, x, ˙ x ¨) ∈ x2 , xx, ˙ x ¨x, x˙ 2 , x ¨x, ˙ x ¨2 . (2)

Clearly, eq.s (1)-(2) represent six different families of simple nonlinear systems depending on the selection of g(x, x, ˙ x ¨). Hereafter, these families will be denoted by Fi , i = 1, . . . , 6, according to the order in (2). The aim of the paper is to provide, for each of the six families, the subset of the parameter space, hereafter denoted by AH ⊆ R4 , where a Hopf bifurcation occurs. Moreover, we are interested in determining the sets AHsup ⊆ AH and AHsub ⊆ AH which contain all the supercritical and subcritical Hopf bifurcations, respectively. Clearly, the parameter vectors belonging to these subsets correspond to simple nonlinear systems which are in general “close” to ones displaying rich dynamical behaviours. In particular, vectors a ∈ AHsup are often used as starting points in numerical tools for parameter bifurcation analysis. In this respect, it is to remark that information on how system parameters affect the bifurcating periodic solution plays a key role in these numerical procedures. The standard way to compute AHsup and AHsub is to rewrite eq.s (1)-(2) in state-space form and then to apply the (state-space) Hopf bifurcation Theorem (Farkas, 1994; Marsden and McCracken, 1976). In particular, the bifurcation stability coefficient, which discriminates between supercritical and subcritical bifurcation, can be efficiently computed via the algorithm proposed in (Howard, 1979) (see also (Hassard et al., 1981; Fu and Abed, 1993; Kuznetsov, 1995)).

0

+ - m −6

-

- y

L(D)

Linear subsystem

G◦y

G



Nonlinear subsystem

Fig. 1. Feedback representation of the family F of simple nonlinear systems. Family F F1 F2 F3 F4 F5 F6

g(x, x, ˙ x ¨) x2 xx ˙ x ¨x x˙ 2 x ¨x˙ x ¨2

y x x x x˙ x˙ x ¨

G◦y y2 y2 y¨y y2 y2 y2

L(D) L0 (D) 1 DL0 (D) 2 L0 (D) DL0 (D) 1 2 D L0 (D) 2 D2 L0 (D)

Table 1. Expressions of y, G◦y and L(D) for each family F. In this paper we pursue a different way to characterize AH , AHsup , and AHsub which is based on the Harmonic Balance (HB) version of the Hopf bifurcation theorem (Mees, 1981; Allwright, 1977; Moiola and Chen, 1996). A first motivation is that system (1) can be represented as a feedback system to which HB tools can be easily applied. Indeed, it is not difficult to verify that (1) admits the feedback interconnection of Fig. 1, where the linear subsystem is described by a linear timeinvariant operator L(D) and the nonlinear subsystem by a scalar time-invariant nonlinear operator G (not necessarily causal), which is explicit, i.e., the output corresponding to any given input function can be directly computed (Tesi et al., 1996). Table 1 summarizes the expressions of y, G◦y and L(D) pertaining to any family F, where L0 (D) has the following expression: a4 L0 (D) = 3 . (3) D + a1 D2 + a2 D + a3 Note that G is non-causal for F3 , while F2 and F5 are diffeomorph to F4 and F6 respectively, according to the relations (x → y, Dx → y) for the first case and (Dx → y, D2 x → y) for the second one. A second motivation to employ the HB approach, is that it provides a natural ground to develop approximate descriptions on how the system parameters affect the bifurcating periodic solutions.

3. COMPLETE CHARACTERIZATION OF HOPF BIFURCATIONS VIA HB In the HB approach the key observation to locate Hopf bifurcations is that if a limit cycle, whose amplitude is collapsing to zero, is represented through a truncated Fourier series, then the approximation error becomes as more negligible as the cycle becomes smaller (Mees, 1981; All-

wright, 1977). In other words, the contribution of a single high harmonic tends to zero more rapidly than those of lower harmonics. Since the starting amplitude of a Hopf generated limit cycle is equal to zero, the HB approach therefore looks for periodic solutions with the smallest number of harmonic terms, nominally only one, and determines the conditions under which such periodic solutions collapse to a single point. Unfortunately, it turns out that the first order approximation is in general not sufficient to locate the bifurcated limit cycle, since it can happen that the related first order HB problem is not solvable. However, it has been shown (Mees, 1981) that, for a system in the form of Fig. 1 with G sufficiently smooth, the second order approximation is always sufficient to completely characterize Hopf bifurcation. More specifically, it turns out that the related second order HB problem always admits a unique solution, which therefore represents a suitable approximation of the bifurcated limit cycle. According to the above reasoning, consider the following HB second order approximation of period 2π/ω: yp (t) = A + B cos ωt + P cos 2ωt + Q sin 2ωt. (4) For the subsequent developments, we find it convenient to introduce the complex valued signal y¯p (t) = A + Beωt + (P − Q) e2ωt , which clearly satisfies yp (t) = ℜ [¯ yp (t)] . Now, the output of the explicit nonlinearity G can be written as (G ◦ yp ) (t) = ℜ [¯ zp (t)] + ∆z(t), where z¯p (t) is the complex valued signal containing up to the second order harmonics and ∆z(t) collects the higher order harmonics. Specifically, z¯p (t) can be expressed as z¯p (t) = N0 A + N1 Beωt + N2 (P − Q) e2ωt , where N0 = N0 (A, B, P, Q, ω) ∈ R N1 = N1R (A, B, P, Q, ω) + N1I (A, B, P, Q, ω) ∈ C N2 = N2R (A, B, P, Q, ω) + N2I (A, B, P, Q, ω) ∈ C (5) are given functions, which can be easily computed for any G in Table 1. For sake of notation, in the sequel the explicit dependence on A, B, P , Q, ω will be sometimes dropped. Hence, it is straightforward to check that, by balancing the second order approximations along the feedback loop, i.e.,

By introducing the notation V1 = V1 (ω) = ℜ [L(ω)] ∈ R W1 = W1 (ω) = ℑ [L(ω)] ∈ R V2 = V2 (ω) = ℜ [L(2ω)] ∈ R W2 = W2 (ω) = ℑ [L(2ω)] ∈ R

(7)

and, observing that we can assume without loss of generality that B 6= 0 and P − Q 6= 0, it is not difficult to show that the second order HB equations (6) can be rewritten in the simplified form: (1 + L(0)N0 (A, B, P, Q, ω)) A = 0 V1 (ω) N1R (A, B, P, Q, ω) = − 2 V1 (ω) + W12 (ω) W1 (ω) N1I (A, B, P, Q, ω) = 2 (8) V1 (ω) + W12 (ω) V2 (2ω) N2R (A, B, P, Q, ω) = − 2 V2 (2ω) + W22 (2ω) W2 (2ω) . N2I (A, B, P, Q, ω) = 2 V2 (2ω) + W22 (2ω) We are interested in determining under which conditions equations (8) are solvable as B, P and Q tend to zero, i.e., for a small amplitude HB second order approximation (4). Obviously, these solvability conditions provide a series of constraints on the system parameter vector a ∈ R4 , which are only necessary for the existence of Hopf bifurcation points. Indeed, according to the Hopf bifurcation Theorem (Farkas, 1994; Marsden and McCracken, 1976), to guarantee sufficiency the stability properties of the equilibrium point to which the cycle collapses have to be investigated. This can be easily done by determining the conditions on the system parameters under which all the eigenvalues of the linearized system at the equilibrium point have a negative real part (except for the couple of pure imaginary ones). By proceeding in this way, we are able to provide a complete analytical description of the sought sets AH , AHsup , and AHsub of all the Hopf bifurcations, of the only supercritical ones and of the only subcritical ones, respectively, for each family F of Table 1. Proposition 1. Let   a3 A = a ∈ R4 : a1 > 0, a3 > 0, a2 = , a4 6= 0 a1 (9)   4 3 4 B = a ∈ R : 8a3 > 3a1 , C = a ∈ R : 8a3 < 3a31 be subsets of the parameters space. Then,

we arrive at the second order HB problem which amounts to solving the following equations

AH ≡ AHsup ≡ A if F = F1 , F2 , F4 AH ≡ AHsub ≡ A if F = F5 , F6 AHsup ≡ A ∩ B, AHsub ≡ A ∩ C if F = F3

A = −L(0)N0 A B = −L(ω)N1 B (P − Q) = −L(2ω)N2 (P − Q).

Proof. The proof is basically the same for each family F and therefore, for space limitation, we

y¯p = −L(D)¯ zp ,

(6)

focus on F4 for the supecritical case and F6 for the subcritical one, which are the two cases considered in the examples in Section 4. Consider family F4 . We have to obtain the specific form of the second order HB equations (8), taking into account that from Table 1 we have L(D) = DL0 (D) and G ◦ y = y 2 . According to (7), it is straightforward to check that the following relations hold L(0) = 0 V1 V12 + W12 W1 2 V1 + W12 V2 2 V2 + W22 W2 V22 + W22


1 ω 2 − a2 a4
1 = a3 − a1 ω 2 a4 ω
1 =− 4ω 2 − a2 a4
1 = a3 − 4a1 ω 2 , 2ωa4

a2 = µ0 + µ ,

=−

(10)

where a4 has been obviously assumed different from zero and the trivial case ω = 0, which corresponds to the constant solution, has been neglected. Also, it turns out that N0 , N1 , and N2 in (5) are such that:
1 1 P 2 + Q2 N0 A = A2 + B 2 + 2 2 N1R = 2A + P N1I = −Q 1 (P N2R + QN2I ) = 2AP + B 2 2 (QN2R − P N2I ) = 2AQ .

Notice that (17) also ensures that equation (16) is satisfied and that the right side of (15) vanishes. However, the latter condition is not sufficient to guarantee solvability also for B tending to zero, because of the presence of the term B 2 . Indeed, for solvability it must be verified that the right side of (15) tends to zero from positive values for some parameter vector a close to the limiting conditions (18). To investigate this issue, we find it convenient to rewrite a3 /a1 = µ0 and a2 as

(11)

Exploiting (10) and (11), the second order HB equations (8) boil down to: A=0 (12)
1 P = ω 2 − a2 (13) a4
1 Q= a1 ω 2 − a3 (14) a4 ω

B2 1 = 2ω 2 ω 2 − a2 4ω 2 − a2 + 2 2 2 2ω a4


+ a1 ω 2 − a3 a3 − 4a1 ω 2 (15)

1 2 a1 ω 2 − a3 4ω 2 − a2 + 2ωa24


− ω 2 − a2 a3 − 4a1 ω 2 = 0 . (16)

Now, we have to determine the conditions under which the above equations are solvable in the limiting case where B, P , and Q tend to zero. From equations (13) and (14), it turns out that it is possible to solve the HB problem for P and Q tending to zero only if a3 ω 2 → a2 = , (17) a1 which in turn implies that a candidate Hopf bifurcation parameter vector a ∈ R4 must necessarily satisfy the following conditions: a3 a4 6= 0 and a2 = >0. (18) a1

(19)

where µ ∈ R. Note that for small µ the corresponding parameter vector a is close to (18). With this notation, it is not difficult to verify that the solution of (16) with respect to ω 2 as a function of µ amounts to: r 5µ0 + 2µ 4µ2 + 4µ0 µ + 9µ20 2 ω (µ) = + = 8µ 64
= µ0 + + O |µ|2 . 3 (20) Exploiting this expression in (15), after some tedious though straightforward manipulations, we get:  

4µ µ 3µ0 + 4µ0 + a21 +O |µ|2 . a24 ω 2 (µ)B 2 = − 3 3 Thus, near a candidate Hopf bifurcation parameter vector, i.e., for small µ, we have:
sgn B 2 = −sgn(µ) .

In turn, this implies that conditions (18) ensures solvability of HB equations (8), since for small µ < 0, i.e., a2 < a3 /a1 , the candidate bifurcating periodic solution may exist. Now, it remains to investigate the stability property of the underlying equilibrium point, corresponding to the constant solution y(t) = A, in the limiting case. Since A = 0, it is straightforward to check that the eigenvalues of the equilibrium point are given by the roots of the polynomial:   a3 λ3 + a1 λ2 + + µ λ + a3 = 0 . (21) a1 Therefore, to guarantee that for µ = 0 the real eigenvalue is negative, the further constraint a1 > 0

(22)

must be considered, in addition to conditions (18), which in turn proves that AH ≡ A. To show that AHsup ≡ A, it is enough to observe that (21) possesses a pair of complex-conjugate roots in which the real part is negative for µ > 0 and positive for µ < 0. Consider now family F6 . It is not difficult to verify that the second order HB equations (8) amount to:

A=0

1 (24) a3 − a1 ω a4 ω 2
1 Q= ω 2 − a2 (25) a4 ω

B2 1 = 2 4 a3 − a1 ω 2 a3 − 4a1 ω 2 + 2 4a4 ω


+2ω 2 ω 2 − a2 a2 − 4ω 2 (26)

1 2 a1 ω 2 − a3 4ω 2 − a2 + 4a24 ω 3


(27) − ω 2 − a2 a3 − 4a1 ω 2 = 0 .

0.6 0.4 0.2

y¨

P =

0.8

(23)
2

0 -0.2 -0.4

It turns out that (A, B 2 , P, Q, ω) is a solution of (23)-(27) if and only (A, −2B 2 ω 2 , Qω, −P ω, ω) solves (12)-(16). This yields that the unique difference with the analysis developed for F4 , is that
sgn B 2 = sgn(µ),

-0.6 -0.8

4. EXAMPLES AND DISCUSSION We first provide a numerical confirmation of the results given in Section 3. For family F4 , according

0

y˙

0.2

0.4

0.6

0.8

1

0.6 0.4 0.2

y¨

Remark 3. The HB approach also provides an analytical approximation of the bifurcated periodic solutions in both supercritical and subcritical cases, as a function of ε. For example, in case of family F4 , it is sufficient to substitute in (13)-(15) the expression of ω(µ) in (20), thus obtaining an analytical expression of B, P , and Q as a function of µ. This yields the sought approximation since, according to (28) and (19), ε = a1 µ.

-0.2

0.8

The HB approach provides a natural way to locate the bifurcated periodic solutions in both supercritical and subcritical bifurcations. Indeed, consider the following subset of the parameter space  Aε = a ∈ R4 : a1 > 0, a3 > 0, a1 a2 − a3 = ε, a4 6= 0 (28) which for ε = 0 is equivalent to A in (9). The next result is a direct consequence of the proof of Proposition 1.

(1) F1 , F2 and F4 have a stable periodic solution for a ∈ Aε , ε < 0, and F3 for a ∈ Aε ∩ B, ε < 0; (2) F5 and F6 have an unstable periodic solution for a ∈ Aε , ε > 0, and F3 for a ∈ Aε ∩ C, ε > 0.

-0.4

Fig. 2. Stable limit cycles (solid line) and their approximations (dotted line) for decreasing values of a2 .

which means that now the candidate bifurcating periodic solution can exist for small µ > 0. Hence, the proof follows by observing that the equilibrium point and its related eigenvalues are the same of the case F4 . 

Corollary 2. Suppose that |ε| is sufficiently small. Then,

-0.6

0 -0.2 -0.4 -0.6 -0.8 -1 -1

-0.8

-0.6

-0.4

-0.2

y˙

0

0.2

0.4

0.6

0.8

Fig. 3. Unstable limit cycles (dashed line) and their approximations (dotted line) for increasing values of a2 . to Proposition 1 and Corollary 2, a supercritical Hopf bifurcation occurs for any a ∈ Aε , ε = 0, and a stable limit cycle exists for any a ∈ Aε , ε < 0, |ε| sufficiently small. Moreover, as outlined in Remark 3, an analytical approximation of these bifurcated periodic solutions can be obtained. As an example, Fig. 2 reports the limit cycles and the relative approximations pertaining to the parameter vectors such that a1 = 2.15, a3 = 1, a4 = 1, and a2 = (1 + ε)/2.15, with ε ∈ (−0.196, 0). A similar behaviour can be shown for family F6 , where the Hopf bifurcations are subcritical. Figure 3 displays the unstable limit cycles and their relative approximations obtained for a1 = 1, a3 = 1, a4 = 1, and a2 = 1 + ε, ε ∈ (0, 0.222). Now, we illustrate how the knowledge of AHsup can be fruitfully employed in searching for complex dynamics in simple systems. More specifically, it is shown how it is possible to recover the known Sprott system (see e.g. (Sprott, 2003)), ... y + 2.017¨ y + y + y˙ 2 = 0. (29)

5. CONCLUSION

2 1 0 -1

y

-2 -3 -4 -5 -6 -7 -8 -0.1

0

0.1

0.2

a2

0.3

0.4

0.5

0.6

A complete characterization of supercritical and subcritical Hopf bifurcations for families of simple third-order autonomous nonlinear systems, has been provided via Harmonic Balance (HB) techniques. A distinguished feature of the HB approach with respect to the commonly employed state-space one, is that it provides a natural way to locate and even to approximate the bifurcated periodic solutions. Moreover, it has been shown how the parameters corresponding to supercritical bifurcations can be exploited as starting points by bifurcation analysis tools to locate very simple system exhibiting complex dynamics, as illustrated for the Sprott system.

Fig. 4. Bifurcation diagram with respect to parameter a2 . REFERENCES 1 0 -1 -2

y

-3 -4 -5 -6 -7 -8 2.02

2.04

2.06

2.08

a1

2.1

2.12

2.14

Fig. 5. Bifurcation diagram of system (30) with respect to parameter a1 . Consider again family F4 , and assume the supercritical Hopf bifurcation considered above, i.e., a1 = 2.15, a2 = 1/2.15, a3 = 1, a4 = 1, as a starting point for parameter bifurcation analysis. Figure 4 reports the bifurcation diagram obtained by decreasing a2 . It turns out that the bifurcated limit cycle undergoes a cascade of period-doubling bifurcations, leading to chaotic dynamics for a2 ∈ (0.035, 0.062). As a2 is further decreased, chaos disappears via an inverse period-doubling bifurcation cascade. In particular, it turns out that for a2 = 0 the system displays a stable periodic solution. Hence, parameters a1 = 2.15, a2 = 0, a3 = 1, a4 = 1 can be used as starting point for bifurcation analysis of the simpler family: ... y + a1 y¨ + y + y¨2 = 0. (30) Figure 5 reports the corresponding bifurcation diagram for decreasing values of a1 . Again, a period-doubling bifurcation cascade occurs and the system exhibits chaotic motion for a1 ∈ (2.0168, 2.0575). In particular, the Sprott system (29) is recovered for a1 = 2.017.

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