Degenerate fold – Hopf bifurcations in a Rössler – type system

x ∈ R3, α = (α1, α2) ∈ R2, f smooth, which has an equilibrium x = O (0, 0, 0) for all ||α|| = √ α2 1 + α 2 2 small enough. Assume that for (α1, α2) = (0, 0) , the Jacobian matrix J0 of the system (1) at O has two purely imaginary eigenvalues ±iω0 and one real eigenvalue equal to 0, for some ω0 > 0. This is known in the literature as the fold-Hopf (or zero–Hopf) bifurcation. Since the eigenvalues of J0 are assumed simple, the eigenvalues of the Jacobian matrix Jα of the system (1) at O for ||α|| small enough are of the form λ1 = ν(α), λ = μ(α) + iω(α) and λ̄, such that ν(0) = μ(0) = 0 and ω(0) = ω0 > 0. The study of this bifurcation, that is, the study of the behavior of the system (1) when ||α|| is small enough, is not trivial. In order to study it, the author of [Kuznetsov, 1995] imposed five other conditions,

The study of this bifurcation, that is, the study of the behavior of the system (1) when ||α|| is small enough, is not trivial.In order to study it, the author of [Kuznetsov, 1995] imposed five other conditions, G1-G5, to be able to describe some generic properties of the system (1) when ||α|| is small enough.He used an approach based on the normal forms theory (see also [Algaba et al., 1998]).We explained briefly in the paper's Appendix how the G1-G5 generic conditions arise.When the conditions are satisfied we say the fold-Hopf bifurcation is non-degenerate, otherwise, degenerate.The bifurcation was well studied in the non-degenerate context in [Kuznetsov, 1995].All five generic conditions are crucial for the results obtained in this case.A natural question arises: what happens when one or more of the G1-G5 conditions are broken, that is, when the fold-Hopf bifurcation becomes degenerate?The motivation of the present work is to try to bring a partial answer to this question by studying the existence of periodic orbits in a particular three-dimensional system that has a degenerate fold-Hopf bifurcation.Moreover, because this kind of bifurcation appears in many differential systems in R 3 , we illustrate how it can be studied using the averaging theory.More exactly, we introduce in this paper a Rössler-type system of the form where a, b, c are real parameters and b = 0.
We say that the system (2) is of Rössler-type because it has a single nonlinear term and the first two equations of the system (2) are identical to the Rössler system [Rössler, 1976], which is well-known in the literature and has been used successfully in applications (secure communications, synchronizations, etc.) because it has one of the simplest analytical forms but displays complex dynamics, including chaos.We assume here b = 0, otherwise the differential system (2) becomes linear, and consequently it would have a trivial dynamics.
As we shall see in section 2 the classical theory for studying the fold-Hopf bifurcation does not work for the Rössler-type differential system (2), but we shall study it using the averaging theory for studying the bifurcations of periodic solutions from a zero-Hopf bifurcation.This tool for studying the zero-Hopf bifurcation is very useful and can be adapted to many different kind of differential equations, see for instance [Castellanos et al., 2013;Euzébio et al., 2015;Llibre et al., 2014;Wei et al., 2016].

Degeneracy of the fold-Hopf bifurcation
The equilibria of (2) are O(0, 0, 0) and A(−ac/b, c/b, −c/b) because b = 0.The points O and A collide when the parameter c = 0 and the coinciding equilibria have the eigenvalues 0 and a/2±iω, where ω = √ 4 − a 2 /2 whenever −2 < a < 2. It follows that the system (2) may undergo a fold-Hopf bifurcation at a = 0, when the eigenvalues are 0, ±i.
Denote further by α = (a, c) .Writing (2) in form (15) (see section 5), one can compute effectively the eigenvectors of the Jacobian matrix J (α) of system (2) at O for all α.Indeed, the eigenvalues of J (α) are λ 1 = −c, respectively λ ± = a/2 ± iω, with the corresponding eigenvectors The adjoint eigenvectors p 0 (α) and p ± (α) are Using the transformation u = p 0 (α), X , v = p 1 (α), X , where X = x y z T , the system (2) December 28, 2016 13:48 "deg-fold-Hopf -revised" Degenerate fold-Hopf bifurcations in a Rössler system (3) At α = (0, 0) we have B (0) = −b = 0, D(0) = b/2 and G 011 (0) = 0.The expressions of these coefficients are given in section 5. Hence, the fold-Hopf bifurcation of system (2) is degenerate with respect to (G.2) G 011 (0) = 0. Therefore, system (2) cannot be put in the normal forms ( 19), ( 20) or ( 21).Yet system (2) can be brought to the Poincaré normal form (18), because this uses only the first generic condition, (G.1) g 200 (0) = −b = 0, which is still valid.However, the Poincaré form does not offer significant tools in studying the system.The implications on the system's dynamics of this degeneracy cannot be inferred from what is known presently in the literature, what makes the system worth studying as a case study for this degeneracy.But we can use the averaging theory as it was done in the paper [Llibre, 2014] for a partial study of this degenerate fold-Hopf bifurcation.
The next result provides a first order approximation for the periodic solutions of a non-autonomous periodic differential system using the averaging theory for computing periodic solutions, for a proof see Theorems 11.5 and 11.6 of [Verhulst, 1991].
with x ∈ D, where D is an open subset of R n , t ≥ 0. We suppose that both F (t, x) and G(t, x, ε) are T −periodic in t.We define the averaged function Theorem 1. Assume that (i) F , its Jacobian ∂F/∂x, its Hessian ∂ 2 F/∂x 2 , G and its Jacobian ∂G/∂x are defined, continuous and bounded by a constant independent of ε in [0, ∞) × D and ε ∈ (0, ε 0 ].(ii) F and G are T −periodic in t (T independent of ε).
Then the following statements hold.
(a) If p is a zero of the averaged function f (x) and then there exists a T −periodic solution x(t, ε) of equation (4) such that x(0, ε) → p as ε → 0. (b) If all the eigenvalues of the Jacobian matrix (∂f /∂x) have negative real part, then the periodic solution x(t, ε) is asymptotically stable.If some of these eigenvalues have positive real parts, this periodic orbit is unstable.
Our first main result is the following.
Proof.If (a, b, c) = (εα, b, εγ) with ε > 0 a sufficiently small parameter, then the Rössler-type system becomes ẋ = −y − z, ẏ = x + εα y, ż = −εγz + b yz. (7) Doing the rescaling of the variables (x, y, z) = (εX, εY, εZ), system (7) in the new variables (X, Y, Z) Now we shall write the linear part at the origin of the differential system (8) when ε = 0 into its real Jordan normal form, i.e. as For doing that we consider the linear change (X, Y, Z) → (u, v, w) of variables given by X = v, Y = −u−w, Z = −w.In these new variables (u, v, w) the differential system (8) writes Now we pass the differential system (9) to cylindrical coordinates (r, θ, w) defined by u = r cos θ and v = r sin θ, and we obtain Now taking as new independent variable the variable θ the previous differential system writes We shall apply the averaging theory described in Theorem 1 to the differential system (11).Using the notation introduced in Theorem 1 we have t = θ, T = 2π, x = (r, w) T and It is easy to check that system (9) satisfies all the assumptions of Theorem 1.Now we compute the integrals (5), i.e.

Local dynamics at equilibria
The eigenvalues at the equilibrium O are real when a 2 − 4 ≥ 0, namely −c and a/2 ± √ a 2 − 4/2.Hence, O is asymptotically stable whenever c > 0 and a ≤ −2, and unstable on c < 0 or a ≥ 2. On a 2 − 4 < 0 the eigenvalues are complex and O is stable whenever c > 0 and −2 < a ≤ 0 and unstable on c < 0 or 0 < a ≤ 2.
When c = 0 a Hopf bifurcation may occur at O when a = 0 because the eigenvalues are −c and ±i.Indeed, the first condition of Hopf bifurcation is fulfilled, Re (dλ/da)| a=0,λ=i = 1/2 = 0. We need further to determine the first Lyapunov coefficient.To this end, write system (2) at a = 0 in the form Ẋ = J 0 X + F (X) where X = x y z T , J 0 = J(0, c) and for any two vectors x = x 1 x 2 x 3 and y = y 1 y 2 y 3 .A complex eigenvector q corresponding to the eigenvalue i, J 0 q = iq, respectively, an adjoint complex eigenvector p, J T 0 p = −ip, can be determined from the above eigenvectors q + and p + for a = 0 and are given by which satisfy p, q = 1.Here we use u, v = n i=1 ūi v i .With these notations, the first Lyapunov coefficient (see [Kuznetsov, 1995]) is 1 (0) given by 1 2ω 0 Re −2 p, B q, J −1 0 B (q, q) + p, B q, (2iω 0 I 3 − J 0 ) −1 B (q, q) , which leads to 1 (0) = 0.This means that the Hopf bifurcation is degenerate.Moreover, the second Lyapunov coefficient is also zero, 2 (0) = 0, which implies that the degeneracy of the Hopf bifurcation does not give rise to a non-degenerate Bautin bifurcation.Hence, no conclusion using this analysis can be drawn on the existence of periodic orbits in the system (2) bifurcating from 0 when a = 0 and c = 0.The second paper's result is the following theorem.At c = a we have , where J a q = iq and J T a p = −ip, respectively.Since an eigenvalue of J a is λ 1 = a, a Hopf bifurcation may occur only when a < 0. Proceeding as above, we find that B (x, y) has the same form as in ( 12) which leads to Since 1 (0) = 0 system (2) undergoes a Hopf bifurcation for all b = 0 and a < 0 at the equilibrium A. A periodic solution bifurcates from the equilibrium A when c crosses a with a < 0 and |c − a| small enough.More exactly, because 1 (0) < 0 and Re (dλ/dc)| c=a,λ=i < 0, a unique and stable periodic solution exists for c < a and |c − a| is sufficiently small.In addition, the equilibrium A is unstable on c < a and stable on c > a.The periodic solution in the normalized system to (2) is near a circle of radius r − 1 2(a 2 +1) (c − a) when c − a < 0 is sufficiently small.From Theorem 3 the local system's dynamics around the equilibrium A in terms of periodic orbits is well characterized for all a < 0 and |c − a| small enough.This is not the case for the equilibrium O for the values of the parameters (a, c) around (0, c) .When c = 0 and a = 0 the two equilibria O and A are still different but 1 (0) = 0, so the Hopf bifurcation does not exist anymore, while for a = c = 0 a degenerate fold-Hopf bifurcation occurs studied in Theorem 2.
Remark 3.1.Notice that we have proved that a Hopf bifurcation occurs at c = a ≤ 0. In fact, at c = a = 0 we have a fold-Hopf bifurcation.We also note that for c = a < 0 if one chooses other eigenvectors, for example q = 1 a (ia + 1) − i a 1 T and p = a 2(a+i) i 1 1 T , we find which is different from ( 14) but has the same sign since a < 0.

Conclusions
In this work we studied the behavior of a three-dimensional Rössler-type differential system in terms of the existence of periodic orbits emerging from degenerate fold-Hopf bifurcations.While the local behavior of a general differential system undergoing degenerate fold-Hopf bifurcations is not known in the literature, we showed that the averaging theory for detecting periodic orbits is a viable approach that can be applied at least for particular differential systems undergoing the bifurcation.Two open problems arise from this work related to the local behavior at the equilibria of system (2).More exactly, the system's behavior is not completely known when the parameters satisfy: 1) a = 0 and c is small enough around 0, respectively, 2) c = 0 and a is small enough around 0.

Theorem 3 .
Let c = a < 0 and b = 0, then the Rössler-type system (2) has a Hopf bifurcation at the equilibrium point (−ac/b, c/b, −c/b), and a stable periodic orbit borns at this equilibrium for a − c > 0 sufficiently small, which in the normalized system to (2) is near a circle of radius1 2(a 2 +1) (a − c).Proof.Consider in the following system (2) in a neighborhood of the equilibrium point A (−ac/b, c/b, −c/b) of course with b = 0. Translating A to the origin by x → x + ac/b, y → y − c/b and z → z + c/b, system (2) becomes ẋ = −y − z, ẏ = x + ay, ż = −cy + byz.(13)The characteristic polynomial of system (13) at O is P (λ) = λ 3 − aλ 2 + λ − c, which has the roots a and ±i if and only if c = a.bifurcation arises when ac > 0 at c = a.