Sliding Hopf bifurcation in interval systems

In this paper, the equivariant degree theory is used to analyze the occurrence of the Hopf bifurcation under effectively verifiable mild conditions. We combine the abstract result with standard interval polynomial techniques based on Kharitonov's theorem to show the existence of a branch of periodic solutions emanating from the equilibrium in the settings relevant to robust control. The results are illustrated with a number of examples.


Introduction
Subject and goal. Many problems in population dynamics, neural networks, fluid dynamics, solid mechanics, elasticity, chemistry, mechanical and electrical engineering lead to studying the socalled Hopf bifurcation (more precisely, Poincaré-Andronov-Hopf bifurcation) in dynamical systems parameterized by a real parameter (see, for example, [5,20,26,34,39] and references therein). To be more specific, given a parameterized familẏ where f : [α − , α + ]×R d → R d is a continuous map and (α, 0) is a curve of trivial stationary solutions, the Hopf bifurcation is a phenomenon occurring when α crosses some critical value α o (for which the linearization D x f (α, 0) admits a purely imaginary eigenvalue) and resulting in appearance of a branch of small amplitude periodic solutions near the curve (α, 0). In his original work [23], E. Hopf studied system (1) under the following assumptions: (a) f is analytic in both variables; (b) for α = α o , exactly two complex conjugate characteristic roots µ(α) and µ(α) intersect the imaginary axis (absence of multiple/resonant roots); (c) µ(0) = 0 (exclusion of steady-state bifurcation); and, (d) Re µ (0) = 0 (transversality). Hopf's theorem includes conditions for the occurrence of the bifurcation (i.e., the existence result) and conditions for stability of small cycles bifurcating from the stationary point. After this pioneering work, a substantial effort was made in order to relax conditions (a)-(d) (see, for example, [5,13,20,26,34,36,39] and references therein). One objective of this paper is to present an abstract result on the occurrence of the Hopf bifurcation in (1) under very mild (and effectively verifiable) hypotheses containing many known occurrence results as a particular case (cf. Theorem 3.2, Theorem 3.5 and Remark 3.6). It should be stressed that we do not study stability of bifurcating periodic solutions. Our choice of the conditions on the nonlinearity f and its derivative D x f (α, 0) is essentially determined by the following observations. In analysis and design, it is customary to deal with approximations of complex models that have some degree of uncertainty (one can think of the socalled nominal systems widely used in robust control; see, for example, [7]). Considering a model with uncertain parameters, one can expect that the entries of the matrix D x f (α, 0) belong to some known intervals of values rather than being represented by fixed numbers. This suggests to study the Hopf bifurcation phenomenon for a class of systems (1) where coefficients of the linearization are limited to known intervals. In this setting, the characteristic polynomial of D x f (α, 0) that defines the stability properties of the linearization also becomes an interval polynomial (see, for example, [7]). Importantly, this setting includes the scenario when the characteristic values of the linearization of a representative system (1) slide along the imaginary axis when the bifurcation parameter is varied (see Figure 1a). The main goal of the present paper is to propose a method for analysis of the occurrence of the Hopf bifurcation in the presence of such sliding. As a matter of fact, the sliding phenomenon makes the problem non-local. Namely, it does not allow one to localize a bifurcation point on the basis of the knowledge of the linearization, that is based on the condition Re µ(α) = 0 (see Figure 1 a,b).
To study the Hopf bifurcation in this setting, one needs to deal with the whole interval of sliding that consists of potential bifurcation points. Thus, sliding is in sharp contrast to the transversality condition (d) above. At the same time, to the best of our knowledge, all the existing results on the occurrence of Hopf bifurcation identify explicitly a critical value of the parameter α at which Re µ(α) changes its sign (for the least restrictive condition of this type, we refer to [36]). Some conditions for the existence of a branch of cycles that are non-local with respect to the parameter can be found in [29][30][31].
The simplest scenario which includes sliding and is covered by our results is the following. Suppose that system (1) has an equilibrium x = 0 for all values of the parameter α ∈ [α − , α + ]. Assume that the linearization D x f (α, 0) of the right hand side is invertible and has at most one pair of purely imaginary eigenvalues for any α ∈ [α − , α + ]. Finally, assume that the zero equilibrium is hyperbolic for α = α ± and the dimension of the stable manifold of the linearization of (1) at zero is different for α = α − and α = α + . Then there is a Hopf bifurcation point on the interval (α − , α + ). Theorem 3.2 presented below also covers more complex scenarios including multiple and resonant eigenvalues of the linearization on the imaginary axis.
Method. In [23], the Hopf bifurcation in (1) was studied based on the series expansion of f . The further progress was related to the methods rooted in the singularity theory: assuming that the system satisfies several regularity and genericity conditions, one can combine the normal form classification with Center Manifold Theorem/averaging method/Lyapunov-Schmidt reduction. For a detailed exposition of these concepts and related techniques, we refer to [20,21,39].
Being very effective in the settings they are usually applied to, the singularity theory based methods meet difficulties if a setting is not regular/generic enough. For example, dynamical systems with hysteresis components admit linearization at the origin while any small neighborhood of the origin contains non-differentiability points which makes the Center Manifold Reduction impossible (see [2,4,9,32,33,35,42]) for details). As long as the stability of bifurcating solutions is not questioned, one can use homotopy theory based methods. Important steps in this direction were done in [1] (framed bordism theory), [13] (Fuller index), [36] (parameter functionalization method combined with the Leray-Schauder degree), to mention a few.
During the last twenty years the equivariant degree theory emerged in non-linear analysis (for the detailed exposition of this theory, including historical remarks, we refer to recent monographs [5,26] and surveys [3,6,25]; for the prototypal invariants, see [14,15,18,38]). The equivariant degree, being the main topological tool used in this paper, is an instrument that allows "counting" orbits of solutions to symmetric equations in the same way as the usual Brouwer degree does, but according to their symmetry properties. In particular, the equivariant degree theory has all the attributes allowing its application in non-smooth and non-generic equivariant settings related to equivariant dynamical systems having, in general, infinite dimensional phase spaces with lack of linear structure (cf. [4]). We refer to [5,26] and references therein for the equivariant degree treatment of the (symmetric) Hopf bifurcation in different environments (see also [28]). In the present paper, we use the S 1 -degree with one free parameter (see [5] for the axiomatic approach).
Theorem 3.7 below explicitly refers to the verification of stability properties of interval polynomials (cf. conditions (R3) and (R4)). Among very few results on the connection between perturbations of the coefficient and root locations, Kharitonov's theorem ( [27], see also [7,22]) takes a firm position. To be more specific, V. L. Kharitonov showed that given a family of interval polynomials with real coefficients, it is necessary and sufficient to test just four canonically defined members of the family in order to decide that all polynomials are Hurwitz stable. The main topological ingredient of Kharitonov's proof is the so-called Zero Exclusion Principle (in short ZEP) which can be traced back to the classical Argument principle in Complex Analysis. In this paper, combining ZEP with simple combinatorial arguments, we establish a Kharitonov type result for the so-called k-stable interval polynomials (cf. Lemma 2.4 and Definition 2.2). In particular, it shows that Kharitonov's approach is sensitive not only to Hurwitz stability, but also to the change of the dimension of the stable manifold in families of interval polynomials which is crucial for studying the Hopf bifurcation phenomenon.
The paper is organized as follows. In the next section, we present some background related to the Hopf bifurcation and interval polynomials. In Section 3, main results are formulated (see Theorems 3.2, 3.5 and 3.7). Some examples illustrating Theorems 3.5 and 3.7 are given in Section 4. Section 5 contains the proof of Theorem 3.2 which is close in spirit to the proofs of Theorems 9.18 and 9.24 from [5]. In Section 6, we provide the proofs of remaining results. A brief summary of properties of the S 1 -equivariant degree is presented in Appendix.

Hopf bifurcation
The Hopf bfurcation being the main subject of the present paper is formalized in the following definition (cf. [5,36]).
Definition 2.1. Consider a non-empty set Γ of non-constant periodic solutions (α, p, x(t)) of system (1) (where p is the minimal period of x(t)) such that p ∈ [p − , p + ] ⊂ (0, ∞). The set Γ is called a branch bifurcating from the trivial solution if the union of Γ and the set of trivial solutions, If Γ is a branch of non-constant periodic solutions bifurcating from the trivial solution, then the interval [α − , α + ] contains at least one Hopf bifurcation point α 0 in the weak sense of [36]. In other words, there are converging sequences α k → α 0 and p k → p 0 > 0 such that system (1) with α = α k has a non-constant periodic solution x k (t) with the minimal period p k and x k C → 0. If the necessary condition for the Hopf bifurcation (see, Section 5.1) is satisfied at exactly one point α 0 ∈ (α − , α + ), then Definition 2.1 reduces to the definition of the Hopf bifurcation used in [5, p. 260]. However, the setting of Definition 2.1 does not exclude a possibility of more complex behavior of the branch shown in Figure 1b in the case of an eigenvalue sliding along the imaginary axis as in Figure 1a.
Definition 2.2. (i) A polynomial with real coefficients is called monic if its highest coefficient is one. An interval polynomial S is called monic if any P ∈ S is monic.
(ii) Let S be a monic interval polynomial of degree n. We say that S is q-stable (resp., qunstable) if for any P ∈ S, P has exactly q roots with Re (z) < 0 and n − q roots with Re (z) > 0 (resp., q roots with Re (z) > 0 and n − q roots with Re (z) < 0).
We will use a q-unstable variant of Kharitonov's theorem.
The main topological ingredient of the proof of both statements is the so-called Zero Exclusion Principle. If some polynomial P o ∈ S is q-unstable and for any P ∈ S and any ω > 0, P (iω) = 0, then the interval polynomial S is q-unstable.
Proof of Lemma 2.4: As an immediate consequence of inequalities (4) one has that, for any P ∈ S, The result then follows from the Zero Exclusion Principle. 2

Interval polynomials and Descartes' Criterion
Recall the following classical result.
Descartes' criterion. If the terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is less than or equal to the number of sign differences between consecutive nonzero coefficients.
As an immediate consequence, we have Proposition 2.5. Given a polynomial P with real coefficients, assume that there exist polynomials Q and R such that the coefficients of the polynomial have at most one sign change. Then, P may have at most one pair of purely imaginary roots.
Indeed, for ω > 0, if iω is a root of P , then ω is a (positive) root of S(P, Q, R).
In what follows, we use an interval polynomial variant of Proposition 2.5. For the precise formulation, we need the following definition. Given an interval polynomial S, we say that the coefficients of S have at most one sign change if, for some j, Notice that if the coefficients of S have at most one sign change then the coefficients of any polynomial P ∈ S have at most one sign change. Set Lemma 2.6. Assume that there exist Q, R such that the coefficients of T (S, Q, R) have at most one sign change. Then, any polynomial P ∈ S has at most one pair of purely imaginary roots.
Proof: Suppose, for the contrary, that some P ∈ S has more than one pair of purely imaginary roots. By (6), S(P, Q, R)(ω) ∈ T (S, Q, R). Therefore, S(P, Q, R)(ω) has at least two distinct positive real roots. Hence, by Descartes' criterion, the coefficients of S(P, Q, R)(ω) have more than one sign change, which is a contradiction.

Abstract result
is a map satisfying the following properties: (P0) f is continuous; (P1) The Jacobi matrix D x f (α, 0) exists for all α, depends continuously on α and (P2) f (α, 0) = 0 for all α; To formulate the next condition, take the map Λ : where R + denotes the non-negative semi-axis. We will denote by ∂Ω the boundary of a domain Ω and by Ω the closure of Ω.
Remark 3.1. Conditions (P0) and (P1) reflect the minimal regularity that we require from system (1). Condition (P2) guarantees the existence of a branch of zero equilibria from which we expect the occurrence of the Hopf bifurcation, while (P3) excludes steady-state bifurcation. The domain P provided by (P4) acts as a "trap" catching the roots of Λ, which may potentially contribute to the Hopf bifurcation. Condition (P4)(ii) guarantees that the roots may only escape P through the planes {α = α − }, {α = α + } and {τ = 0}. Condition (P4)(iii) is an analog of the standard non-zero crossing number assumption.
On the other hand, the sets D k provided by (P5) form the domain on which we will compute the topological invariant. Property (P5)(iii) (which is a kind of non-resonance condition) ensures that the topological invariant is well-defined, while (P5)(ii) (which says that all the roots in D k are precisely those "exiting" P) ensures that the invariant is non-trivial and thus that the Hopf bifurcation takes place. Several versions of conditions (P4) and (P5) directly related to the classical setting for the Hopf bifurcation are discussed in the next subsection.
The following statement is our main abstract result.

Corollaries
Let us consider some corollaries of Theorem 3.2 based on variations of conditions (P4) and (P5) which are more relaxed but easier to verify. To this end, we introduce the following notation: Notice that R(f ) is the set of purely imaginary characteristic roots lying between α − and α + , while S(f ) is the set of points an integer multiple of which lies in R(f ).
We use a few variants of conditions (P4) and (P5).
(P4 ) There exist α − , α + , for which x = 0 is a hyperbolic equilibrium of (1) and the dimension of the unstable manifold of the linearization of (1) at 0 is different for α − and α + .
Remark 3.4. Observe that (P4 ) is a non-zero crossing number condition; in particular, the classical Routh-Hurwitz criterion (see, for example, [41]) can be useful for its verification. Condition (P5 ) is a slight modification of (P5), adjusted to the case when (P4 ) holds. Condition (P5 ) is the classical non-resonance condition. Condition (P5 ), although much more restrictive than condition (P5 ), can be verified using Descartes' criterion (see also Proposition 2.5). Finally, (P5 ) is the standard isolated center condition (see, for example, [5]).
The following statement is based on Theorem 3.2 and is used below to obtain sufficient conditions for the Hopf bifurcation in interval systems.  Under the assumption that f is of class C 1,1 , Theorem 3.5(e) was established in [24] (see also [1,5,13,19]). On the other hand, by taking a sufficiently small neighborhood (α − , α + ), one can deduce the main result of [36] from Theorem 3.5(d) (without extra "simplicity" assumptions on the corresponding eigenvalues).

Theorem 3.5 and interval polynomials
In this section, we address families of one-parameter systems for which every member is undergoing the Hopf bifurcation. To be more precise, denote by A a map from [α − , α + ] to the set of interval matrices of size d × d and by r a set of maps r : we mean the family of all systems of the forṁ satisfying the following conditions: Denote by Q the map from R to the set of monic interval polynomials such that for any α ∈ R, is the collection of all possible characteristic polynomials corresponding to each member of the family A (α) (in fact, this collection constitutes an interval polynomial). To generalize Theorem 3.5(a,b,c) to the interval setting, we need "interval analogs" of notations (10). Given a family of systems (11) with interval characteristic equation (13), put (cf. (3) and (5)) w 1 (α, β) = Re(g 1 (Q(α), iβ)) · Re(g 2 (Q(α), iβ)), Here R is the set of all the purely imaginary zeros of all polynomials P that belong to the family (13). We make the following assumptions.
(R0) r is continuous in both variables for any r ∈ r; (R1) For any r ∈ r, (R4) Q(α + ) is q 2 -unstable with q 1 = q 2 ; (R5 ) There exists a finite collection of disjoint sets D k ⊂ [α − , α + ] × R + such that: (i) each D k is homeomorphic to a closed disk; (R5 ) For any α ∈ [α − , α + ] and for any P ∈ Q, P has at most one pair of purely imaginary roots.
We are now in a position to formulate our main result on the Hopf bifurcation in interval systems.

Examples
Below we present three examples illustrating Theorem 3.7 with one of the conditions (R5 ) -(R5 ) in each of them. To simplify the exposition, we are dealing with higher order scalar equations rather than with equivalent first order systems. The class of nonlinearities r in each example is assumed to satisfy conditions (R0) and (R1). Example 4.1 (Theorem 3.7 with (R5 )). Fix ε = 0.28 and, for any real α, define four intervals as follows: Consider the following forth order interval differential equation J 0 (α)y + J 1 (α)y + J 2 (α)y + J 3 (α)y + y = r(α, y, y , y , y ).

Normalization of the period
We are looking for periodic solutions, with unknown period p, of the differential equatioṅ Following the standard scheme, let us introduce the unknown period p as an additional parameter. Define β = 2π p and apply the change of variables We are now in a position to reformulate the original problem as an operator equation in the appropriate space of 2π-periodic functions and apply the equivariant degree method.

S 1 -representations
We will use the first Sobolev space of functions on the unit circle equipped with the natural structure of S 1 -representation induced by the shift in time. Let us recall some standard facts related to S 1representations. As is well-known (see, for example, [12]), any real irreducible S 1 -representation is of dimension 1 or 2 and can be described as follows. Take an integer l > 0 and define the S 1 -action on C R 2 by (e iϕ , z) → e ilϕ · z, where "·" stands for complex multiplication,(denote this representation V l ); also, denote by V 0 the trivial one-dimensional S 1 -representation.
Define V = R n . Denote by W = H 1 (S 1 ; V ) the first Sobolev space of functions from S 1 to V . Observe that W admits the "Fourier decomposition" where the subspace of zero Fourier modes (i.e., constant functions) is identified with V , while the subspace of the l-th Fourier modes W l is identified with the complexification of V (denoted V c ). In particular, any function u ∈ W l can be written in the form e ilt · (x l + iy l ) for some x l , y l ∈ V . There is a natural orthogonal S 1 -representation on W given by Formula (25) gives rise to the trivial action on V and the action (e iϕ , u)(t) → e ilϕ · u(t) on W l .

Reformulation in the functional space
Take the first Sobolev space W and define the orthogonal projector K : W → L 2 (S 1 ; V ) by We can now rewrite (23) as the following operator equation in [α − , α + ] × R + × W : where . Formula (25) gives rise to the S 1 -action on [α − , α + ] × R + × W (we assume that S 1 acts trivially on [α − , α + ] × R + ). Moreover, it is easy to see that F given by (26) and (27) is S 1 -equivariant.

Reducing the problem to computing S 1 -degree
In order to apply the equivariant degree method, we need to localize potential bifurcating branches in a cylindric box Ω ⊂ [α − , α + ] × R + × W in such a way that the operator (26) is Ω-admissible.
To this end, consider the sets D k provided by condition (P5) and put where D u F denotes the derivative of F with respect to u (cf. (27)).
(ii) This part trivially follows from the compactness of Σ and condition (P5)(iii) combined with the standard linearization argument.
Take D k given by (28) and B r (0) provided by Lemma 5.2. Define Clearly, Ω is S 1 -invariant. By the existence of the invariant Urysohn function, one can take an invariant function ς : Ω → R satisfying the properties Consider the map F ς : Ω → R ⊕ W given by By definition, any solution to the equation F ς (α, β, u) = 0 is also a solution to (23). In addition, F ς is an S 1 -equivariant Ω-admissible map for which S 1 -Deg (F ς , Ω) is correctly defined. The next statement provides a sufficient condition for the existence of a branch of periodic solutions bifurcating from the trivial solution (cf. Definition 2.1). We follow the scheme suggested in [5] (see Theorem 9.18) with several modifications making the argument more transparent.
As in [5], the following statement is the main topological ingredient in the proof of Proposition 5.4 (cf. Theorem 3 in [37], p. 170).
Proposition 5.5 (Kuratowski). Let X be a metric space, A, B ⊂ X two disjoint closed sets in X, and K a compact set in X such that K ∩ A = ∅ = K ∩ B. If the set K does not contain a connected component Proof of Proposition 5.4. Put Consider the family of invariant functions ς q : Ω → R given by Suppose for contradiction, there does not exist a compact connected set To apply Proposition 5.5, we need to show that K ∩ ς −1 0 (0) = ∅ = K ∩ ς −1 r (0). Notice that for any q ∈ (0, r), ς q satisfies properties (32), so S 1 -Deg (F ςq , Ω = 0 (cf Remark 5.3 and the assumptions of Proposition 5.4. By the existence property of S 1 -degree, for each q ∈ (0, r) there exists (α q , β q , u q ) ∈ K with u q = q. Since K is compact, it follows that there exist (α 0 , β 0 , u 0 ) ∈ K ∩ ς −1 0 (0) and (α r , β r , u r ) ∈ K ∩ ς −1 r (0).
and let us, first, show that Z is S 1 -invariant. Notice that A is invariant. Suppose for contradiction that K ∩ N is not invariant. Then, there exist u ∈ K ∩ N and γ ∈ S 1 such that (γ, u) / ∈ K ∩ N . However, since K is invariant and K ⊂ N ∪ N , it follows that (γ, u) ∈ K ∩ N . We now have S 1 (u) ⊂ N ∪ N with S 1 (u) ∩ N = ∅ = S 1 (u) ∩ N and N ∩ N = ∅, which contradicts the connectedness of S 1 (u). Thus, Z is invariant as the union of invariant sets. Similarly, Z is also invariant.

5.6
Computation of S 1 -Deg (F ς , Ω) via deformations Proposition 5.4 reduces the proof of Theorem 3.2 to the computation of S 1 -Deg (F ς , Ω) and showing that this degree is non-zero. Our goal now is to connect S 1 -Deg (F ς , Ω) to spectral properties of D x f (α, 0) (cf. condition (P1)). This will be done in several steps.
Step II: Computation of the degree. For any m ∈ N, put W m = V ⊕ W 1 ⊕ ... ⊕ W m (cf. (24)). Combining the compactness of the operator a with the suspension property of the S 1 -equivariant degree (see Appendix), one can find a sufficiently large m such that the field F k is equivariantly homotopic to the compact field F 2 k : Ω Fix some α between α − and α + . By condition (P3), the map a 0 k : N k → GL(V ) is homotopic to the constant map a : N k → GL(V ) given by a(α, β) ≡ −D x f (α, 0). Now, we are going to use formula (43) presented in Appendix. To this end, one needs to separate the "contribution" of the zero Fourier mode to the S 1 -degree from other modes. Define where B is the unit ball in V . Also, defineF k : Combining the suspension property of the S 1 -degree with the product formula (see [5], Theorem 6.8), one obtains Further, by applying formula (43), Finally, applying the additivity property of S 1 -Deg and the Brouwer degree, we get where a l (α, β) = a(α, β)| W l .
(b) To prove Part (b), it suffices to deduce (P5 ) from (P5 ). Notice that R(f ) is the set of roots of polynomials with coefficients parameterized by α ∈ [α − , α + ]. Hence, the coefficients of these polynomials are uniformly bounded. Observe also that the leading coefficient of these polynomials is identically equal to 1, therefore R(f ) is a compact set. For any ε > 0, there exists a sufficiently large m such that and D x f (·, 0) is non-singular, it follows that R(f ) is uniformly separated from [α − , α + ] × {β < ε} provided that ε is small enough. On the other hand, the sets m−1 k=2 S i (f ) and R(f ) are compact and disjoint (see condition (P5 )), so they can be uniformly separated. Hence there exists a neighborhood N ε (R(f )) of R(f ) in [α − , α + ]×R + such that N ε (R(f ))∩S(f ) = ∅. Without loss of generality, one can assume that N ε (R(f )) is a finite union of discs, therefore, the complement to N ε (R(f )) in [α − , α + ] × R + has finitely many bounded connected components, say, By construction, D is a finite collection of disjoint sets homeomorphic to closed discs (denoted D k ) and ∂D k ⊂ ∂N ε (R(f )), thus D k satisfies condition (P5 ). Hence, the result follows from Theorem 3.5(a).

Proof of Theorem 3.5(d)
Let P be the set provided by condition (P4). Our first goal is to construct P ⊃ P such that (a) P satisfies (P4); and, (b) P 0 is a disjoint union of finitely many sets homeomorphic to a closed disc (cf. (9)). To this end, without loss of generality (use a small perturbation of P if necessary), one can assume that P 0 ⊂ [α − , α + ] × R + is a disjoint union ∪ m i=1 K i , where K i is a (ν i + 1)-connected compact domain. Using the same surgery argument as in the proof of Alexander's tame sphere Theorem (see, for example, [11], Theorem 4.34), one can construct P satisfying (a) and (b).
Our next goal is to construct a finite collection of discs D k ⊂ [α − , α + ] × R + satisfying (P5). Take R and S given by (10). Using the same argument as in the proof of Theorem 3.5(b) above, one can construct a sufficiently small neighborhood N ε (R ∩ P 0 ) of the intersection R ∩ P 0 such that N ε (R ∩ P 0 ) ∩ S = ∅ and N ε (R ∩ P 0 ) ⊂ P 0 (cf. condition (P5 )). Take C = [α − , α + ] × R + \ N ε (R ∩ P 0 ). By the standard compactness argument, without loss of generality, assume that C splits into finitely many connected components C = C 0 ∪ C 1 ∪ ... ∪ C r , where C 0 stands for the (unique) unbounded component. Put D = ∪ r i=1 C i ∪ N ε (R ∩ P 0 ). Let us show that D is a finite union of discs. By construction, D = ∪ k i=1 D k is a finite disjoint union of regular closed subsets (i.e., each D k is a closure of its interior). To show that each D k is contractible, take a closed curve γ ⊂ D and assume that it is not contractible to a point inside D k . Then, there exists a set K ⊂ [α − , α + ]×R + \D k bounded by γ. However, this contradicts the construction of D. Therefore, D k satisfies condition (P5)(i). Also, since ∂D k ⊂ ∂N ε (R∩P 0 ) for any k, it follows that D k satisfies (P5)(iii). Finally, to show that D k satisfies (P5)(ii), observe that R ∩ D ⊃ R ∩ P 0 . Since, D ⊂ P 0 , one has R ∩ D = R ∩ P 0 and (P5)(ii) follows.
called an admissible pair. Similarly, a continuous map h : [0, 1] × Ω → V is called an admissible (equivariant) homotopy if h(t, ·) is admissible for any t ∈ [0, 1]. It is possible to axiomatically define a unique function S 1 -Deg which assigns to each admissible pair a formal sum of finite cyclic groups with integer coefficients (cf. [5], pp. 109, 113 ). The following is a partial list of the axioms: (A1) (Existence) If S 1 -Deg (f, Ω) = k l=1 n l k (Z l k ) and n l k = 0 for some k, then there exists an x ∈ Ω such that f (x) = 0 and Z l k ⊂ G x .
(A5)(Suspension) Suppose that A is an orthogonal S 1 -representation and U is an open bounded invariant neighborhood of zero in A. Then, Using the equivariant version of the standard Leray-Schauder projection, one can define the S 1degree to S 1 -equivariant compact vector fields (see [5,26] for details). Combining the axioms of the S 1 -degree with some standard homotopy theory techniques, one can reduce the computation of the S 1 -degree of the maps naturally associated with the system undergoing the Hopf bifurcation to the computation of the Brouwer degree. To be more precise, let V be an orthogonal S 1 -representation with V S 1 = {v ∈ V : (γ, v) = v ∀γ ∈ S 1 } = {0}. Take the isotypical decomposition where each V kj is modeled by the k j -th irreducible representation. Define O = {(λ, v) ∈ C ⊕ V : v < 2 , 1/2 < |λ| < 4}. Now, consider a map a : S 1 → GL S 1 (V ) and define a j : S 1 → GL S 1 (V kj ) by the formula a j (λ) = a(λ) |V k j (see, [5, p. 284]). Let f a : O → R ⊕ V be an S 1 -equivariant map defined by The following formula plays an important role in the proof of Theorem 3.2: where B stands for the unit ball in C (cf. [5], Theorem 4.23).