On Takens’ Last Problem: tangencies and time averages near heteroclinic networks

On Takens’ Last Problem:
tangencies and time averages near heteroclinic networks

Isabel S. Labouriau  Alexandre A. P. Rodrigues
Centro de Matemática da Universidade do Porto
and Faculdade de Ciências, Universidade do Porto
Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
islabour@fc.up.pt  alexandre.rodrigues@fc.up.pt
CMUP (UID/MAT/00144/2013) is supported by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the Fundação para a Ciência e a Tecnologia (FCT) under the partnership agreement PT2020. A.A.P. Rodrigues was supported by the grant SFRH/BPD/84709/2012 of FCT. Part of this work has been written during AR stay in Nizhny Novgorod University partially supported by the grant RNF 14-41-00044.
July 15, 2019
Abstract

We obtain a structurally stable family of smooth ordinary differential equations exhibiting heteroclinic tangencies for a dense subset of parameters. We use this to find vector fields -close to an element of the family exhibiting a tangency, for which the set of solutions with historic behaviour contains an open set. This provides an affirmative answer to Taken’s Last Problem (F. Takens (2008) Nonlinearity, 21(3) T33–T36). A limited solution with historic behaviour is one for which the time averages do not converge as time goes to infinity. Takens’ problem asks for dynamical systems where historic behaviour occurs persistently for initial conditions in a set with positive Lebesgue measure.

The family appears in the unfolding of a degenerate differential equation whose flow has an asymptotically stable heteroclinic cycle involving two-dimensional connections of non-trivial periodic solutions. We show that the degenerate problem also has historic behaviour, since for an open set of initial conditions starting near the cycle, the time averages approach the boundary of a polygon whose vertices depend on the centres of gravity of the periodic solutions and their Floquet multipliers.

We illustrate our results with an explicit example where historic behaviour arises -close of a SO(2)-equivariant vector field.

Keywords: Heteroclinic cycle, Time averages, Historic behaviour, Heteroclinic tangencies, Newhouse phenomena.

2010 — AMS Subject Classifications

Primary: 34C28; Secondary: 34C37, 37C29, 37D05, 37G35

1 Introduction

Chaotic dynamics makes it difficult to give a geometric description of an attractor in many situations, when probabilistic and ergodic analysis becomes relevant. In a long record of a chaotic signal generated by a deterministic time evolution, for suitable initial conditions the expected time average exists — see [30, 31]. However, there are cases where the time averages do not converge no matter how long we wait. This historic behaviour is associated with intermittent dynamics, which happens typically near heteroclinic networks.

The aim of this article is to explore the persistence of this behaviour for a deterministic class of systems involving robust heteroclinic cycles, leading to an answer to Taken’s Last Problem [35]. More precisely, we study non-hyperbolic heteroclinic attractors such that the time averages of all solutions within their basin of attraction do not converge, and for which this holds persistently.

This is done by first studying a one-parameter family of vector fields having periodic solutions connected in a robust cycle. We show that under generic conditions there are parameter values for which the invariant manifolds of a pair of periodic solutions have a heteroclinic tangency. This implies the Newhouse property of existence of infinitely many sinks. Results by Kiriki and Soma [18] may then be used to provide an affirmative answer to the problem proposed by Takens in [35].

1.1 Takens’ last problem

Let be a compact three-dimensional manifold without boundary and consider a vector field defining a differential equation

(1.1)

and denote by , with , the associated flow with initial condition . The following terminology has been introduced by Ruelle [30] (see also Sigmund [31]).

Definition 1

We say that the solution , , of (1.1) has historic behaviour if there is a continuous function such that the time average

(1.2)

fails to converge.

A solution , with historic behaviour retains informations about its past. This happens, in particular, if there are at least two different sequences of times, say and , such that the following limits exist and are different:

The consideration of the limit behaviour of time averages with respect to a given measure has been studied since Sinai [32], Ruelle [29] and Bowen [4]. Usually, historic behaviour is seen as an anomaly. Whether there is a justification for this belief is the content of Takens’ Last Problem [18, 34, 35]: are there persistent classes of smooth dynamical systems such that the set of initial conditions which give rise to orbits with historic behaviour has positive Lebesgue measure? In ergodic terms, this problem is equivalent to finding a persistent class of systems admitting no physical measures [11, 30], since roughly speaking, these measures are those that give probabilistic information on the observable asymptotic behaviour of trajectories.

The class may become persistent if one considers differential equations in manifolds with boundary as in population dynamics [11, 13]. The same happens for equivariant or reversible differential equations [10]. The question remained open for systems without such properties until, recently, Kiriki and Soma [18] proved that any Newhouse open set in the -topology, , of two-dimensional diffeomorphisms is contained in the closure of the set of diffeomorphisms which have non-trivial wandering domains whose forward orbits have historic behaviour. As far as we know, the original problem, stated for flows, has remained open until now.

1.2 Non-generic historic behaviour

In this section, we present some non-generic examples that, however, occur generically in families of discrete dynamical systems depending on a small number of parameters. The first example has been given in Hofbauer and Keller [12], where it has been shown that the logistic family contains elements for which almost all orbits have historic behaviour. This example has codimension one in the space of endomorphisms of the interval; the regularity is due to the use of the Schwarzian derivative operator.

The second example is due to Bowen, who described a codimension two system of differential equations on the plane whose flow has a heteroclinic cycle consisting of a pair of saddle-equilibria connected by two trajectories. As referred by Takens [34, 35], apparently Bowen never published this result. We give an explicit example in 7.2 below. The eigenvalues of the derivative of the vector field at the two saddles are such that the cycle attracts solutions that start inside it. In this case, each solution in the domain has historic behaviour. In ergodic terms, it is an example without SRB measures. Breaking the cycle by a small perturbation, the equation loses this property. This type of dynamics may become persistent for dynamical systems in manifolds with boundary or in the presence of symmetry. We use Bowen’s example here as a first step in the construction of a generic example. Other examples of high codimension with heteroclinic attractors where Lebesgue almost all trajectories fail to converge have been given by Gaunersdorfer [8] and Sigmund [31].

Ergodicity implies the convergence of time averages along almost all trajectories for all continuous observables [17]. For non-ergodic systems, time averages may not exist for almost all trajectories. In Karabacak and Ashwin [17, Th 4.2], the authors characterise conditions on the observables that imply convergent time averages for almost all trajectories. This convergence is determined by the behaviour of the observable on the statistical attractors (subsets where trajectories spend almost all time). Details in [17, §4].

1.3 General examples

The paradigmatic example with persistent historic behaviour has been suggested by Colli and Vargas in [6], in which the authors presented a simple non-hyperbolic model with a wandering domain characterised by the existence of a two-dimensional diffeomorphism with a Smale horseshoe whose stable and unstable manifolds have persistent tangencies under arbitrarily small perturbations. The authors of [6] suggest that this would entail the existence of non-wandering domains with historic behaviour, in a robust way. This example has been carefully described in [18, §2.1].

For diffeomorphisms, an answer has been given by Kiriki and Soma [18], where the authors used ideas suggested in [6] to find a nontrivial non-wandering domain (the interior of a specific rectangle) where the diffeomorphism is contracting. In a robust way, they obtain an open set of initial conditions for which the time averages do not converge. Basically, the authors linked two subjects: homoclinic tangencies studied by Newhouse, Palis and Takens and non-empty non-wandering domains exhibiting historic behaviour. An overview of the proof has been given in §2 of [18]. We refer those that are unfamiliar with Newhouse regions to the book [24].

1.4 The results

The goal of this article is twofold. First, we extend the results by Takens [34] and by Gaunersdorfer [8] to heteroclinic cycles involving periodic solutions with real Floquet multipliers. The first main result is Theorem 8, with precise hypotheses given in Section 3:

1 result:

Consider an ordinary differential equation in having an attracting heteroclinic cycle involving periodic solutions with two-dimensional heteroclinic connections. Any neighbourhood of this cycle contains an open set of initial conditions, for which the time averages of the corresponding solutions accumulate on the boundary of a polygon, and thus, fail to converge. The open set is contained in the basin of attraction of the cycle and the observable is the projection on a component.

This situation has high codimension because each heteroclinic connection raises the codimension by one, but this class of systems is persistent in equivariant differential equations. The presence of symmetry creates flow-invariant fixed-point subspaces in which heteroclinic connections lie — see for example the example constructed in [28, §8]. Another example is constructed in Section 7.4 below. The second main result, Theorem 11, concerns tangencies:

2 result:

Consider a generic one-parameter family of structurally stable differential equations in the unfolding of an equation for which the 1 result holds. Then there is a sequence of parameter values for which there is a heteroclinic tangency of the invariant manifolds of two periodic solutions.

We use this result to obtain Theorem 12:

3 result:

Consider a generic one-parameter family of structurally stable differential equations in the unfolding of an equation for which the 1 result holds. Therefore, for parameter values in an open interval, there are vector fields arbitrarily -close to an element of the family, for which there is an open set of initial conditions exhibiting historic behaviour.

In other words, we obtain a class, dense in a -open set of differential equations and elements of this class exhibit historic behaviour for an open set of initial conditions, which may be interpreted as the condition required in Takens’ Last Problem. The idea behind the proof goes back to the works of [15, 16], combined with the recent progress on the field made by [18]. The proof consists of the followingsteps:

  1. use the 3 result to establish the existence of intervals in the parameters corresponding to Newhouse domains;

  2. in a given cross section, construct a diffeomorphim (-close to the first return map) having historic behaviour for an open set of initial conditions;

  3. transfer the historic behaviour from the perturbed diffeomorphism of 2. to a flow -close to the original one.

Furthermore, in the spirit of the example by Bowen described in [34], we obtain:

4 result:

We construct explicitly a class of systems for which historic behaviour arises -close to the unfolding of a fully symmetric vector field, we may find an open set of initial conditions with historic behaviour. In contrast to the findings of Bowen and Kleptsyn [19], our example is robust due the hyperbolicity of the periodic solutions and the transversality of the local heteroclinic connections.

The results in this article are stated for vector fields in , but they hold for vector fields in a three-dimensional Riemannian manifold and, with some adaptation, in higher dimensions.

1.5 An ergodic point of view

Concerning the first result, the outstanding fact in the degenerate case is that the time averages diverge precisely in the same way: they approach a -polygon. This is in contrast with ergodic and hyperbolic strange attractors admitting a physical measure, where almost all initial conditions lead to converging time averages, in spite of the fact that the observed dynamics may undergo huge variations.

If a flow admits an invariant probability measure that is absolutely continuous with respect to the Lebesgue measure and ergodic, then is a physical measure for , as a simple consequence of the Birkhoff Ergodic Theorem. In other words if is a -integrable function, then for -almost all points in the time average:

exists and equals the space average . In the conservative context, historic behaviour has zero Lebesgue measure.

Physical measures need not be unique or even exist in general. When they exist, it is desirable that the set of points whose asymptotic time averages are described by physical measures be of full Lebesgue measure. It is unknown in how much generality do the basins of physical measures cover a subset of of full Lesbegue measure. There are examples of systems admitting no physical measure but the only known cases are not robust, ie, there are systems arbitrarily close (in the Whitney topology) that admit physical measures. In the present article, we exhibit a persistent class of smooth dynamical systems that does not have global physical measures. In the unfolding of an equation for which the first result holds, there are no physical measures whose basins intersect the basin of attraction of an attracting heteroclinic cycle. Our example confirms that physical measures need not exist for all vector fields. Existence results are usually difficult and are known only for certain classes of systems.

1.6 Example without historic behaviour

Generalised Lotka-Volterra systems has been analysed by Duarte et al in [7]. Results about the convergence of time averages are known in two cases: either if there exists a unique interior equilibrium point, or in the conservative setting (see [7]), when there is a heteroclinic cycle. In the latter case, if the solution remains limited and does not converge to the cycle, then its time averages converge to an equilibrium point. The requirement is that the heteroclinic cycle is stable but not attracting, and the limit dynamics has been extended to polymatrix replicators in [25]. This is in contrast to our findings in the degenerate case, emphasising the importance of the hypothesis that the cycle is attracting in order to obtain convergence to a polygon.

1.7 Framework of the article

Preliminary definitions are the subject of Section 2 and the main hypotheses are stated in Section 3. We introduce the notation for the rest of the article in Section 4 after a linearisation of the vector field around each periodic solution, whose details are given in Appendix A. We use precise control of the times of flight between cross-sections in Section 5, to show that for an open set of initial conditions in a neighbourhood of asymptotically stable heteroclinic cycles involving non-trivial periodic solutions, the time averages fail to converge. Instead, the time averages accumulate on the boundary of a polygon, whose vertices may be computed from local information on the periodic solutions in the cycle. The proofs of some technical lemmas containing the computations about the control of the flight time between nodes appear in Appendix B, to make for easier reading.

In Section 6, we obtain a persistent class of smooth dynamical systems such that an open set of initial conditions corresponds to trajectories with historic behaviour. Symmetry-breaking techniques are used to obtain a heteroclinic cycle associated to two periodic solutions and we find heteroclinic tangencies and Newhouse phenomena near which the result of [6, 18] may be applied. This is followed in Section 7 by an explicit example where historic behaviour arise in the unfolding of an SO(2)-equivariant vector field.

2 Preliminaries

To make the paper self-contained and readable, we recall some definitions.

2.1 Heteroclinic attractors

Several definitions of heteroclinic cycles and networks have been given in the literature. In this paper we consider non-trivial periodic solutions of (1.1) that are hyperbolic and that have one Floquet multiplier with absolute value greater than 1 and one Floquet multiplier with absolute value less than 1. A connected component of , for a periodic solution , will be called a branch of , with a similar definition for a branch of . Given two periodic solutions and of (1.1), a heteroclinic connection from to is a trajectory contained in , that will be denoted .

Let be a finite ordered set of periodic solutions of saddle type of (1.1). The notation for is cyclic, we indicate this by taking the index , ie . Suppose

A heteroclinic cycle associated to is the union of the saddles in with a heteroclinic connection for each . We refer to the saddles defining the heteroclinic cycle as nodes. A heteroclinic network is a connected set that is the union of heteroclinic cycles. When a branch of coincides with a branch of , we also refer to it as a two-dimensional connection .

2.2 Basin of attraction

For a solution of (1.1) passing through , the set of its accumulation points as goes to is the -limit set of and will be denoted by . More formally,

It is well known that is closed and flow-invariant, and if is compact, then is non-empty for every . If is a flow-invariant subset for (1.1), the basin of attraction of is given by

Note that, with this definition, the set is not contained in .

3 The setting

3.1 The hypotheses

Our object of study is the dynamics around a heteroclinic cycle associated to periodic solutions, , , for which we give a rigorous description here. Specifically, we study a one-parameter family of -vector fields in whose flow has the following properties (see Figure 1):

  1. For , there are hyperbolic periodic solutions of , , of minimal period . The Floquet multipliers of are real and given by and where .

  2. For each , the manifolds and are smooth surfaces homeomorphic to a cylinder – see Figure 2.

  3. For each , and for , one branch of coincides with a branch of , forming a heteroclinic network, that we call , and whose basin of attraction contains an open set.

  4. [Transversality] For and for each , a branch of the two-dimensional manifold intersects transverselly a branch of at two trajectories, forming a heteroclinic network , consisting of two heteroclinic cycles.

For , any one of the two trajectories of (P4) in will be denoted by . A more technical assumption (P5) will be made in Section 4.1 below, after we have established some notation. For , define the following constants:

(3.3)

Also denote by the centre of gravity of , given by

Without loss of generality we assume that the minimal period , for all . It will be explicitly used in system (4.4) below.

Figure 1: Configuration of for (left) and (right). The representation is done for .

3.2 The dynamics

The dynamics of this kind of heteroclinic structures involving periodic solutions has been studied before in [1, 3, 22, 28], in different contexts.

Since satisfies (P1)–(P3) then, adapting the Krupa and Melbourne criterion [20, 21], any solution starting sufficiently close to will approach it in positive time; in other words is asymptotically stable. As a trajectory approaches , it visits one periodic solution, then moves off to visit the other periodic solutions in the network. After a while it returns to visit the initial periodic solution, and the second visit lasts longer than the first. The oscillatory regime of such a solution seems to switch into different nodes, at geometrically increasing times.

For , by (P4), the invariant manifolds of the nodes meet transversally, and the network is no longer asymptotically stable due to the presence of suspended horseshoes in its neighbourhood. As proved in [28], there is an infinite number of heteroclinic and homoclinic connections between any two periodic solutions and the dynamics near the heteroclinic network is very complex. The route to chaos corresponds to an interaction of robust switching with chaotic cycling. The emergence of chaotic cycling does not depend on the magnitude of the multipliers of the periodic solutions. It depends only on the geometry of the flow near the cycle.

In Table 1, we summarise some information about the type of heteroclinic structure of and the type of dynamics nearby.

Structure of Dynamics near References
zero torus of genus Attractor [22, 28]
non-zero torus of genus Chaos (Switching and Cycling) [1, 3, 28]
Table 1: Heteroclinic structure of , for and .

4 Local and global dynamics near the network

Given a heteroclinic network of periodic solutions with nodes , , let be a compact neighbourhood of and let be pairwise disjoint compact neighbourhoods of the nodes , such that each boundary is a finite union of smooth manifolds with boundary, that are transverse to the vector field everywhere, except at their boundary. Each is called an isolating block for and, topologically, it consists of a hollow cylinder. Topologically, may be seen as a solid torus with genus (see Table 1).

4.1 Suspension and local coordinates

For , let be a cross section transverse to the flow at . Since is hyperbolic, there is a neighbourhood of in where the first return map to , denoted by , is conjugate to its linear part. Moreover, for each there is an open and dense subset of such that, if the eigenvalues lie in this set, then the conjugacy is of class — see [33] and Appendix A. The eigenvalues of are and . Suspending the linear map gives rise, in cylindrical coordinates around , to the system of differential equations:

(4.4)

which is -conjugate, after reparametrising the time variable, to the original flow near . In these coordinates, the periodic solution is the circle defined by and , its local stable manifold, , is the plane defined by and is the surface defined by as in Figure 2.

We will work with a hollow three-dimensional cylindrical neighbourhood of contained in the suspension of given by:

When there is no ambiguity, we write instead of . Its boundary is a disjoint union

such that :

  • is the union of the walls, defined by , of the cylinder, locally separated by . Trajectories starting at go inside the cylinder in small positive time.

  • is the union of two anuli, the top and the bottom of the cylinder, defined by , locally separated by . Trajectories starting at go inside the cylinder in small negative time.

  • The vector field is transverse to at all points except possibly at the four circles: .

The two cylinder walls, are parametrised by the covering maps:

where , . In these coordinates, is the union of the two circles . The two anuli are parametrised by the coverings:

for and and where is the union of the two circles . In these coordinates is the union of the four circles defined by and .

The portion of the unstable manifold of that goes from to without intersecting will be denoted . Similarly, will denote the portion of the stable manifold of that is outside and goes directly from to . With this notation, we formulate the following technical condition:

Figure 2: Local coordinates on the boundary of the neighbourhood of a periodic solution where . Double bars mean that the sides are identified.
  1. For , and close to zero, the manifolds intersect the cylinders on a closed curve. Similarly, intersects the annulus on a closed curve.

The previous hypothesis complements (P4) and corresponds to the expected unfolding from the coincidence of the manifolds and at , see Chilingworth [5]. Note that (P4) and (P5) are satisfied in an open subset of the set of unfoldings of satisfying (P1)–(P3).

In order to distinguish the local coordinates near the periodic solutions, we sometimes add the index with .

4.2 Local map near the periodic solutions

For each , we may solve (4.4) explicitly, then we compute the flight time from to by solving the equation for the trajectory whose initial condition is , with . We find that this trajectory arrives at at a time given by:

(4.5)

Replacing this time in the other coordinates of the solution, yields:

(4.6)

The signs depend on the component of we started at, for trajectories starting with and for . We will discuss the case , , the behaviour on the other components is analogous.

4.3 Flight times for

Here we introduce some terminology that will be used in Section 5; see Figure 3. For , let be the smallest such that . For , , we define inductively as the smallest such that , where

is the remainder in the integer division by and is the greatest integer less than or equal to . Recall that the index in lies in , so that and represent the same periodic solution.

In order to simplify the computations, we may assume that the transition from to is instantaneous. This is reasonable because, as , the time of flight inside each tends to infinity, whereas the time of flight from to remains limited. In the proof of Proposition 1 below, we will see that this assumption does not affect the validity of our results. With this assumption, the time of flight inside at the -th pass of the trajectory through will be

thus extending the notation introduced in 4.2 above to and any index .

Figure 3: For , the solution remains in for a time interval of length , then spends units of time near , and , after full turns, stays again in for units of time, and so on. The representation is done for .

For each , and for , we define the transition map

(4.7)

The transition maps for will being discussed in Section 6.1.

5 The -polygon at the organising centre

Let be a vector field in satisfying (P1)–(P3). All the results of this section assume . Suppose, from now on, that is a solution with initial condition in , the basin of attraction of .

5.1 The statistical limit set of

The statistical limit set associated to the basin of attraction of is the smallest closed subset where Lebesgue almost all trajectories spend almost all time. More formally, following Ilyashenko [14] and Karabacak and Ashwin [17], we define:

Definition 2

For an open set and a solution of (1.1) with :

  1. the frequency of the solution being in is the ratio:

    where denotes the Lebesgue measure in .

  2. the statistical limit set, denoted by , is the smallest closed subset of for which any open neighbourhood of of satisfies the equality:

Since the transitions between the saddles of are very fast compared with the times of sojourn near the periodic solutions , (see 4.3) we may conclude that:

Proposition 1

Let be a vector field in satisfying (P1)–(P3). Then:

Proof: The flow from to is non-singular as in a flow-box. Since both and are compact sets, the time of flight between them has a positive maximum. On the other hand, for each , the time of flight inside from to tends to infinity as approach the stable manifold of , , or equivalently as the trajectory accumulates on .

Remark 1

It follows from Proposition 1 that, for each , the time intervals in which trajectories are travelling from to do not affect the accumulation points of the time averages of a solution that is accumulating on . This result will be useful in the proof of the Theorem 8 because it shows that the duration of the journeys between nodes may be statistically neglected.

5.2 Estimates of flight times

In this section, we obtain relations between flight times of a trajectory in consecutive isolating blocks as well as other estimates that will be used in the sequel.

Lemma 2

For all and any initial condition we have:

(5.8)

In particular the ratio does not depend on .

Proof: Given , let . Using the expressions (4.5), (4.6) and the expression for in (4.7), we have:

Thus

Recall from (3.3) that , . With this notation we obtain:

Corollary 3

For such that , and for any , we have:

  1. .

  2. .

We finish this section with a result comparing the two sequences of times and . The proof is very technical and is given in Appendix B.1.

Lemma 4

For , and for any , the following equalities hold:

  1. .

5.3 The vertices of the -polygon

In this section, we show that in the time averages fail to converge, by finding several accumulation points for them. For each , define the point

(5.9)

Note that and lie in and . Later we will see that these points are the vertices of a polygon of accumulation points. First we show that they are accumulation points for the time averages.

Proposition 5

Let , let be a vector field in satisfying (P1)–(P3) and let a solution of with . Then

In order to prove Proposition 5, first we show that it is sufficient to consider the limit when of the averages over one turn around and then we prove that these averages tend to . The proof is divided in two technical lemmas, which may be found in Appendix B.2.

5.4 The sides of the -polygon

In Section 5.3 we have shown that for , the time average over the sequences of times accumulate, as in the . In this section we describe accumulation points for intermediate sequences of times . For this, it will be useful to know how and are related:

Lemma 6

For all , the following equalities hold:

(5.10)
Proposition 7

The point lies in the segment connecting to .

Proof: We use Lemma 6 to obtain

and hence

Again from Lemma 6 we have , and therefore

hence lies in the line through and . From the expression in Lemma 6 it follows that , hence and thus lies in the segment from to , proving the result.

Figure 4: Representation of the sequence of times , where , is fixed and .

We now come to the main result of this section:

Theorem 8

If is a vector field in satisfying (P1)–(P3), then for any , the set of accumulation points of the time average is the boundary of the -polygon defined by in (5.9). Moreover, when the polygon collapses into a point.

Proof: First we show that all points in the boundary of the polygon are accumulation points. Given and , consider the sequence , we want the accumulation points of as . For this we write

where

Since both and are limited, each one of them contains a converging subsequence. We analyse separately each of the terms in the expression for above.

We have already seen in Proposition 5 that, if , then . In particular, if , then , and .

We claim that if , then . To see this, note that for . Moreover, since , then for large , we have that is much larger than , the period of . Since , then , with , tends to when and the average of tends to , the average of .

At this point we have established that any accumulation point of lies in the segment connecting to . We have shown in Proposition 7 that this segment also contains . By Proposition 5 we have that for . On the other hand, is an increasing function of , so, as increases from 0 to 1, the accumulation points of move from to in the segment connecting them.

Conversely, any accumulation point lies on the boundary of the polygon. To see this, let be an accumulation point of the time average. This means that there is an increasing sequence of times , tending to infinity, and such that , where . Since tends to infinity, then it may be partitioned into subsequences of the form for each , and some , as shown in Figure 4. The arguments above, applied to this subsequence, show that the accumulation points of lie in the segment connecting to . Therefore, since converges, there are two possibilities. The first is that all the (except possibly finitely many) are of the form above for a fixed , and hence lies in the the segment connecting to . The second possibility is that all the (except maybe a finite number) are of one of the forms

and that . In both cases, the accumulation point of the time average will lie on the boundary of the polygon. Finally, when , the expressions (5.10) in Lemma 6 become and , hence

and the polygon collapses to a point at the same time as stops being attracting.

Figure 5: The polygon in Theorem 8 with : the accumulation points of the time average lie on the boundary of the triangle defined by and

Taking the observable as the projection on any component, the first main result of this paper may be stated as:

Corollary 9

If is a vector field in satisfying (P1)–(P3), then all points in the basin of attraction of have historic behaviour. In particular the set of initial conditions with historic behaviour has positive Lebesgue measure.

The points of do not have historic behaviour. Indeed, if then either or accumulates on for some . In both cases, .

The previous proofs have been done for a piecewise continuous trajectory; when , the trajectory jumps from to , whereas the real solutions have a continuous motion from to along the corresponding heteroclinic connection, during a bounded interval of time. As shown in Proposition 1, the statistical limit set of is meaning that trajectories spend Lebesgue almost all time near the periodic solutions, and not along the connections. Therefore, the intervals in which the transition occurs do not affect the accumulation points of the time averages of the trajectories and the result that was shown for a piecewise continuous trajectory holds.

6 Persistence of historic behaviour

From now on, we discuss he differential equation satisfying (P1)–(P5), with . In this case it was shown in Rodrigues et al [28] that the simple dynamics near jumps to chaotic behaviour near .

6.1 Invariant manifolds for

Figure 6: For close to zero, both and are closed curves, given in local coordinates as the graphs of periodic functions; this is the expected unfolding from the coincidence of the invariant manifolds at .

We describe the geometry of the two-dimensional local invariant manifolds of and for , under the assumptions (P1)–(P5). For this, let be an unfolding of