RESEARCH ARTICLE | JULY 05 2023 Zero-Hopf bifurcation in the Chua’s circuit Jean-Marc Ginoux ; Jaume Llibre  J. Math. Phys. 64, 072701 (2023) https://doi.org/10.1063/5.0137020 CrossMark   View Export Online Citation 28 November 2023 14:30:34 Journal of ARTICLE pubs.aip.org/aip/jmp Mathematical Physics Zero-Hopf bifurcation in the Chua’s circuit Cite as: J. Math. Phys. 64, 072701 (2023); doi: 10.1063/5.0137020 Submitted: 30 November 2022 • Accepted: 6 June 2023 • Published Online: 5 July 2023 Jean-Marc Ginoux1,a) and Jaume Llibre2,b) AFFILIATIONS 1 Laboratoire LSIS, CNRS, UMR 7296, Université de Toulon, BP 20132, F-83957 La Garde cedex, France 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Catalonia, Spain a) E-mail: ginoux@univ-tln.fr b) Author to whom correspondence should be addressed: jaume.llibre@uab.cat ABSTRACT An equilibrium point of a differential system in R3 such that the eigenvalues of the Jacobian matrix of the system at the equilibrium are 0 and ± ωi with ω > 0 is called a zero-Hopf equilibrium point. First, we prove that the Chua’s circuit can have three zero-Hopf equilibria varying its three parameters. Later, we show that from the zero-Hopf equilibrium point localized at the origin of coordinates can bifurcate one periodic orbit. Moreover, we provide an analytic estimation of the expression of this periodic orbit and we have determined the kind of the stability of the periodic orbit in function of the parameters of the perturbation. The tool used for proving these results is the averaging theory of second 28 November 2023 14:30:34 order. © 2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0137020 I. INTRODUCTION AND STATEMENT OF THE MAIN RESULTS In 1993, Chua et al.1 proposed to replace the piecewise linear function of his famous circuit by a cubic one. The system they designed then became a relaxation oscillator with a cubic nonlinear characteristic elaborated from a circuit comprising a harmonic oscillator for which the operation was based on a field-effect transistor, coupled to a relaxation–oscillator composed of a tunnel diode. The modeling of the circuit used a capacity preventing from abrupt voltage drops and allowing to describe the fast motion of this oscillator. This gave rise to Eq. (1), which constitutes a singularly perturbed system and is since considered the paradigm of complex dynamics.2 The Chua’s circuit is analyzed using the Kirchhoff’s laws. Then, the dynamics of the Chua’s circuit is modeled by means of the following system of three nonlinear ordinary differential equations in the variables x(t), y(t), and z(t): ẋ = a(y − cx − x3 ), ẏ = x − y + z, (1) ż = −by, where a, b, and c are real parameters and the dot indicates derivative with respect to the time t. For more details on the Chua’s circuit, see Refs. 3 and 4. Note that this differential system is invariant under the symmetry (x, y, z) → (−x, −y, −z). Then, if (x(t), y(t), z(t)) is a solution of the differential system (1), then (−x(t), −y(t), −z(t)) is another solution of this differential system, eventually both solutions coincide. The study of the periodic orbits of a differential system is one of the main objectives of the qualitative theory of the differential systems. In general, this is not an easy task, mainly if we want to study the periodic orbits analytically. A way of finding periodic orbits is through a Hopf bifurcation, which in R3 takes place when a periodic orbit bifurcates from an equilib- rium point whose linear part has eigenvalues λ ≠ 0 and ±ωi with ω > 0, and moving the parameters of the differential system this equilibrium changes its kind of stability (for more details, see Ref. 5 and an application in Ref. 6). While there is a well-developed theory for studying the Hopf bifurcation when λ ≠ 0, this is not the case when λ = 0. J. Math. Phys. 64, 072701 (2023); doi: 10.1063/5.0137020 64, 072701-1 © Author(s) 2023 Journal of ARTICLE pubs.aip.org/aip/jmp Mathematical Physics Here, we shall use the averaging theory of second order for studying the periodic orbits that bifurcate from the zero-Hopf equilibria of the differential system (1). We note that this technique applied for studying the zero-Hopf bifurcations of the Chua’s system can be used for studying arbitrary zero-Hopf bifurcations of other differential systems. Sometimes the averaging theory of first order for studying zero-Hopf bifurcations is sufficient (see, for instance, Ref. 7), but here this is not the case. A zero-Hopf equilibrium p of a differential system in R3 is an equilibrium point with eigenvalues 0 and ±ωi with ω > 0. A zero-Hopf bifurcation takes place when one or several periodic orbits bifurcate from the equilibrium p when the parameters of the system move. Easy computations show that the equilibria of the differential system (1) are √ √ √ √ E1 = (0, 0, 0), E2 = ( −c, 0, − −c), E3 = (− −c, 0, −c). Of course, the last two equilibria exist only if the parameter c < 0. Proposition 1. The following statements hold. (a) The point E1 is a zero-Hopf equilibrium if and only if b = 0, c = −1/a, and a < −1. (b) The points E2 and E3 are zero-Hopf equilibria if and only if b = 0, c = 1/(2a), and a < −1. Proposition 1 is proved in Sec. III. Theorem 2. The equilibrium point E1 of the Chua’s circuit (1) has a zero-Hopf bifurcation when b = 0, c = −1/a, and a = −1 − a21 < −1 with a1 > 0 and it is perturbed as follows: 1 b = b2 ε2 , c = − + c2 ε2 , (2) a where b2 c2 ≠ 0 and b2 − a21 c2 (1 + a21 ) > 0. Then, for ε ≠ 0 sufficiently small the periodic solution (x(t, ε), y(t, ε), z(t, ε)) equal to 28 November 2023 14:30:34 1 1 ε(r0 cos (a1 t), r0 (a1 + ) sin (a1 t) + cos (a1 t), O(1)) + O(ε2 ) (3) a1 a1 bifurcates from E1 (see Fig. 1). Moreover, this periodic solution is asymptotically unstable if b2 > 0. It has a local stable and a local unstable manifold each one formed by two topological cylinders if b2 < 0. Theorem 2 is proved in Sec. III. We note that in the expressions of b and c given in (2), there are no terms in b1 ε and c1 ε, which is due to the fact that these terms do not contribute to the existence of periodic orbits when we compute such periodic orbits for this differential system using the averaging theory. Of course, we see that doing all the computations of the averaging theory of first and second order. But we have omitted them in (2) because otherwise the expressions of the Proof of Theorem 2 become longer without providing any new information. FIG. 1. The periodic orbit of Theorem 2 for the values ε = 0.005, a = −2, a1 = 1, b = 0, b2 = 3, c = 0.5, and c2 = 1. J. Math. Phys. 64, 072701 (2023); doi: 10.1063/5.0137020 64, 072701-2 © Author(s) 2023 Journal of ARTICLE pubs.aip.org/aip/jmp Mathematical Physics The averaging theory does not provide information about the possible orbits bifurcating from the zero-Hopf equilibria E2 and E3 . This is due to the fact the first averaged function f 1 (r, u) = ( f 11 (r, u), f 12 (r, u)) at these equilibria is √ πr(2Aa21 c1 + 3 2u) f11 (r, u) = , A3/2 a31 2πb1 u f12 (r, u) = , a31 √ where A = 1/ a21 + 1. The unique solution of the system f 1 (r, u) = (0, 0) is (r, u) = (0, 0), but this solution only provides an equilibrium point, instead of a periodic solution. Then, since the averaging theory of first order does not provide information, we must do the averaged function f 1 (r, u) of first order identically zero and compute the averaged function of second order. But in this case, we cannot do identically √ zero the averaged function of first order because we cannot vanish the coefficient 3 2π/(A3/2 a31 ) of the term ru in the function f 11 (r, u). For more details on the averaging theory, see Sec. II. II. THE AVERAGING THEORY OF FIRST AND SECOND ORDER In this section, we present the averaging theory of second order for finding periodic orbits that we need for proving our main theorem. A proof of the following result can be found in Ref. 8 or 9, and in Theorem 11.6 of Ref. 10, it is studied the kind of stability of the periodic solutions obtained using the averaging theory. For more details on the general theory of averaging theory, see the book 11, and for another application to the zero-Hopf bifurcation, see Ref. 12. Theorem 3. Let ẋ(t) = εF1 (t, x) + ε2 F2 (t, x) + ε3 R(t, x, ε) (4) be a non-autonomous differential system such that D is an open subset of R , where the functions F 1 , F2 : R × D → R , and R : R × D n n × (−ε f , ε f ) → Rn are continuous and periodic of period T in the variable t. Assume that the next conditions are satisfied. (i) The functions F 1 (t, ⋅) ∈ C1 (D) for all t ∈ R, F 1 , F 2 , R, and Dx F 1 are locally Lipschitz with respect to x, and the function R is C1 with respect 28 November 2023 14:30:34 to ε. The functions f1 , f2 : D → Rn are defined as T f1 (z) = ∫ F1 (s, z)ds, 0 (5) T s f2 (z) = ∫ [Dz F1 (s, z)∫ F1 (t, z)dt + F2 (s, z)]ds. 0 0 (ii) For a bounded and open set V ⊂ D and for every ε ∈ (−ε f , ε f )/{0}, there is a ∈ V satisfying f 1 (a) + εf 2 (a) = 0. Moreover the Brouwer degree of the function f 1 + εf 2 at a is not zero. Then, for ∣ε∣ > 0 sufficiently small, there is a periodic solution x(t, ε) of period T of the differential system (4) verifying that x(0, ε) → a when ε → 0. Furthermore the eigenvalues of the Jacobian matrix D( f 1 (a) + εf 2 (a)) provides the kind of stability of the Poincaré map associated with the periodic solution x(t, ε). Assume that a is a fixed point of a function f . When the Jacobian of the function f at a (if it is defined) does not vanish, then the Brouwer degree of function f at a is non-zero; for a proof, see Ref. 13. When the first averaged function f 1 is not identically zero, for ε sufficiently small the zeros of f 1 + εf 2 are essentially the zeros of f 1 . Then, we say that Theorem 3 provides the averaging theory of first order. When the first averaged function f 1 is identically zero and the second averaged function f 2 is not identically zero, the zeros of the function f 1 + εf 2 are essentially the zeros of f 2 , and then we say that Theorem 3 provides the averaging theory of second order. III. PROOF OF THE RESULTS Proof of Proposition 1. The Jacobian matrix of the vector field associated to system (1) is ⎛−a(c + 3x2 ) a 0⎞ ⎜ ⎟ M=⎜ ⎜ 1 −1 1⎟ ⎟. ⎜ ⎟ ⎝ 0 −b 0⎠ In order that the eigenvalues of the matrix M at some equilibrium Ek for k = 1, 2, 3 be 0 and ±ωi with ω > 0, the characteristic polynomial of M must be −λ(λ2 + ω2 ). Imposing this fact for every equilibrium Ek for k = 1, 2, 3, we obtain the results stated in the statement of the proposition. ◻ J. Math. Phys. 64, 072701 (2023); doi: 10.1063/5.0137020 64, 072701-3 © Author(s) 2023 Journal of ARTICLE pubs.aip.org/aip/jmp Mathematical Physics Proof of Theorem 2. We consider system (1) for the values of the parameters given in (2), so the equilibrium E1 is zero-Hopf. Now we shall do several changes of variables until to write the differential system into the normal form (4) for applying the averaging theory of second order, and in this way to study using this theory the possible periodic orbits bifurcate from the equilibrium E1 . We shall write the matrix of the linear part of the differential system (1) at the equilibrium E1 in its real Jordan normal form ⎛0 −a1 0⎞ ⎜ ⎟ ⎜a1 0⎟ ⎜ 0 ⎟. ⎜ ⎟ ⎝0 0 0⎠ For doing this, we do the change of variables, which pass the variables (x, y, z) to the variables (X, Y, Z), through 1 a1 Y + X Z (x, y, z) = (X − ( + 1)Z, − 2 , Z). (6) a21 a21 + 1 a1 Then, system (1) in the new variables (X, Y, Z) becomes 3 (a21 + 1)(a21 (X − Z) − Z) Ẋ = −a1 Y + a61 b2 (a1 Y + X) (a21 + 1)Z(b2 − a21 (a21 + 1)c2 ) + ε2 (− 2 + (a21 + 1)c2 X + ) + O(ε3 ), a1 a41 2 4 3(a21 + 1) a21 XZ(a21 X − (a21 + 1)Z) + (a21 + 1) Z3 − (a21 + 1)a61 X 3 (7) Ẏ = a1 X + a71 (a 2 1 + 1)c 2 (a 2 1 (Z − X) + Z) + ε2 + O(ε3 ), a31 28 November 2023 14:30:34 b2 Z b2 (a1 Y + X) Ż = ε2 ( 2 − ) + O(ε3 ). a1 a21 + 1 Now, we change the coordinates (X, Y, Z) to the variables (r, θ, u) taking X = εr sin θ, Y = εr cos θ and Z = εu. Finally, taking θ as the new independent variable the differential system (7) writes in the new variables (r, θ) as r′ = ε2 F21 + O(ε3 ), (8) u′ = ε2 F22 + O(ε3 ), where the prime denotes derivative with respect to the variable θ, and 1 F21 = ((a21 + 1)a71 r3 cos4 θ − (a21 + 1)a51 r2 cos3 θ(3(a21 + 1)u + a1 r sin θ) a81 − a1 cos θ(a21 (a21 + 1)u(a21 (a21 + 1)c2 − b2 ) + (a21 + 1)4 u3 + a1 r sin θ(3(a21 + 1)3 u2 + a41 (a21 c2 + b2 + c2 ))) + (a21 + 1)2 u sin θ(a41 c2 + (a21 + 1)2 u2 ) + a31 r cos2 θ(a61 c2 + a41 c2 − a21 b2 + 3(a21 + 1)2 a1 ru sin θ + 3(a21 + 1)3 u2 )), b2 a2 r(a1 sin θ + cos θ) F22 = 3 (u − 1 ). a1 a21 + 1 Differential system (8) is written into the normal form (4) for applying the averaging theory, where using the notation of Sec. II, we have t = θ, x = (r, u), T = 2π, n = 2, F 1 = (F 11 , F 12 ) = (0, 0), and F 2 = (F 21 , F 22 ). Since all the assumptions of Theorem 3 of Sec. II are satisfied, we can apply it to the differential system (8). Then, the first averaged function f 1 (r, u) defined in (5) is identically zero, and the second averaged function f 2 (r, u) = ( f 21 (r, u), f 22 (r, u)) is 1 3 2 7 3 9 7 5 3 3 2 f21 (r, u) = 8 ( π(a1 + 1)a1 r + πr(a1 c2 + a1 c2 − a1 b2 + 3(a1 + a1 ) u )), a1 4 2πb2 u f22 (r, u) = . a31 J. Math. Phys. 64, 072701 (2023); doi: 10.1063/5.0137020 64, 072701-4 © Author(s) 2023 Journal of ARTICLE pubs.aip.org/aip/jmp Mathematical Physics The unique zero of the second averaged function f 2 (r, u), which going back through the changes of variables, is not associated with an equilibrium point of system (2) and, which has r ≥ 0, is ¿ ⎛ Á b2 − a2 c2 (1 + a2 ) ⎞ (r, u) = ⎜2ÁÀ 1 1 , 0⎟ = (r0 , 0), ⎝ 3a21 (1 + a21 ) ⎠ which is real because by assumptions b2 − a21 c2 (1 + a21 ) > 0. Since the Jacobian of the function f 2 (r, u) at that zero is 4π2 b2 (b2 − a21 c2 (1 + a21 ))/a61 ≠ 0 by assumptions, it follows from Theorem 3 that the differential system (8) has a periodic solution (r(θ, ε), u(θ, ε)) such that (r(0, ε), u(0, ε)) = (r0 , 0) + O(ε). Since the eigenvalues of the Jacobian matrix D( f 21 , f 22 )(r0 , 0) are 2π(b2 − a21 c2 (1 + a21 )) 2πb2 >0 and . a31 a31 From Theorem 3, the corresponding periodic solution (r(θ, ε), u(θ, ε)) is asymptotically unstable if b2 > 0. If b2 < 0 the fixed point of the Poincaré map associated with the periodic orbit (r(θ, ε), u(θ, ε)) is a saddle, so this periodic orbit has a local unstable manifold and a local stable manifold each one formed by two topological cylinders. Going back through the changes of variables the periodic solution (r(θ, ε), u(θ, ε)) of system (8) becomes the periodic solution (r(t, ε), θ(t, ε), u(t, ε)) of the differential system (ṙ, θ̇, u̇), where as usual the dot denotes derivative with respect to the time t. We did not write explicitly the differential (ṙ, θ̇, u̇), which is in the middle of the differential systems (7) and (8). The periodic solution (r(t, ε), θ(t, ε), u(t, ε)) satisfies (r(0, ε), θ(0, ε), u(0, ε)) = (r0 , a1 t, 0) + O(ε). This periodic solution becomes the periodic solution (X(t, ε), Y(t, ε), Z(t, ε)) of system (7) such that 28 November 2023 14:30:34 (X(0, ε), Y(0, ε), Z(0, ε)) = ε(r0 cos (a1 t), r0 sin (a1 t), 0) + O(ε2 ). Finally, going back to the coordinates (x, y, z) undoing the change (6), we obtain the periodic solution of system (1) described in the statement of the theorem. ◻ IV. CONCLUSIONS We have proved that from the zero-Hopf equilibrium point localized at the origin of coordinates of the Chua’s circuit can bifurcate one periodic orbit. Moreover, we have provided an analytic estimation of the expression of this periodic orbit, and additionally we have determined its kind of the stability in function of the perturbation of the parameters b = 0, c = −1/a, and a < −1, for which the zero-Hopf equilibrium at the origin of coordinates exists. The tool used for proving these results has been the averaging theory of second order. Unfortunately, the averaging theory does not provide information about the possible periodic orbits bifurcating from the other two zero-Hopf equilibria that the Chua’s circuit can have. ACKNOWLEDGMENTS The second author was partially supported by the Agencia Estatal de Investigación (Grant No. PID2019-104658GB-I00) and the H2020 European Research Council (Grant No. MSCA-RISE-2017-777911). AUTHOR DECLARATIONS Conflict of Interest The authors have no conflicts to disclose. Author Contributions Jean-Marc Ginoux: Investigation (equal); Methodology (equal). Jaume Llibre: Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). DATA AVAILABILITY Our paper has no data. J. Math. Phys. 64, 072701 (2023); doi: 10.1063/5.0137020 64, 072701-5 © Author(s) 2023 Journal of ARTICLE pubs.aip.org/aip/jmp Mathematical Physics REFERENCES 1 A. I. Khibnik, D. Roose, and L. O. Chua, “On periodic orbits and homoclinic bifurcations in Chua’s circuit with a smooth nonlinearity,” Int. J. Bifurcation Chaos 3(2), 363–384 (1993). 2 C. Mira, “Chua’s circuit and the qualitative theory of dynamical systems,” Int. J. Bifurcation Chaos 9(7), 1911–1916 (1997). 3 L. O. Chua, “Chaos synchronization in Chua’s circuit” (Electronics Research Laboratory, College of Engineering, University of California, OCLC, Berkeley, 1992), p. 44107698. 4 L. Fortuna, M. 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