Generalized Pitchfork Bifurcations in D-Concave Nonautonomous Scalar Ordinary Differential Equations

The global bifurcation diagrams for two different one-parametric perturbations ($+\lambda x$ and $+\lambda x^2$) of a dissipative scalar nonautonomous ordinary differential equation $x'=f(t,x)$ are described assuming that 0 is a constant solution, that $f$ is recurrent in $t$, and that its first derivative with respect to $x$ is a strictly concave function. The use of the skewproduct formalism allows us to identify bifurcations with changes in the number of minimal sets and in the shape of the global attractor. In the case of perturbation $+\lambda x$, a so-called global generalized pitchfork bifurcation may arise, with the particularity of lack of an analogue in autonomous dynamics. This new bifurcation pattern is extensively investigated in this work.


Introduction
The interest that the description of nonautonomous bifurcation patterns arouses in the scientific community has increased significantly in recent years, as evidenced by the works [1], [2], [3], [6], [11], [12], [16], [17], [20], [21], [24,25], [27,28], [30,31], [32], and references therein.This paper constitutes an extension of the work initiated in [11], were we describe several possibilities for the global bifurcation diagrams of certain types of one-parametric variations of a coercive equation.We make use of the skewproduct formalism, which allows us to understand bifurcations as changes in the number of minimal sets and in the shape of the global attractor, which of course give rise to substantial changes in the global dynamics.
Let us briefly describe the skewproduct formalism.Standard boundedness and regularity conditions ensure that the hull Ω of a continuous map f 0 : R × R → R, defined as the closure of the set of time-shifts {f 0 •t : t ∈ R} in a suitable topology of C(R × R, R), is a compact metric space, and that the map σ : R × Ω → Ω, (t, ω) → ω•t (where, as in the case of f 0 , (ω•t)(s, x) = ω(t + s, x)) defines a global continuous flow.The continuous function f (ω, x) = ω(0, x) provides the family of equations which includes x ′ = f 0 (t, x): it corresponds to ω 0 = f 0 ∈ Ω. When, in addition, f 0 satisfies some properties of recurrence in time, the flow (Ω, σ) is minimal, which means that Ω is the hull of any of its elements.If u(t, ω, x) denotes the solution of (1.1) ω with u(0, ω, x) = x, then τ : R × Ω × R → Ω × R, (t, ω, x) → (ω•t, u(t, ω, x)) defines a local flow on Ω×R of skewproduct type: it projects over the flow (Ω, σ).If f 0 is coercive with respect to x uniformly in t ∈ R, so is f uniformly in ω ∈ Ω, and this ensures the existence of the global attractor and of at least one minimal compact τ -invariant subset of Ω × R. In the simplest nonautonomous cases, the minimal subsets are (hyperbolic or nonhyperbolic) graphs of continuous functions, and thus they play the role performed by the critical points of an autonomous equation; but there are cases in which both the shape of a minimal set and the dynamics on it are extremely complex, without autonomous analogues, and therefore impossible bifurcation scenarios for a autonomous equation can appear in the nonautonomous setting.So, we take as starting points a (global) continuous minimal flow (Ω, σ) and a continuous map f : Ω × R → R, assume that f is coercive in x uniformly on Ω, and define the dissipative flow τ .Throughout this paper, we also assume that the derivatives f x and f xx globally exist and are jointly continuous on Ω × R, as well as the fundamental property of strict concavity of f x with respect to x: d-concavity.Not all these conditions are in force to obtain the results of [11], but, for simplicity, we also assume them all to summarize part of the properties there proved.
The first goal in [11] is to describe the possibilities for the global µ-bifurcation diagram of the one-parametric family x ′ = f (ω•t, x) + µ, with global attractor A µ .In particular, it is proved that, if there exist three minimal sets for a value µ 0 ∈ R of the parameter, then: A µ contains three (hyperbolic) minimal sets if and only if µ belongs to a nondegenerate interval (µ − , µ + ); the two upper (resp.lower) minimal sets collide on a residual invariant subset of Ω when µ ↓ µ − (resp.µ ↑ µ + ); and A µ reduces to the (hyperbolic) graph of a continuous map on Ω if µ / ∈ [µ − , µ + ].That is, the global bifurcation diagram presents at µ − and µ + two local saddlenode bifurcation points of minimal sets and two points of discontinuity of A µ : is the nonautonomous analogue of the bifurcation diagram of x ′ = −x 3 + x + µ.
A second type of perturbation is considered in [11], namely x ′ = f (ω•t, x) + λx, with global attractor A λ , under the additional assumption f (•, 0) ≡ 0. Now, M 0 = Ω × {0} is a minimal set for all λ, and its hyperbolicity properties are determined by the Sacker and Sell spectrum [−λ + , −λ − ] of the map ω → f x (ω, 0).Two possible global bifurcation diagrams are described, and some conditions ensuring their occurrence are given.The first one is the classical global pitchfork bifurcation diagram, with unique bifurcation point λ + : M 0 is the unique minimal set for λ ≤ λ + , and two more (hyperbolic) minimal sets occur for λ > λ + , which collide with M 0 as λ ↓ λ + .An autonomous analogue is the diagram of x ′ = −x 3 + λ.The second one is the local saddle-node and transcritical bifurcation diagram, with a local saddle node bifurcation of minimal sets at a point λ 0 < λ − and a so-called generalized transcritical bifurcation of minimal sets around M 0 .We will describe this diagram in detail in the next pages, pointing out now the most remarkable fact: M 0 collides with another (hyperbolic) minimal set as λ ↑ λ − and as λ ↓ λ + , and it is the unique minimal set lying on a band Ω × [−ρ, ρ] for a ρ > 0 if λ ∈ [λ − , λ + ].This local transcritical bifurcation becomes classical if λ − = λ + , being x ′ = −x 3 + 2x 2 + λx an autonomous example of this situation.
This analysis of the family x ′ = f (ω•t, x) + λx initiated in [11] is far away to be complete.The goal of this paper is to describe all the possibilities for its global bifurcation diagram.Besides the two described ones, only a third situation may arise: a global generalized pitchfork bifurcation diagram, just possible when λ − < λ + .It is characterized by the existence of two bifurcation points, λ 0 ∈ [λ − , λ + ) and λ + : M 0 is the unique minimal set for λ < λ 0 , there are two of them for λ ∈ (λ 0 , λ + ], and there are three for λ > λ + .The lack of an autonomous analogue raises a nontrivial question: does this bifurcation diagram correspond to some actual family?We also answer it, explaining how to construct nonautonomous patterns fitting at each one of the described possibilities.Furthermore, we prove that, given λ − < λ + , any λ 0 ≤ λ + is the first bifurcation point of a suitable family x ′ = g λ0 (ω•t, x) + λx with Sacker and Sell spectrum of (g λ0 ) x (•, 0) given by [−λ + , −λ − ], and that the three possible diagrams actually occur: they correspond to λ 0 < λ − , λ 0 = λ + and λ 0 ∈ [λ − , λ + ).As a tool to prove of this last result, we analyze the bifurcation possibilities for a new one-parametric family, namely x ′ = f (ω•t, x) + ξx 2 .In order not to lengthen this introduction too much, we omit here the (self-interesting) description of the bifurcation possibilities for this case, and refer the reader to Section 6.
These are the main results of this paper, which presents more detailed descriptions in some particular cases.Its contents are organized in five sections.Section 2 contains the basic notions and properties required to start with the analysis.Section 3 is devoted to the description of the three mentioned possibilities for the bifurcation diagrams of x ′ = f (ω•t, x) + λx.In Section 4, we focus on the case of a cubic polynomial f (ω, x) = −a 3 (ω)x 3 + a 2 (ω)x 2 + a 1 (ω) with strictly positive a 3 , and show how some suitable properties of the coefficients a 1 , a 2 and a 3 and some factible relations among them either preclude or ensure each one of the three different bifurcation diagrams.Section 5 extends these results to more general functions f (ω, x) = (−a 3 (ω) + h(ω, x))x 3 + a 2 (ω)x 2 + a 1 (ω), describing in this way other patterns fitting each one of the possibilities.And Section 6 begins with the description of the casuistic for the bifurcation diagrams of x ′ = f (ω•t, x) + ξx 2 to conclude with the consequence mentioned at the end of the previous paragraph.

Preliminaries
Throughout the paper, the map σ : R × Ω → Ω, (t, ω) → σ t (ω) = ω•t defines a global continuous flow on a compact metric space Ω, and we assume that the flow (Ω, σ) is minimal, that is, that every σ-orbit is dense in Ω.This paper will be focused on describing the bifurcation diagrams of simple parametric variations of the family where f : Ω × R → R is assumed to be jointly continuous, f x and f xx are supposed to exist and to be jointly continuous (which we represent as f ∈ C 0,2 (Ω × R, R)), and f (ω, 0) = 0 for all ω ∈ Ω (that is, x ≡ 0 solves the equation).If only f and f x are assumed to exist and to be jointly continuous, then we shall say that f ∈ C 0,1 (Ω × R, R).Additional coercivity and concavity properties will be assumed throughout the paper.In Section 4, we focus on the case in which f (ω, x) is a cubic polynomial in the state variable x with strictly negative cubic coefficient.We develop our bifurcation theory through the skewproduct formalism: as explained in the Introduction, our bifurcation analysis studies the variations on the global attractors and on the number and structure of minimal sets for the corresponding parametric family of skewproduct flows.In the next subsections, we summarize the most basic concepts and some basic results required in the formulations and proofs of our results.The interested reader can find in Section 2 of [11] more details on these matters, as well as a suitable list of references.
2.2.Functions of bounded primitive.Throughout this paper, the space of continuous functions from Ω to R will be represented by C(Ω), the subspace of functions a ∈ C(Ω) such that Ω a(ω) dm = 0 for all m ∈ M erg (Ω, σ) will be represented by C 0 (Ω), the subspace of functions a ∈ C(Ω) such that the map t → a ω (t) = a(ω•t) is continuously differentiable on R will be represented by C 1 (Ω) (in this case we shall represent a ′ (ω) = a ′ ω (0)), and the subspace of functions a ∈ C(Ω) with continuous primitive, that is, such that there exists b ∈ C 1 (Ω) with b ′ = a, will be represented by CP (Ω).
It is frequent to refer to a function a ∈ CP (Ω) as "with bounded primitive".Let us explain briefly the reason.Recall that (Ω, σ) is minimal.Then, a ∈ CP (Ω) if and only if there exists ω 0 ∈ Ω such that the map a 0 : R → R, t → a(ω 0 •t) has a bounded primitive b 0 (t) = t 0 a 0 (s) ds, in which case this happens for all ω ∈ Ω (see e.g.Lemma 2.7 of [15] or Proposition A.1 of [19]).
Given a Borel measure m on Ω, we shall say that β : Ω → R is m-measurable if it is measurable with respect to the m-completion of the Borel σ-algebra, and we shall say that β : . Note that any τ -equilibrium is C 1 along the base orbits.We shall say that β : Ω → R is a semicontinuous equilibrium (resp.semiequilibrium) if it is an equilibrium (resp.semiequilibrium) and a bounded semicontinuous map.A copy of the base for the flow τ is the graph of a continuous τ -equilibrium.
Let β : Ω → R be C 1 along the base orbits.The map β shall be said to be a global upper (resp.lower for every ω ∈ Ω, and to be strict if the previous inequalities are strict for all ω ∈ Ω.Some comparison arguments prove the following facts (see Sections 3 and 4 of [23]): if every forward τ -semiorbit is globally defined, then β is a τ -superequilibrium (resp.τ -subequilibrium) if and only if is a global upper (resp.lower) solution of (2.1), and it is strong as superequilibrium (resp.subequilibrium) if it is strict as global upper (resp.lower) solution.Analogously, if every backward τ -semiorbit is globally defined, then β is a time-reversed τsubequilibrium (resp.time-reversed τ -superequilibrium) if and only if it is a global upper (resp.lower) solution of (2.1), and it is strong as time-reversed subequilibrium (resp.time-reversed superequilibrium) if it is strict as global upper (resp.lower) solution.

Minimal sets, coercivity and global attractor
is composed by globally defined τ -orbits, and it is minimal if it is compact, τ -invariant and it does not contain properly any other compact τ -invariant set.Let us recall some properties of compact τ -invariant sets and minimal sets for the local skewproduct flow (Ω × R, τ ) over a minimal base (Ω, σ).Let K ⊂ Ω × R be a compact τ -invariant set.Since (Ω, σ) is minimal, K projects onto Ω, that is, the continuous map π : K → Ω, (ω, x) → ω is surjective.In addition, where α K (ω) = inf{x ∈ R : (ω, x) ∈ K} and β K (ω) = sup{x ∈ R : (ω, x) ∈ K} are, respectively, lower and upper semicontinuous τ -equilibria whose graphs are contained in K.In particular, the residual sets of their continuity points are σinvariant.They will be called the lower and upper delimiter equilibria of K.The compact τ -invariant set K is said to be pinched if there exists ω ∈ Ω such that the section (K) ω = {x : (ω, x) ∈ K} is a singleton.A τ -minimal set M ⊂ Ω × R is said to be hyperbolic attractive (resp.repulsive) if it is uniformly exponentially asymptotically stable at ∞ (resp.−∞).Otherwise, it is said to be nonhyperbolic.
uniformly on Ω.A stronger definition of coercivity will be needed in part of Section 5 and in Section 6: then it is (Co).The arguments leading to Theorem 16 of [8] (see also Section 1.2 of [9]) ensure that, if f ∈ C 0,1 (Ω × R, R) is (Co), then the flow τ is globally forward defined and admits a global attractor.That is, a compact τ -invariant set A which satisfies lim t→∞ dist(τ t (C), A) = 0 for every bounded set dist Ω×R ((ω 1 , x 1 ), (ω 2 , x 2 )) .
In addition, the attractor takes the form ) and is composed by the union of all the globally defined and bounded τ -orbits.And, as proved in Theorem 5.1(iii) of [11], any global (strict) lower solution κ satisfies κ ≤ β A (κ < β A ) and a (strict) upper solution The set of λ ∈ R such that the family x ′ = (a(ω•t) − λ) x does not have exponential dichotomy over Ω is called the Sacker and Sell spectrum of a ∈ C(Ω), and represented by sp(a).Recall that Ω is connected, since (Ω, σ) is minimal.The arguments in [18] and [33] show the existence of m l , m u ∈ M erg (Ω, σ) such that sp(a) = γ a (Ω, m l ), γ a (Ω, m u ) , and also that Ω a(ω) dm ∈ sp(a) for any m ∈ M inv (Ω, σ).We shall say that a has band spectrum if sp(a) is a nondegenerate interval and that a has point spectrum if sp(a) reduces to a point.As seen in Subsection 2.2, sp(a) = {0} if a ∈ C 0 (Ω).
On the other hand, assume that f ∈ C 0,1 (Ω × R, R), where f is the function on the right hand side of (2.1).The Lyapunov exponent of a compact τ -invariant set We will frequently omit the subscript f x if no confusion may arise.We will refer to the Sacker and Sell spectrum of f x : K → R as the Sacker and Sell spectrum of f x on a compact τ -invariant set K ⊂ Ω × R. Since (Ω, σ) is a minimal flow, a τ -minimal set M ⊂ Ω × R is nonhyperbolic if and only if 0 belongs to the Sacker and Sell spectrum of f x on M.Moreover, Proposition 2.8 of [5] proves that M is an attractive (resp.repulsive) hyperbolic copy of the base if and only if all its Lyapunov exponents are strictly negative (resp.positive).
Theorems 1.8.4 of [4] and 4.1 of [14] provide a fundamental characterization of the set M erg (K, τ ) given by the τ -ergodic measures concentrated on a compact τinvariant set K ⊂ Ω × R: for any ν ∈ M erg (K, τ ), there exists an m-measurable τ -equilibrium β : Ω → R with graph contained in K such that, for every continuous function where m ∈ M erg (Ω, σ) is the ergodic measure on which ν projects, given by m(A) = ν((A × R) ∩ K).In particular, the Lyapunov exponent on K for (2.1) with respect to any τ -ergodic measure projecting onto m is given by an integral of the form Ω f x (ω, β(ω)) dm.The converse also holds: any m-measurable τ -equilibrium β : Ω → R with graph in K defines ν ∈ M erg (K, τ ) projecting onto m by (2.3).Note that β 1 and β 2 define the same measure if and only if they coincide m-a.e.
2.6.Strict d-concavity.We shall say that f ∈ C 0,1 (Ω × R, R) is d-concave ((DC) for short) if its derivative f x is concave on R for all ω ∈ Ω.With the purpose of measuring the degree of strictness of the concavity of f x , the standardized ǫmodules of d-concavity of f on a compact interval J were introduced in [11], and several subsets of strictly d-concave functions of C 0,1 (Ω × R, R) were defined in terms of these modules.In this paper, we will only be interested in the set (SDC) * of strictly d-concave functions with respect to every measure (see Definition 3.8 of [11]).Proposition 3.9 of [11] gives a characterization of this set of functions which will be sufficient for the purposes of this paper: •) is strictly decreasing on J}) > 0 for every compact interval J and every m ∈ M erg (Ω, σ).In particular, it can be easily checked that any polynomial of the form p(ω, x) = −a 3 (ω)x 3 + a 2 (ω)x 2 + a 1 (ω)x + a 0 (ω), where the coefficients are continuous and Ω and a 3 is nonnegative and nonzero, is (SDC) * , since p xx (ω, •) is strictly decreasing on R for every ω on an open subset of Ω (recall that the minimality of (Ω, σ) ensures that every open set has positive m-measure for all m ∈ M erg (Ω, σ)).
Assume that the function f of (2.1) is (SDC) * .Following the methods of [29] and [35], Theorems 4.1 and 4.2 of [11] state relevant dynamical properties of the local skewproduct flow τ in terms of the previous properties.Let K ⊂ Ω × R be a compact τ -invariant set.Then, there exist at most three distinct τ -invariant measures of M erg (K, τ ) which project onto m.Moreover, if there exist three such measures ν 1 , ν 2 and ν 3 projecting onto m, and they are respectively given by the m-measurable equilibria β 1 , β 2 and β 3 (see (2.3)) with β 1 (ω) < β 2 (ω) < β 3 (ω) for m-a.e.ω ∈ Ω, then γ fx (K, ν 1 ) < 0, γ fx (K, ν 2 ) > 0 and γ fx (K, ν 3 ) < 0 (see the proof of Theorem 4.1 of [11]).In addition, K contains at most three disjoint compact τ -invariant sets, and if it contains exactly three, then they are hyperbolic copies of the base: attractive the upper and lower ones, and repulsive the middle one.These properties will be often combined with those established in Proposition 5.3 of [11]: if f is coercive and either if there exists a repulsive hyperbolic τ -minimal set or if there exist two hyperbolic τ -minimal sets, then there exist three τ -minimal sets.
The possibilities for the global bifurcation diagram are symmetric with respect to the horizontal axis to those described if α λ collides with 0 as λ ↓ λ + .
The proof is analogous in the other case.
Proof of Theorem 3.1.The Sacker and Sell spectrum of which ensures the stated hyperbolicity properties of M 0 (see Subsection 2.5).As in Theorem 6.3 of [11], we define which, as proved there, belong to (−∞, λ + ].This property guarantees the stated structure of the τ λ -minimal sets for λ > λ + , since there exist at most three τ λminimal sets (see Subsection 2.6).Theorem 6.3(ii) of [11] also ensures that at least one of these two parameters µ − , µ + coincides with λ + , which proves the stated collision properties for α λ or for β λ as λ ↓ λ + .As in the statement, we assume that this is the case for β λ , i.e., that µ + = λ + .Then, since λ → β λ (ω) is nondecreasing for all ω ∈ Ω and the intersection of two residual sets is also residual, M 0 is the upper minimal set for all λ ≤ λ + .If also µ − = λ + , then Theorem 6.3(iii) of [11] ensures that the bifurcation diagram is that of (ii).If µ − < λ − , then Theorem 6.4 of [11] shows that the diagram is that of (i), with λ 0 = µ − .The remaining case is, hence, µ − ∈ [λ − , λ + ).We will check that, in this case, the situation is that of (iii), which will complete the proof.
Let us call λ 0 = µ − .Notice that, if λ ∈ (λ 0 , λ + ], then there exist only two τ λ -minimal sets, as otherwise the nonhyperbolicity of M 0 would be contradicted (see Subsection 2.6); and, as explained before, M l λ is hyperbolic attractive (and given by the graph of α λ ).Consequently, it only remains to prove that α λ (ω) = 0 on the residual σ-invariant set of its continuity points for λ < λ 0 .This will ensure that M 0 is the unique τ λ -minimal set for λ < λ 0 , and the hyperbolic attractiveness of M 0 for λ < λ − will ensure that A λ = M 0 for λ < λ − (see Theorem 3.4 of [5]).Recall also that α λ vanishes at all its continuity points if it vanishes at one of them (see e.g.Proposition 2.5 of [11]).
First, let us assume that M l λ0 = M 0 , which means that α λ0 (ω) = 0 on the residual set of its continuity points (see Subsection 2.4).Therefore, the same happens with α λ if λ < λ 0 , since α λ0 ≤ α λ ≤ 0 and the intersection of two residual sets is also residual.This proves the result in this case.
There are simple autonomous examples giving rise to situations (ii) (as x ′ = −x 3 + λx, with λ ± = 0 as bifurcation point, of classical pitchfork type) and (i) (as x ′ = −x 3 ± 2x 2 + λx, with λ 0 = −1 as local saddle-node bifurcation point and λ ± = 0 as local classical transcritical bifurcation point; the two possibilities of (i) correspond to the two signs of the second-order term).Clearly, case (iii) cannot occur in an autonomous (and hence uniquely ergodic) case.We will go deeper in this matter in Sections 4, 5 and 6, where we will show that all the possibilities realize for suitable families (3.1).

Criteria for cubic polynomial equations
Let us consider families of cubic polynomial ordinary differential equations where In addition, f is (SDC) * (see Subsection 2.6).Then, Theorem 3.1 describes the three possible λ-bifurcation diagrams for (4.1).Our first goal in this section, achieved in Subsections 4.1 and 4.2, is to describe conditions on the coefficients a i determining the specific diagram.
The last subsection is devoted to explain how to get actual patterns satisfying the previously established conditions.

4.1.
The case of a 1 with continuous primitive.Throughout this subsection, we assume that a 1 ∈ CP (Ω).Since CP (Ω) ⊆ C 0 (Ω) (see Subsection 2.2), the Sacker and Sell spectrum of a 1 is sp(a 1 ) = {0}.Hence, the bifurcation diagram of (4.1) fits in (i) or (ii) of Theorem 3.1, and our objective is to give criteria ensuring each one of these two possibilities.The relevant fact in terms of which the criteria will be constructed is that the number of τ 0 -minimal sets distinguishes the type of bifurcation: there is either one τ 0 -minimal set in (ii) or two τ 0 -minimal sets in (i).
Proposition 4.2 provides a simple classification of the casuistic for (4.1) in this case.It is based on the previous bifurcation analysis of made in Proposition 4.1.These two results extend Proposition 6.6 and Corollary 6.7 of [11] to the case of strictly positive a 3 (instead of a 3 ≡ 1), since the case of a 1 ≡ 0 is trivially covered by Proposition 4.2 with b ≡ 0. We will call τξ the local skewproduct flow induced by (4.2) ξ on Ω × R.
The main results of this subsection are stated in Propositions 4.7, 4.8 and 4.9, whose proofs use the next technical results.The first one shows that one of the conditions required in Proposition 4.7 always holds if a 1 has band spectrum (in which case a 1 is not a constant function).Lemma 4.5.Let a ∈ C(Ω).Then, the next three assertions are equivalent: a is nonconstant; min ω∈Ω a(ω) < inf sp(a); max ω∈Ω a(ω) > sup sp(a).
) for all ω ∈ Ω.Hence, Lemma 4.6 ensures the existence of a τ λ−−δ -minimal set , and this situation only arises in the stated case of Theorem 3.1(i).
Recall that λ + < k 2 if a 1 has band spectrum: see Lemma 4.5.The following two results refer to the case that a 1 has band spectrum: λ − < λ + .(This is ensured in Proposition 4.9 by its condition (4.4)).

Proposition 4.8 (A criterium ensuring pitchfork bifurcation). If λ
for all ω ∈ Ω, then (4.1) does not exhibit the saddle-node and transcritical bifurcations of minimal sets described in Theorem 3.1(i).
Proof examples by choosing a suitable a 2 once fixed a 1 and a 3 .In the same line, Proposition 4.9 establishes conditions ensuring the generalized pitchfork case of Theorem 3.1(iii).But the existence of polynomials satisfying these last conditions is not so obvious.Therefore, our next objective is to develop systematic ways of constructing third degree polynomials giving rise to families (4.1) for which the global bifurcation diagram is that of Theorem 3.1(iii).Hence, all the situations described in that theorem actually realize.Lemma 4.10.Let m 1 , . . ., m n be different elements of M erg (Ω, σ) with n ≥ 1, and let 0 < ǫ < 1 be fixed.For every i ∈ {1, . . ., n}, there exists a continuous c i : Ω → [0, 1] with min ω∈Ω c i (ω) = 0 and max ω∈Ω c i (ω) = 1 such that c i c j ≡ 0 and for every i, j ∈ {1, . . ., n} with j = i.
Proof.It is easy to check that 0 < ǫ 1 < 1/n.In addition, according to Lemma 4.10, So, it is enough to check that the right-hand side is strictly positive; that is, And this follows from ǫ < ǫ 1 , since ǫ 1 is the lowest root of the polynomial.
Note that every function a 1 constructed by the procedure of Proposition 4.11 takes positive and negative values.But this is not a real restriction to get a generalized pitchfork bifurcation diagram, since that corresponding to a 1 + µ for any constant µ ∈ R is of the same type.
Proposition 4.11 shows that the occurrence of families (4.1) with generalized pitchfork bifurcation diagram only requires the existence of two different ergodic measures.The functions a 1 constructed as there indicated are intended to satisfy (4.4); that is, their extremal Lyapunov exponents are near its maximum and minimum.But in fact this is not a necessary condition for a function a 1 to be the first order coefficient of a polynomial giving rise to a generalized pitchfork bifurcation.Theorem 4.14 proves this assertion in the case of a finitely ergodic base flow.Its proof in based on Proposition 4.12 and Corollary 4.13.Proposition 4.12.Assume that M erg (Ω, σ) = {m 1 , . . ., m n } with n ≥ 1.There exists ǫ 2 > 0 such that, if 0 < ǫ ≤ ǫ 2 and c 1 , . . ., c n : Ω → R are the functions constructed in Lemma 4.10 for m 1 , . . ., m n and ǫ, then as topological sum of vector spaces, where C(Ω) is endowed with the uniform topology, given by a = max ω∈Ω |a(ω)|.In particular, the Sacker and Sell spectrum of a ∈ C(Ω) coincides with that of its projection onto c 1 , . . ., c n .
Proof.Let M n×n (R) be the linear space of n × n real matrices, which we endow with the norm C ∞ = max 1≤i,j≤n |c ij |, where C = {c ij } 1≤i,j≤n .The set of regular n×n real matrices GL n (R) is an open subset of M n×n (R), and the identity matrix I belongs to GL n (R).Hence, there exists ǫ 2 ∈ (0, 1) such that, if C −I ∞ ≤ ǫ 2 , then C is regular.Therefore, if ǫ ∈ (0, ǫ 2 ], then the corresponding functions c 1 , . . ., c n of Lemma 4.10 provide a regular matrix Let us consider the continuous linear functionals . ., n}, and note that Ker(T i ) has codimension 1.Therefore, the codimension of the set C 0 (Ω), which coincides with i∈{1,...,n} Ker(T i ), is at most n.
In addition, the linear space c 1 , . . ., c n has dimension n, since the supports of c 1 , . . ., c n are pairwise disjoint.Let us check that c 1 , . . ., dm j for every j ∈ {1, . . ., n}.These n equations provide a homogeneous linear system for α 1 , . . ., α n with regular coefficient matrix C; so α 1 = • • • = α n = 0 and hence c = 0. Consequently, C(Ω) is the algebraic direct sum of c 1 , . . ., c n and C 0 (Ω).We will check that the projections of C(Ω) onto each one of the subspaces are continuous, which will complete the proof of the first assertion.Given a ∈ C(Ω), its projection P c1,...,cn a = n i=1 α i c i onto c 1 , . . ., c n is given by  a for every i ∈ {1, . . ., n}, and hence . ., c n is continuous.Finally, as P C0(Ω) a = a − P c1,...,cn a, also the projection P C0(Ω) is continuous, as asserted.The second assertion is an easy consequence of the first one.
Proof.We take any strictly positive ã3 ∈ C(Ω) and 0 < r 1 ≤ r 2 with r 1 ≤ ã3 (ω) ≤ r 2 for all ω ∈ Ω, and call r = r 2 /r 1 .There is no loss of generality in assuming that λ − < 0 < λ + , since the bifurcation diagrams for a 1 and a 1 + µ coincide for any µ ∈ R. We associate ã1 to a 1 and r by Corollary 4.13.Note that there exists . Now, in order to apply Proposition 4.9, we take ã2 ∈ C(Ω) Hence, the parametric family presents a generalized pitchfork bifurcation of minimal sets.As explained in the proof of Proposition 4.2, the family of changes of variables y(t) = e b(ω•t) x(t) takes (4.9) to without changing the global structure of the bifurcation diagram.That is, the strictly positive functions a 3 = e −2b ã3 and a 2 = e −b ã2 fulfill the statement.

Criteria in a more general framework
The ideas of Subsection 4.2 and 4.3 can be used to construct examples of all the three possible types of global bifurcation diagrams described in Theorem 3.1 for families of differential equations of a more general type.Let us consider where a i ∈ C(Ω) for i ∈ {1, 2, 3}, a 3 is strictly positive, h ∈ C 0,2 (Ω × R, R), and h(ω, 0) = 0 for all ω ∈ Ω.Throughout the section, we will represent the Sacker and Sell spectrum of a 1 as sp(a In the line of part of the results of Section 4, we fix a 1 , a 3 and h satisfying the mentioned hypotheses as well as a fundamental extra condition which relates the behavior of h for small values of x to the properties of a 1 and a 3 , and such that the function f (ω, x) = (−a 3 (ω) + h(ω, x))x 3 + a 2 (ω)x 2 + a 1 (ω)x (5.2) is (Co) and (SDC) * .The goal is to describe conditions on a 2 determining each one of the possible bifurcation cases described in Theorem 3.1 for (5.1).The function a 2 will be sign-preserving under all these conditions.Note the sp(a 1 ) is the Sacker and Sell spectrum of f x on M 0 .Proposition 5.1.Assume that the function f given by (5.2) is (Co) and (SDC) * , and that (H) there exist ρ 0 > 0 and 0 for all ω ∈ Ω, then (5.1) exhibits the local saddle-node and transcritical bifurcations of minimal sets described in Theorem 3.1(i), with α λ (resp.β λ ) colliding with 0 on a residual σ-invariant set as λ ↓ λ + .
for all ω ∈ Ω, then (5.1) does not exhibit the classical pitchfork bifurcation of minimal sets described in Theorem 3.1(ii).
for all ω ∈ Ω, then (5.1) does not exhibit the local saddle-node and transcritical bifurcations of minimal sets described in Theorem 3.
for all ω ∈ Ω, then (5.1) exhibits the generalized pitchfork bifurcation of minimal sets described in Theorem 3.1(iii).
Regarding (H), notice that the included inequality (−λ − − k 1 )/(r 2 + ǫ 0 ) < ρ 0 is fulfilled by taking a large enough upper bound r 2 for a 3 (although the smaller r 2 is, the less restrictive the conditions in points (ii), (iii) and (v) of Proposition 5.1 are).The following results indicate three ways to get the rest of the conditions in (H).Recall that h is always assumed to belong to C 0,2 (Ω × R, R) and to satisfy h(ω, 0) = 0 for all ω ∈ Ω.And recall also the meaning of λ − , λ + , k 1 , k 2 , r 1 and r 2 .

A second bifurcation problem
The ideas and methods developed in [11] and in the previous sections of this paper allow us to classify and describe all the possibilities for the bifurcation diagram of a problem different from that analyzed in Sections 3, 4 and 5, namely Besides its own interest, this analysis allows us to go deeper in the construction of patterns for the three bifurcation possibilities described in Theorem 3.1, as explained at the end of this section.
The proof of Theorem 6.3, which describes the possible bifurcation diagrams for (6.1), requires the next technical result, similar to Proposition 4.4 of [11].
Proof.We call M l µ and M u µ the lower and upper τ µ -minimal sets, defined as in Section 3, and recall that they are attractive if they are hyperbolic, in which case they respectively coincide with the graphs of the continuous maps α µ and β µ .
(i) Since every Lyapunov exponent of M 0 is strictly positive for any µ ∈ R (see Subsection 2.5), M 0 is a repulsive hyperbolic τ µ -minimal set for every µ ∈ R. Consequently, there exist three different hyperbolic τ µ -minimal sets M l µ < M 0 < M u µ (see Subsection 2.6), with M l µ and M u µ given respectively by the graphs of α µ < β µ , which are continuous.The hyperbolic continuation of minimal sets (see Theorem 3.8 of [26]) guarantees the continuity of the maps R → C(Ω), µ → β µ and R → C(Ω), µ → α µ in the uniform topology.
Note that the model analyzed in Proposition 4.2 fits in the situation of Theorem 6.3(iii), and that in that case we can determine the values of µ 1 and µ 2 .Autonomous cases x ′ = f (x) + λx 2 fitting the possibilities described in the previous theorem are very easy to find, since they just depend on the sign of f ′ (0).For example, x ′ = −x 3 + x + λx 2 for (i), x ′ = −x 3 − x + λx 2 for (ii), and x ′ = −x 3 + λx 2 for (iii).
By taking the lower delimiters of the global attractors instead of the upper ones in (6.5), we get a result analogous to Corollary 6.5, with β λ colliding with 0 at the upper bifurcations points.

which contradicts the definition of t 1 . 4 . 2 .
Proposition Let b be a continuous primitive of a 1 .Then, (i) sp(e b a 2 ) ⊂ (0, ∞) if and only if (4.1) exhibits the local saddle-node and classical transcritical bifurcations of minimal sets described in Theorem 3.1(i), with α λ colliding with 0 on a residual σ-invariant set as λ ↓ λ + .In particular, this situation holds if 0 ≡ a 2 ≥ 0. (ii) sp(e b a 2 ) ⊂ (−∞, 0) if and only if (4.1) exhibits the local saddle-node and classical transcritical bifurcations of minimal sets described in Theorem 3.1(i), with β λ colliding with 0 on a residual σ-invariant set as λ ↓ λ + .In particular, this situation holds if 0 ≡ a 2 ≤ 0. (iii) 0 ∈ sp(e b a 2 ) if and only if (4.1) exhibits the classical pitchfork bifurcation of minimal sets described in Theorem 3.1(ii).
[5]the lower (resp.upper)τ-minimal set, and its sections reduce to the points α A (ω) (resp.βA(ω)) at all the continuity points ω of α A (resp.βA ): see Theorem 3.3 of[5].Moreover, it is easy to check by contradiction that, if M l (resp.M u ) is hyperbolic, then it is attractive and it coincides with the graph of α A (resp.β A ), which therefore is a continuous map.
As said after Theorem 3.1, there are autonomous cases presenting either the local saddle-node and classical transcritical bifurcations or the classical pitchfork bifurcation of minimal sets described in cases (i) and (ii) of that theorem.These two possibilities are also the unique ones in nonautonomous examples when a 1 has point spectrum, and we have classified them if a 1 ∈ CP (Ω) in Subsection 4.1, where in addition we have shown simple ways to construct examples fitting in each one of these two situations.
. Condition (4.4) ensures that a 1 has band spectrum and that the intervals in which a 2 can take values are nondegenerate.Propositions 4.8 and 4.7(ii) respectively preclude situations (i) and (ii) of Theorem 3.1, and Proposition 4.4(ii) ensures the stated collision property for α λ (resp.for β λ ).4.3.Cases of generalized pitchfork bifurcation.