Robust Chaos and the Continuity of Attractors
Paul A. Glendinning David J. W. Simpson


𝒜
ℬ
𝒞
𝒟
𝒳
𝒴
ε

theoremTheorem[section]
corollary[theorem]Corollary
lemma[theorem]Lemma
proposition[theorem]Proposition
conjecture[theorem]Conjecture

definition
definitionDefinition[section]
example[definition]Example

remark
remarkRemark[section]











Robust Chaos and the Continuity of Attractors.

    
P.A. Glendinning^† and D.J.W. Simpson^

^†School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK.

^School of Fundamental Sciences, Massey University, Palmerston North, New Zealand

    December 30, 2023
=========================================================================================================================================================================================================================








As the parameters of a map are varied an attractor may vary continuously in the Hausdorff metric.
The purpose of this paper is to explore the continuation of chaotic attractors.
We argue that this is not a helpful concept for smooth unimodal maps for which periodic windows fill parameter space densely,
but that for piecewise-smooth maps it provides a way to
delineate structure within parameter regions of robust chaos
and form a stronger notion of robustness.
We obtain conditions for the continuity of an attractor
and demonstrate the results with coupled skew tent maps, the Lozi map, and the border-collision normal form.












§ INTRODUCTION



Let f_μ be a family of maps on ℝ^p
that vary continuously with respect to a parameter μ∈ M ⊂ℝ^q,
where M is compact with non-empty interior.
We say that f_μ exhibits robust chaos in M if f_μ has a chaotic attractor for each μ∈ M
and there exist μ_1, μ_2 ∈ M such that f_μ_1 is not topologically conjugate to f_μ_2 <cit.>.
This last stipulation prohibits the trivial case that f_μ undergoes no topological change as μ is varied.
While robust chaos does not occur for generic smooth maps of the interval <cit.>,
it appears to be typical for maps that are piecewise-smooth <cit.>.


In applications that utilise chaos, such as mixing <cit.>,
spacecraft trajectories <cit.>,
and encryption <cit.>,
robust chaos is often a desired property.
It seems reasonable that chaotic attractors at nearby parameter values should be in some way related
because the map varies continuously even if the details of the dynamics can change.
The aim of this paper is to give mathematical meaning to this sense of sameness
by using continuity in the Hausdorff metric.
We show how this adds structure to parameter regimes of robust chaos,
and in fact a layered structure when multiple attractors coexist.




The continuity of attractors in the Hausdorff metric has been useful in a number of problems.
Stuart and Humphries <cit.> use it with the semi-distance (see <ref>)
to assess the numerical approximation of dynamical systems via the geometry of attractors rather than their dynamics.
This is a natural extension to continuation techniques for periodic orbits.
There are also fairly general results for the continuity of global attractors of semi-flows <cit.>.
In particular Hoang et. al. <cit.> develop a concise and effective characterisation of the continuity of global attractors
and we follow their approach in <ref>.

Outside of <ref> we investigate the continuity of attractors through a series of examples.
Our motivation is to develop ideas which can be used easily.
This, together with the fact that our definition of attractors (see <ref>) is local,
means we need a slightly more complicated continuity argument than that of <cit.>, although the underlying principles are the same.







§ DEFINITIONS




Let d be a metric on ℝ^p.
The (asymmetric) semi-distance between sets X, Y ⊆ℝ^p is defined as

    d_a(X,Y) = sup_x ∈ Xinf_y ∈ Y d(x,y),

see Fig. <ref> for a visualisation.
The Hausdorff distance is the following symmetric version of this semi-distance:

    d_H(X,Y) = max[ d_a(X,Y), d_a(Y,X) ].

We also write

    B_r(X) = { y ∈ℝ^p  |  d_a(y,X) ≤ r },

to denote the closed ball of radius r > 0 around a set X ⊆ℝ^p.








Let f be a continuous map on ℝ^p.
A compact set ⊂ℝ^p is an attractor of f if



  * f() =,

  *  contains a dense orbit, and

  * there exists r > 0 such that
d_a ( f^n(x), ) → 0 
as n →∞ for all x ∈ B_r().

If in addition Lyapunov exponents of typical points are positive then we say  is a chaotic attractor.


Now suppose a family of maps f_μ has an attractor _μ for all μ∈ M.
To say that _μ is continuous in the Hausdorff metric at some μ∈ M means the following:
for all > 0 there exists δ > 0 such that
d_H ( _μ, _ν) < whenever ν∈ M and |μ - ν| < δ.














§ TENT MAPS



Here we consider the tent map

    T_s(x) = 
    s x,     0 ≤ x ≤1/2, 
    
    s(1-x),    1/2≤ x ≤ 1.

If 1 < s ≤ 2 the tent map has a unique attractor on [0,1],
see Fig. <ref>.
If √(2)≤ s ≤ 2, the attractor is the interval
I_0(s) = [ s ( 1 - s/2), s/2].
Otherwise it is the union of 2^n disjoint closed intervals where
√(2)≤ s^2^n-1 < 2, see <cit.>.
Lemma <ref> below shows that, despite having different numbers of connected components,
the attractor is continuous in the Hausdorff metric.
In a similar way stable periodic solutions in period-doubling cascades are continuous
because, despite a change in the period at period-doubling bifurcations,
the attractor does not experience a jump in phase space.
In this and the next section we use d(x,y) = |x-y|.








The attractor of the tent map (<ref>) is continuous for 1 < s ≤ 2.




If √(2) < s ≤ 2 the attractor is
I_0 = [ T_s^2 ( 1/2), T_s ( 1/2) ]
which is continuous since its endpoints vary continuously.

Next we verify continuity at s = √(2).
As s →√(2) from above the attractor is I_0(s)
and simply converges to I_0 ( √(2)).
As s →√(2) from below the attractor is the disjoint union I_1(s) ∪ I_2(s) where

    I_1(s)    = [ T_s^2 ( 12), T_s^4 ( 12) ],    
    I_2(s)    = [ T_s^3 ( 12), T_s ( 12) ],

see Fig. <ref>.
The lower endpoint of I_1(s) and the upper endpoint of I_2(s) converge to the endpoints of
I_0 ( √(2)),
so it remains to show that the size of the gap between I_1(s) and I_2(s)
converges to zero as s →√(2).
Indeed a direct calculation produces

    T_s^3 ( 12) - T_s^4 ( 12) =
    s (s - 1) ( 1 - s^22),

which vanishes at s = √(2),
This shows that d_H ( I_0 ( √(2)), I_1(s) ∪ I_2(s) ) → 0
as s →√(2) from below,
so the attractor of (<ref>) is continuous at s = √(2).

If s < √(2) then T_s^2 restricted to I_1(s) or I_2(s)
is a linear rescaling of T_s^2 restricted to I_0 ( s^2 )
and so the proof can be completed by considering higher iterates
in an inductive fashion <cit.>.












§ QUADRATIC MAPS



The aim of this section is to argue that the continuation of chaotic attractors is not helpful for smooth maps
(at least in one dimension).
To do this we rely on a number of standard results for quadratic families which can be found, for example, in <cit.>.
Smooth families of unimodal maps f_μ with a single maximum occurring at c (the critical point)
can be described in terms of symbolic dynamics.
Every orbit defines a sequence of C's, L's and R's by seeing if the
n^ th point of the orbit equals c, lies to the left of c, or lies to the right of c, respectively.
There is a natural order on these sequences:
if W is a finite sequence of symbols (a word) then
WL < WC < WR if W has an even number of R's
and the inequality is reversed if W has an odd number of R's
(this is connected with the fact that f^n is decreasing if there are an odd number of R's).
The kneading invariant of f_μ is the symbol sequence K_μ associated with f_μ(c).
There are consistency conditions on the possible sequences that can occur:
if the critical point is not periodic then this condition is simply that K_μ is the maximal element of its shifts:

    σ^k ( K_μ) ≤ K_μ ,   for all  k ≥ 1,

where σ is the shift map.
Families which depend smoothly (C^1) on the parameter are full if all possible consistent kneading invariants
between those of the end-points do actually occur in the family.
A family is monotonic if the kneading invariant is monotonic in the parameter
(this means in particular that certain behaviour is not repeated).
The standard quadratic maps are full and monotonic.
Further conditions such as convexity or negative Schwarzian derivative imply that there is a unique attractor at every parameter value.
Recall that f_μ is in a period-n window if there exists n>0 and an interval J such that f_μ^n restricted to J is a unimodal map.




Suppose that f_μ is a smooth unimodal map (at least C^2 in phase space and C^1 in the parameter) and has a unique attractor at each parameter value μ in some closed interval I. If there is a parameter μ_0∈int(I) with a non-degenerate saddle-node bifurcation of period k>1, then the attractor is not continuous in the Hausdorff metric at μ_0.




We may assume without loss of generality that f_μ_0 is not in a period r window, r<k (otherwise consider f^r).
In particular we can assume that the parameter lies in the single band region (the equivalent of s>√(2) in the tent map)
with kneading invariant greater than RLR^∞.
The kneading invariant at μ_0 is K(μ_0)=(WC)^∞ with W a sequence of L's and R's starting RL.
Choose the sign of μ so that the periodic orbit created by the saddle-node bifurcation exists if μ >μ_0. If μ<μ_0 with |μ -μ_0| small, then the kneading invariant starts (WE)^n…,
with n→∞ as μ→μ_0 from below and

    E =
    
    L,    W has an even number of  R 's, 
    
    R,    W has an odd number of  R 's.

It is now an elementary exercise to show that the sequences (WE)^n WLR^∞ if E=L or (WE)^nWR^∞
if E=R satisfy the consistency conditions (<ref>), and since the points c and its images define a Markov partition
(they are Misiurewicz points) and are not in a period r window the attractor is the interval I_μ_n=[f^2_μ_n (c),f_μ_n(c)].
Thus arbitrarily close to the bifurcation value μ_0 the attractor is an interval.

If μ =μ_0 then there is a non-hyperbolic periodic orbit W_k of period k which is the attractor for
f and k non-trivial close intervals B_i, i=1, … ,k, which are the immediate basins of attraction of the periodic orbit
of period k (one end point of each of these intervals is a point of period k; these are one-sided basins of attraction).

We have already seen that in any small neighbourhood of μ_0 there exists μ
such that the attractor of f_μ is an interval I_μ_n,
and there exists i such that B_i⊂ I_μ_n (as the period k≥ 3).
Hence if |B_i|=2, d_a(I_μ_n ,W_k)> since the closest that W_k can be to the part of the attractor inside B_i is .
Hence the attractor is not continuous at μ_0.




The only non-trivial intervals on which the attractor of the logistic map is continuous
in the Hausdorff metric are the period-doubling cascades of stable periodic orbits.




Between any two topologically distinct chaotic attractors
there exist parameter values with saddle-node bifurcations of periodic orbits.








§ UNIFORM CONTINUITY AND CONTINUATION



In this section we derive general results for the continuity of attractors.
Our approach follows Hoang et. al. <cit.>
with some technical additions required to accommodate local attractors
that will be useful when we come to the Lozi map in <ref>. 

 
Let f_μ be a family of maps on ℝ^p
where μ∈ M and M ⊂ℝ^q is compact.
Assume f_μ varies continuously in phase space and in μ.
Assume that for each μ∈ M, the map f_μ has an attractor _μ
and let r_μ > 0 be a suitable value for part (iii) of Definition <ref>.
Two further assumptions are needed.


  (A1)
There exists compact Ω⊂ℝ^p such that B_r_μ(_μ) ⊆Ω for all μ∈ M.

  (A2)
For all μ∈ M there exists compact N_μ⊆Ω,
continuous (with respect to μ) in the Hausdorff metric,
such that f(N_μ) ⊆ N_μ and
_μ =  cl( ∩_n=0^∞ f^n_μ (N_μ) ).

 
These are the natural generalizations of (L2) and (L3) of <cit.>.
The following lemma shows that at each stage of the construction of the attractor by iterates of N_μ,
the sets remain close (this is equivalent to Lemma 3.1 of <cit.>).




Suppose f_μ is continuous with an attractor _μ and (A1) and (A2) hold.
For each n ≥ 0, f_μ^n(N_μ) is continuous in the Hausdorff metric in M.




Choose any n ≥ 0 and > 0.
Since f^n is continuous in x and μ
and Ω and M are compact,
by the Heine-Cantor theorem f^n is uniformly continuous in x and μ.
Thus there exist δ_Ω, δ_M > 0 such that
for all x, y ∈Ω with d(x,y) < δ_Ω
and all μ, ν∈ M with |μ - ν| < δ_M
we have d ( f_μ^n(x), f_ν^n(y) ) <.

Since N_μ is continuous on the compact set M
it is similarly uniformly continuous and so there exists δ_1 > 0 such that
for all μ, ν∈ M with |μ - ν| < δ_1
we have d_H ( N_μ, N_ν) < δ_Ω.

Let δ = min(δ_1,δ_M).
Choose any μ, ν∈ M with |μ - ν| < δ.
Then

    d_a ( f_μ^n(N_μ), f_ν^n(N_ν) ) =
    sup_x ∈ N_μinf_y ∈ N_ν d ( f_μ^n(x), f_ν^n(y) ) < ,

because for all x ∈ N_μ there exists y ∈ N_ν such that d(x,y) < δ_Ω.
We similarly have d_a ( f_ν^n(N_ν), f_μ^n(N_μ) ) <,
thus d_H ( f_μ^n(N_μ), f_ν^n(N_ν) ) <, as required.





If the conditions of Lemma <ref> hold and d_H ( f_μ^n(N_μ), _μ) → 0
as n →∞ uniformly in M, then _μ is continuous in the Hausdorff metric in M.




Choose any > 0.
There exists n_0 ≥ 0 such that for all μ∈ M and all n ≥ n_0 we have
d_H ( f_μ^n(N_μ), _μ) < /3.
By Lemma <ref>, f_μ^n(N_μ) is continuous in M,
but M is compact so the continuity is uniform,
thus there exists δ > 0 such that for all μ, ν∈ M with |μ - ν| < δ
we have d_H ( f_μ^n(N_μ), f_ν^n(N_ν) ) < /3.
Then for any μ, ν∈ M with |μ - ν| < δ we have

    d_H ( _μ, _ν) ≤
    d_H ( _μ, f_μ^n(N_μ) ) +
    d_H ( f_μ^n(N_μ), f_ν^n(N_ν) ) +
    d_H ( f_ν^n(N_ν), _ν) < .



The most remarkable aspect of Hoang et. al. <cit.>
is their proof that uniform convergence to the attractor with respect to the parameter
implies continuity of the attractor,
and, if the convergence is only pointwise then the continuity at least occurs on a residual set.
Recall, a residual set is the complement of a countable union of nowhere dense sets,
and every residual set is dense.
The uniform case is covered above by Theorem <ref>,
so it remains for us to address pointwise convergence.





The following technical result will be needed, and indeed, contains all the hard work!




Let  be a complete metric space,  be a metric space,
and g_n : → be a family of continuous maps.
If the pointwise limit
g(x) = lim_n →∞ g_n(x)
exists for each x ∈, then g is continuous on a residual subset of .


In our case,  is the parameter space M and  the space of compact subsets of ℝ^p with the Hausdorff metric.




Suppose f_μ is continuous with an attractor _μ and (A1) and (A2) hold.
Then _μ is continuous in the Hausdorff metric on a residual subset of M.




Let g_n(μ) = f_μ^n(N_μ) and g(μ) = _μ.
By Lemma <ref>, each g_n(μ) is continuous,
and by (A2), g_n(μ) → g(μ) as n →∞ for each μ∈ M,
so the result follows by Lemma <ref>.








§ COUPLED SKEW TENT MAPS



In the next three sections we identify continuous chaotic attractors
in three different piecewise-linear maps.
In these sections d is the Euclidean metric on ℝ^2.








Skew tent maps generalise (<ref>) to allow two slopes that differ in absolute value.
Specifically we consider

    T̃_s(z) =
     s z,     0 ≤ z ≤1/s, 
    s/s-1(1-z),    1/s≤ z ≤ 1,

where 1 < s < 2.
Each T̃_s, see Fig. <ref>,
is a skew tent map on [0,1] equivalent to a full shift on two symbols.


As considered originally in <cit.>,
here we use (<ref>) to form the coupled skew tent map

    f_s,ω(x) = [ (1-ω) T̃_s(x_1) + ωT̃_s(x_2); ωT̃_s(x_1) + (1-ω) T̃_s(x_2) ],

where 0 ≤ω≤1/2 is a measure of the coupling strength.
This is a map on [0,1] × [0,1] and we write x = (x_1,x_2) ∈ℝ^2.

The diagonal x_1 = x_2 is an invariant set that is stable for sufficiently large values of ω.
As the value of ω is decreased a `blowout bifurcation' occurs when typical transverse Lyapunov exponents become positive at
ω = 1/2( 1 -  e^-γ),
where γ =  ln(s) - ( 1 - 1/s)  ln(1-s), see <cit.>.
However, some orbits on the diagonal become transversely unstable
before the blowout bifurcation.


This first occurs at ω = 1/2 s and is responsible for the creation of a two-dimensional attractor.








Let √(5) + 1/2 < s < 2.
Let 𝒟 be the closed quadrilateral ORIR' where

    O    = (0,0),    
    R    = ( 2 ω, 1-2ω+2ω^21-ω),    
    I    = (1,1),    
    R^'   = ( 1-2ω+2ω^21-ω, 2 ω),

see Fig. <ref>.
If 0 < ω < 1/2 s
then 𝒟 is the unique attractor of (<ref>),

whilst if 1/2 s < ω < 1/2 then the diagonal x_1 = x_2
is the unique attractor of (<ref>).


The two types of attractor: 𝒟 and the diagonal x_1 = x_2,

are clearly chaotic and vary continuously with s and ω.
Consequently we have the following result.




Let √(5) + 1/2 < s < 2.
Then (<ref>) has robust chaos for 0 < ω < 1/2.
The attractor is continuous in the Hausdorff metric for 0 < ω < 1/2 s
and 1/2 s < ω < 1/2.


The region of robust chaos, Fig. <ref>,
is thus divided into two pieces by the curve ω = 1/2 s
through which the attractor cannot be continued.
In this way our consideration of continuity in the Hausdorff metric
has allowed us to partition the region of robust chaos into two different types in a formal way.

It could be objected that this example is a boundary case
as 𝒟 does not satisfy part (iii) of Definition <ref>.


In this sense the map has the same status as x ↦ 4 x (1-x)
for which the interval [0,1] is the `attractor' although all points outside this interval diverge.
This is a technical nicety that we expect can be circumvented by generalising the skew tent map (<ref>) to

    T̃_s,t(z) =
    
    s z,     0 ≤ z ≤ t, 
    s t/1-t(1-z),     t ≤ z ≤ 1,

where 0 < t < 1/s.
Numerical experiments suggest that the two-dimensional map obtained by replacing T̃_s
with T̃_s,t in (<ref>) exhibits an analogous continuous quadrilateral attractor
that now satisfies part (iii) of Definition <ref> for some r > 0,
but it remains to carefully extend the construction of 𝒟 given in <cit.> to allow t < 1/s.



























§ LOZI MAPS




The Lozi map <cit.>

    L(x) = [ 1 - a|x_1| + x_2;            b x_1 ],

where a, b ∈ℝ are parameters,
is a piecewise-linear version of the Hénon map.
Misiurewicz established robust chaos for (<ref>) in <cit.>.




Suppose

    0    < b < 1,    
    0    < a < 4-b2,    
    a    > b + 2√(2),    
    b    < a^2 - 12 a + 1.

Then the Lozi map (<ref>) has a unique saddle-type fixed point in x_1 > 0 (denoted X)
and the closure of the unstable manifold of this point is a chaotic attractor .


Here we adapt Misiurewicz's construction to show that the chaotic attractor he obtains
varies continuously with a and b.
Our proof uses the results of <ref>
and explains why it was necessary to add the variation
of the fundamental converging sets N_μ in that section.













Throughout the parameter region (<ref>)
the attractor  of Theorem <ref> is continuous in the Hausdorff metric.








Following <cit.>,
let X be the fixed point in x_1 > 0,
let Z be the intersection of the local unstable manifold of X with the x_1-axis,
and let P be the intersection of the local stable manifold of X with the line segment Z L^2(Z),
see Fig. <ref>.
Let H_0 be the compact filled triangle XZP.
The conditions (<ref>) imply that
H_0 is contained in the region x_1 > 0,
the line segments X Z and Z P belong to the unstable manifold of X,
and X P belongs to the local stable manifold of X.
It follows that every point on the boundary of the forward invariant set H = ∪_k=0^∞ L^k(H_0)
belongs to either the unstable manifold of X or the line segment X P.
Moreover every point on the boundary of L^n(H)
belongs to either the unstable manifold of X or the line segment X L^n(P).
Notice d ( X, L^n(P) ) = λ_s^n d(X,P),
where 0 < λ_s < 1 is the stable eigenvalue associated with X.





Misiurewicz <cit.> shows that ∩_k=0^∞ L^k(H) is the attractor  of Theorem <ref>.
By Theorem <ref> it remains to show that d_H ( L^n(H), ) → 0 as n →∞
uniformly in a and b.

The compact filled triangle Z L(Z) L^2(Z) is forward invariant, see <cit.>,
so if K denotes the area of this triangle then Area(H) ≤ K.
For each n, Area( L^n(H) ) = b^n  Area(H)
(because L is invertible and the absolute value of the
determinant of the Jacobian matrix of L is b at all points with x_1  0).
Thus the distance of any x ∈ L^n(H) to the boundary of L^n(H)
is at most √(K/π) b^n/2

(obtained by imagining L^n(H) as a circle with centre x and using the Euclidean metric).
Thus for any x ∈ L^n(H),


    d_a ( L^n(H), ) ≤√(K/π) b^n/2 + λ_s^n d(X,P).

and since ⊆ L^n(H) the same bound applies to d_H ( L^n(H), ).

Now fix any pair of parameters (a_0,b_0) ∈ℝ^2 satisfying (<ref>).
There exists δ > 0 such that (<ref>) is satisfied by
all (a,b) ∈ℝ^2 a distance at most δ from (a_0,b_0),
call this parameter set M.
Denote the supremum values of K, b, λ_s, and d(X,P) over M
by K_ max, b_ max, λ_s, max, and d_ max, respectively.
Then for any (a,b) ∈ M we have

    d_H ( L^n(H), ) ≤√(K_ max/π) b_ max^n/2 + λ_s, max^n d_ max.

Since b_ max, λ_s, max < 1
we conclude that d_H ( L^n(H), ) → 0 as n →∞ uniformly in M_δ.
Thus  is continuous at (a_0,b_0) by Theorem <ref>.













































§ BORDER-COLLISION NORMAL FORM



In this section we describe a numerical example of bifurcations of continuous chaotic attractors in
the two-dimensional border-collision normal form

    x ↦[  τ_L    1; -δ_L    0 ] x +
    [ 1; 0 ],     x_1 ≤ 0, 
    [  τ_R    1; -δ_R    0 ] x +
    [ 1; 0 ],     x_1 ≥ 0,

which has parameters τ_L, δ_L, τ_R, δ_R ∈ℝ.
This map, introduced in <cit.>,
is a generalisation of the Lozi map and can be used to approximate the dynamics
near any generic border-collision bifurcation in two dimensions <cit.>.





We develop an example of <cit.> and fix

    δ_L    = 0.3,    δ_R    = 0.3.

In the (τ_L,τ_R)-plane, see Fig. <ref>,
four codimension-one bifurcation curves, labelled a–d and explained below,
divide parameter space into six regions, labelled 1–6.
Fig. <ref> provides one representative phase portrait for each region.
Numerically we observe three continuous chaotic attractors,
a three or six-piece attractor  (purple) in regions 2 and 5,
a one-piece attractor  (yellow) in regions 1–3,
and a merging of these two attractors  (cyan) in region 6.






Let us first describe the four bifurcation curves. Curve a is the locus of a border collision bifurcation. Below curve a there exist unique LRL and RRL-cycles (these are period-3 solutions with the indicated symbolic itineraries <cit.>).
The LRL-cycle is stable in regions 1 and 4. On curve b the LRL-cycle has an eigenvalue of -1 and there exists a period-6 solution with one point on the switching manifold x_1 = 0. This solution grows continuously into attractor  in regions 2 and 5.
As τ_L increases, crossing curve b, there is a transition from the stable LRL-cycle to  which is not continuous in the Hausdorff metric because the LRL-cycle and period-6 solution do not coincide on curve b.
For a greater description of this type of non-smooth period-doubling bifurcation refer to <cit.>. Curves c and d are the loci of boundary crisis bifurcations which create and destroy the attractor  (curve c) and  (curve d). The intersection of these two curves is a codimension two boundary crisis described by <cit.>, and these curves form the boundary of the region in which attractor  exists.





In all six regions the RRL-cycle is a saddle
and its stable and unstable manifolds, W^s and W^u,
are shown in Fig. <ref>.
In regions 1 and 2, W^s forms the boundary between the basins of attraction of the two coexisting attractors.
The unstable eigenvalue associated with the RRL-cycle is positive
so W^u has two dynamically independent branches (and each branch has three pieces).

Points on the `outer' branch of W^u converge (under forward iteration of (<ref>))
to the stable LRL-cycle in regions 1 and 4
and to the attractor  in regions 2 and 5.
The attractor  is destroyed in a crisis on curve c:
here the outer branch of W^u attains an intersection with W^s.
This is a first homoclinic tangency <cit.> except W^s and W^u are piecewise-linear
so form `corner' intersections <cit.>.
To the right of curve c points on the outer branch converge to the same attractor as points on the inner branch.

In regions 1–3, points on the `inner' branch of W^u converge to the attractor .
This attractor is destroyed in a crisis on curve d:
here the inner branch of W^u attains an intersection with W^s.
Below curve d points on the inner branch converge to the same attractor as points on the outer branch.

In region 6 points converge to attractor  which involves
both parts of phase space associated with  and .
As we cross curves c or d the transition from  or  to  is not continuous in the Hausdorff metric
because the crises cause orbits to suddenly access new areas of phase space.

As an additional visualisation, Fig. <ref> shows numerically computed
maximal Lyapunov exponents of the attractors.
The observation that the Lyapunov exponents of , , and  are positive
and vary continuously in their respective regions supports our conjecture that
these attractors are chaotic and continuous.
The Lyapunov exponent varies continuously as we cross from region 6 to region 3 through curve d
because as we approach curve d the fraction of iterates of attractor 
that dwell near attractor  tends to 1
(the invariant measure changes continuously across curve d),
and similarly from region 6 to region 5 through curve c.





In summary, (<ref>) has robust chaos in all but region 4
and each chaotic attractor appears to be continuous in the Hausdorff metric in the regions in which it exists.
The particular novelty of this example is region 2
where the chaotic attractors  and  coexist.
One may continue each attractor separately,
 may be continued into region 5,
while  may be continued into regions 1 and 3.







§ DISCUSSION



In this paper we have added depth to the phenomenon of robust chaos in piecewise-smooth maps.
Previous works have shown piecewise-smooth maps to exhibit robust chaos
in the sense that a chaotic attractor exists throughout an open region of parameter space.
In several examples we have found this attractor to be continuous in the Hausdorff metric
and in this sense exhibits an extra level of robustness.

In the context of numerical exploration
such an attractor could be continued numerically along a one-dimensional path in parameter space.
Given attractors at two different points in parameter space,
one could ask whether or not there exists a path along which
one attractor can be continued into the other.
We stress that the continuation of invariant sets and attracting sets is more commonplace.
Attractors are more restrictive objects needing, among other things, a dense orbit (see Definition <ref>),
and so, as argued in <ref>, the continuation of a chaotic attractor
may only be useful for piecewise-smooth maps.

Rather than use the Hausdorff metric, one could instead consider the continuity
of an attractor with respect to its Lyapunov spectrum,
the topology of its support (e.g. number of holes),
its invariant probability measure <cit.>,
or, in the case of piecewise-smooth maps,
the fraction of iterates that lie on one side of the switching manifold
(which may have a useful physical interpretation).
Indeed, as evident from Fig. <ref>,
if one continued attractors using the maximal Lyapunov exponent,
attractors  and  in region 2 could be connected by a closed path through regions 5, 6, and 3.




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