Normal form of double-Hopf singularity with 1 : 1 resonance for delayed differential equations

Abstract. In this manuscript, we provide a framework for the double-Hopf singularity with 1:1 resonance for general delayed differential equations (DDEs). The corresponding normal form up to the third-order terms is derived. As an application of our framework, a double-Hopf singularity with 1:1 resonance for a van der Pol oscillator with delayed feedback is investigated to illustrate the theoretical results.


Introduction
In this research, we study the double-Hopf singularity with 1:1 resonance for the following general delayed differential equations (DDEs): where • A and B are real constant n × n matrices; ) satisfies F (0, µ) = 0 and DF X (0, µ) = 0; • µ ∈ R m is the bifurcation parameter.
Obviously, 0 is an equilibrium of system (1), and the characteristic equation of system (1) at 0 is c Vilnius University, 2019 The dynamical behavior, especially, the bifurcation behavior around the equilibrium 0, presented by system (1), is generally determined by the distribution of the roots of Eq. ( 2) and has been studied extensively by many researchers.Some of the bifurcation results in literature regarding system (1) and Eq. ( 2) are summarized in the following.
(i) If Eq. ( 2) has a pair of purely imaginary roots and other roots have negative real parts, system (1) may exhibit a Hopf bifurcation; see [5,6,9] and references therein.(ii) If Eq. ( 2) has a double or triple zero root and other roots have negative real parts, a double or triple zero singularity for system (1) may occur.The mathematical frameworks for these singularities were established in [3] and [19], where the explicit conditions were formulated and the normal forms up to order 3 were derived.(iii) If Eq. ( 2) has a simple zero root and a pair of purely imaginary roots and other roots have negative real parts, a zero-Hopf singularity may occur.Wu and Wang [18] studied this case for system (1) and provided the explicit conditions for system (1) to exhibit zero-Hopf singularity and derived the normal forms up to order 3. (iv) If Eq. ( 2) has two pairs of purely imaginary roots ±ω 1 i and ±ω 2 i and other roots have negative real parts, a double-Hopf bifurcation may occur.For the case ω 1 /ω 2 / ∈ Q, Buono and Bélair [2] computed the corresponding normal form for scalar DDE, and Qesmia and Babram [15] derived the same for systems of DDEs.Recently, Ma et al. [13] studied a similar double-Hopf bifurcation for van der Pol-Duffing oscillator with parametric delayed feedback control.(v) For the case that Eq. ( 2) has two pairs of purely imaginary roots ±ω 1 i and ±ω 2 i with ω 1 = ω 2 , Guo and Wu [7,8] studied the following van der Pol oscillator: where f (x(t − τ )) is the delayed feedback for the position x.They established the explicit conditions such that the corresponding characteristic equation has a pair of purely imaginary roots with multiplicity 2. Zhang and Guo [21] studied the double-Hopf bifurcation with 1:1 resonance for system (3).Using the center manifold reduction method developed in [9], they derived the corresponding normal forms up to order 2 for f (x) = γx and provided the bifurcation diagrams.
For general DDEs (1), to the authors' knowledge, the explicit conditions for the double-Hopf singularity have not been formulated, and the corresponding normal forms have not been given in the literature, perhaps due to the extreme complexity and difficulty.In this manuscript, we will focus on deriving the normal forms for system (1) assuming that a double-Hopf bifurcation occurs.In particular, we study the case that Eq. ( 2) has a pair of purely imaginary roots with algebraic multiplicity 2 and geometric multiplicity 1, namely, double-Hopf singularity with 1:1 resonance.The main contribution of this manuscript is to characterize the center manifold for this singularity and to derive the corresponding normal forms up to order 3. https://www.mii.vu.lt/NAThe rest of this manuscript is organized as follows.In Section 2, we formulate and characterize the double-Hopf singularity for general DDEs with 1:1 resonance.In Section 3, we use the normal form theory developed by Faria and Magalhães [5,6] to compute the normal form for system (1) up to order 3.In Section 4, to illustrate our theoretical results, we study a double-Hopf singularity for the van der Pol oscillator with delayed feedback (3).The normal form up to order 3 is derived.Finally, the manuscript ends with a conclusion in Section 5.
2 The double-Hopf singularity of the general DDEs with 1:1 resonance In this section, we characterize the double-Hopf singularity for the general DDEs with 1:1 resonance.Write system (1) in the following form: where ) function with F (0, µ) = 0 and D X F (0, µ) = 0. Consider the following linear system: Since L is a bounded linear operator, L can be represented by a Riemann-Stieltjes integral by the Riesz representation theorem, where η(θ) (θ ∈ [−1, 0]) is an n × n matrix function of bounded variation.Let I be the n × n identity matrix, and define Let A 0 be the infinitesimal generator for the solution semigroup defined by system (5) such that Define the bilinear form between C and C * = C([0, 1], R n * ) (where R n * is the space of all n-dimensional row vectors by Nonlinear Anal.Model.Control, 24(2):241-260 The adjoint of A 0 is defined by Since LX t = AX(t) + BX(t − 1), η(θ) and ∆(λ) can be expressed, respectively, as Using this, we can rewrite the bilinear form as Note that, for a function ϕ ∈ C, Lϕ = Aϕ(0) + Bϕ(−1).For simplicity, we still use C and C * to represent the vector spaces on [0, 1] to the corresponding complex field, namely, Since we only study the double-Hopf singularity with 1:1 resonance for system (1), we make the following assumption.
Note that A 0 has an eigenspace P , which is invariant under the flow (5).Let P * be the space adjoint to P in C * .Then C can be decomposed as C = P ⊕ Q where Q = {ϕ ∈ C: ψ, ϕ = 0 ∀ψ ∈ P * }.
The following theorem characterizes P and P * .
In fact, we can choose ψ 1 , ψ 2 such that This finishes the proof the theorem.
From the proof of this theorem we can get the following equivalent conditions to assumption (H).
Corollary 1. Assumption (H) is equivalent to the following conditions: https://www.mii.vu.lt/NA 3 The Faria-Magalhães normal forms In this section, we use the idea of Faria and Magalhães [5,6] to conduct a center manifold reduction and to compute the normal form for system (1) for the double-Hopf singularity with 1:1 resonance.We assume that assumption (H) holds.Let and I is the n × n identity matrix.Define the projection π : BC → P by where Φ and Ψ are defined in Section 2. Let F = j 2 F j /j! and X = Φx + y with where J is given in (6) and ).Note that, for each j, the first and the third, and the second and the fourth components of f 1 j (x, 0, µ) are conjugate.On the center manifold, system (9) can be transformed to the following normal form: where g 1 j (x, 0, µ) are homogeneous polynomials of degree j in (x, µ).Let Y be a normed space and j ∈ N. Let V j (Y ) be the space of homogeneous polynomials with degree j in a linear space Y .Define M 1 j , M 2 j to be the operators in V j (C 4 ) and V j (ker π), respectively, by . By the above decompositions, g 1 2 (x, 0, µ), g 1 3 (x, 0, µ) can be expressed as where and where U 1 2 and U 2 2 are determined by where J = iω 1 0 iω .Define M 1 j to be the operators in V j (C 2 ) by Thus system (10) can be written as the following form: where Clearly, for p = (p 1 , p 2 ) T , and write where Now we want to obtain the explicit expressions of α, β, γ, a, b and c in terms of the coefficients of f 1 1 (x, x, 0, µ) and f 1 3 (x, x, 0, µ).Let Nonlinear Anal.Model.Control, 24(2):241-260 Note that, according to [4,8], (10) can be written as the following normal form: and α, β, γ, a, b and c are complex constants.Note that A 1000 , A 0100 , A 0010 , and A 0001 are linear function of µ and Ψ = ψ1 ψ2 .We calculate g 1 1 first.Lemma 1.In fact, , We can see that only A  (2) and hence the proof of the lemma is complete.
From these two lemmas we can see that Next, we compute g 1 3 (x, x, 0, µ).Using the definition of M 1 3 and Mathematica, we have the following result.Lemma 3. , where It is easy to get this, and hence we omit the detail.Now we have , where https://www.mii.vu.lt/NANext, we calculate U 1 2 and U 2 2 , which are determined by x, 0, 0).The expression of U 1 2 (x, x, 0) can be attained from the proof of Lemma 1 for A = f 1 2 (x, x, 0, 0).Now we work on U 2 2 (x, x, 0).This is the most difficult part since its computation involves solving linear systems with singular coefficient matrices.
Define h = h(x, x)(θ) = U 2 2 (x, x, 0) and write where ḣ denotes the derivative of h(θ) relative to θ. Comparing the coefficients of , we have that h0020 = h 2000 , h0002 = h 0200 , h0011 = h 1100 and that h ijkl , i + j + k + l = 2 satisfy the following differential equations, respectively: By (12), we have and hence , where 0110 (−1) + 2C (1) In this section, we apply the framework developed in Sections 2 and 3 to study a double-Hopf bifurcation with 1:1 resonance for Eq. ( 3) in Section 1.For studies of van der Pol equations, please see [1, 10-12, 14, 16, 20].For simplicity, we assume that f : R → R is a C 4 function such that Then the corresponding characteristic equation at 0 is Atay [1] showed that, for small e > 0, Eq. ( 3) possesses both stable and unstable periodic orbits.Wei and Jiang [16] showed that Eq. ( 3) undergoes a Hopf bifurcation at the origin when τ passes through a sequence of critical values and then determined the direction of the Hopf bifurcation and the stability of the periodic solutions by using the normal form theory. Wu and Wang [17] showed that Eq. ( 3) undergoes a zero-Hopf bifurcation at the origin and gave the corresponding bifurcation diagram near critical values of τ and γ.Let {ξ n } ∞ n=1 be the monotonic sequence of positive solutions of the equation x = tan x.

Conclusion
In this manuscript, we studied the double-Hopf singularity with 1:1 resonance for general DDEs.We characterized this complicated singularity and derived some equivalent conditions that will guarantee this singularity to occur.The corresponding normal form up to the third order was derived by using the idea of Faria and Magalhães.The unimaginable complexity and difficulty of some symbolic manipulation during the derivation were made possible by using the powerful symbolic software Mathematica.Our results were applied to a Van der Pol's oscillator with delayed feedback.The existence of a periodic solution and its stability were established.