It has been proposed to make practical use of chaos in communication, in
enhancing mixing in chemical processes and in spreading the spectrum of
switch-mode power suppies to avoid electromagnetic interference. It is however
known that for most smooth chaotic systems, there is a dense set of periodic
windows for any range of parameter values. Therefore in practical systems
working in chaotic mode, slight inadvertent fluctuation of a parameter may take
the system out of chaos. We say a chaotic attractor is robust if, for its
parameter values there exists a neighborhood in the parameter space with no
periodic attractor and the chaotic attractor is unique in that neighborhood. In
this paper we show that robust chaos can occur in piecewise smooth systems and
obtain the conditions of its occurrence. We illustrate this phenomenon with a
practical example from electrical engineering.Comment: 4 pages, Latex, 4 postscript figures, To appear in Phys. Rev. Let

Banerjee, Soumitro

Grebogi, Celso

Yorke, James A.

English

arXiv

Practical applications of chaos require the chaotic orbit to be robust, defined by the absence of periodic windows and coexisting attractors in some neighborhood of the parameter space. We show that robust chaos can occur in piecewise smooth systems and obtain the conditions of its occurrence. We illustrate this phenomenon with a practical example from electrical engineering

Publications of the IAS Fellows

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Robust Chaos
Soumitro Banerjee 1,∗, James A. Yorke 2,† and Celso Grebogi 2,‡
1 Department of Electrical Engineering, Indian Institute of Technology, Kharagpur–721302, India
2 Institute of Physical Science & Technology, University of Maryland, College Park, MD 20742, USA
Practical applications of chaos require the chaotic orbit to be robust, defined by the absence of periodic
windows and coexisting attractors in some neighborhood of the parameter space. We show that robust
chaos can occur in piecewise smooth systems and obtain the conditions of its occurrence. We illustrate this
phenomenon with a practical example from electrical engineering.
It has been proposed to make practical use of chaos in
communication [1], in enhancing mixing in chemical pro-
cesses [2] and in spreading the spectrum of switch-mode
power suppies to avoid electromagnetic interference [3, 4].
In such applications it will be necessary to obtain reliable
operation in the chaotic mode.
It is known that for most smooth chaotic systems (take
the logistic map [5] for example), there is a dense set of pe-
riodic windows for any range of parameter values. There-
fore in practical systems working in chaotic mode, slight
inadvertent fluctuation of a parameter may take the sys-
tem out of chaos. The question is, how to guarantee that
there is no periodic window for a given range of param-
eter values and the maximal Lyapunov exponent remains
positive throughout the range? In this Letter, we address
this problem.
We say a chaotic attractor is robust if, for its param-
eter values there exists a neighborhood in the parameter
space with no periodic attractor and the chaotic attrac-
tor is unique in that neighborhood. It is known that ro-
bust chaos cannot occur in smooth systems. In this Let-
ter we show that such situations can occur in piecewise
smooth maps and obtain the conditions of existence of ro-
bust chaos.
We first give a practical example from electrical engi-
neering to demonstrate robust chaos. The circuit shown in
Fig.1 is known as the boost converter. It consists of a con-
trolled switch S, an uncontrolled switch D, an inductor L,
a capacitor C and a load resistor R. When the controlled
switch is turned on, the current in the inductor increases
and energy is stored in it. When the controlled switch is
turned off, the stored energy in the inductor drops and the
polarity of the inductor voltage changes so that it adds to
the input voltage. The voltage across the inductor and
the input voltage together “boosts” the output voltage to
a value higher than the input voltage. Such circuits are
widely used in regulated dc switch-mode power supplies.
Regulation of the output current is achieved by con-
trolling the switching by current feedback — known as
Figure 1: The current mode controlled boost converter
“current-mode control”. In this control logic, the switch
is turned on by clock pulses that are spaced T seconds
apart. When the switch is closed, the inductor current
increases till it reaches the specified reference value Iref .
The switch opens when i = Iref . Any clock pulse arriv-
ing during the on period is ignored. Once the switch has
opened, the next clock pulse causes it to close.
We obtain a discrete-time model by observing the state
variables at every clock instant. There are two ways in
which a state can evolve from one clock instant to the next.
If the on-time Ton = L(Iref − in)/Vin is less than T , the
evolution between observation instants includes one on pe-
riod and one off period. Since the clock period typically is
much smaller than the characteristic time of the LCR cir-
cuit, we assume the waveforms to be linear between clock
instants. By neglecting the higher order Taylor terms, the
two dimensional map for Ton<T is derived as:
in+1 = Iref +
1
L
(
Vin − vn +
vnTon
CR
)
(T − Ton)
vn+1 = vn−
vnTon
CR
+
(
Iref
C
− vn
CR
+
vnTon
C2R2
)
(T−Ton)
On the other hand, if the clock pulse arrives while
i < Iref , the switch remains on between the observation
1
instants. If Ton ≥ T , then the map takes the form
in+1 = in +
Vin
L
T
vn+1 = vn −
vn
CR
T
The borderline between the two cases is given by the
case where the current reaches Iref exactly at the arrival
of the next clock pulse, i.e., Iborder= Iref − VinT/L. The
resulting map, therefore, is piecewise smooth.
Figure 2: The bifurcation diagram and the Lyapunov spec-
trum of the boost converter. The parameter values are:
c = 220µF , Iref = 0.5A, Vin = 30V , T = 400µs, L = 0.1H
The bifurcation diagram and the Lyapunov spectrum of
the boost converter is presented in Fig.2. It may be noted
that there is no periodic window or coexisting attractor in
the parameter range R= [241, 500]Ω. The chaotic attrac-
tor therefore satisfies the conditions of robustness.
We now obtain the general conditions of occurrence of
robust chaos. Let f(xˆ, yˆ; ρ) be a two-dimensional piece-
wise smooth map which depends on a single parameter ρ.
Let Γρ, given by xˆ = h(yˆ, ρ) denote a smooth curve that
divides the phase plane into two regions Ra and Rb. The
map is given by
f(xˆ, yˆ; ρ) =
{
f1(xˆ, yˆ; ρ) for xˆ, yˆ ∈ Ra,
f2(xˆ, yˆ; ρ) for xˆ, yˆ ∈ Rb (1)
It is assumed that the functions f1 and f2 are both con-
tinuous and have continuous derivatives. The map f is
continuous but its derivative is discontinuous at the line
Γρ, called the “border”. It is further assumed that the
one-sided partial derivatives at the border are finite. We
study the bifurcations of this system as the parameter ρ
is varied.
If a bifurcation occurs when the fixed point of the map is
in one of the smooth regions Ra or Rb, it will be one of the
“standard” types, namely period doubling, saddle-node or
Hopf bifurcation. But if the bifurcation occurs when the
fixed point is on the border, there is a discontinuous change
in the elements of the Jacobian matrix as ρ is varied. A
rich variety of bifurcations have been reported [6, 7, 8]
in this situation, which have been called border collision
bifurcation. We show that under certain conditions border
collision bifurcation results in robust chaos.
It has been shown [6] that by a change of coordinates,
any piecewise smooth map can be reduced to the normal
form (2) in some small neighborhood of the fixed point
undergoing border collision bifurcation.
Gµ =


(
τL 1
−δL 0
)(
x
y
)
+ µ
(
1
0
)
, for x ≤ 0,(
τR 1
−δR 0
)(
x
y
)
+ µ
(
1
0
)
, for x > 0,
(2)
where x and y are the new coordinates for which the border
is along the line x=0, dividing the phase space into two
halves L and R. µ is the new parameter, which is obtained
by scaling ρ. τL and δL are the trace and determinant of
the Jacobian matrix in side L. τR and δR are the corre-
sponding values in side R. Since the trace and determinant
are invariant under change of coordinates, these quantities
are the same as in map f , calculated in the neighborhood
of the point where border collision occurs. As the parame-
ter µ is varied through zero, local bifurcations depend only
on the values of τL, δL, τR, and δR appearing in (2) and
therefore it suffices to study the bifurcations in the normal
form (2) in exploring the border collision bifurcations in
the piecewise smooth map (1).
The fixed points of the system in the two sides are given
by
L∗ =
(
µ
1− τL + δL
,
−δLµ
1− τL + δL
)
R∗ =
(
µ
1− τR + δR
,
−δRµ
1− τR + δR
)
and their stability is determined by the eigenvalues λ1,2=
1
2
(
τ ±
√
τ2 − 4δ).
It may be noted that if
τL>(1 + δL) and τR<(1 + δR) (3)
then there is no fixed point for µ<0 and there are two fixed
points, one each in L andR, for µ>0. The two fixed points
are born on the border at µ = 0. We call this a border
collision pair bifurcation. An analogous situation occurs
if τL<(1 + δL) and τR>(1 + δR) as µ is reduced through
zero. Due to symmetry of the two cases, we consider only
the parameter region (3).
2
If (1 + δR)> τR > −(1 + δR) then for µ> 0, the fixed
point in L is a regular saddle and the one in R is an attrac-
tor. This is like a saddle-node bifurcation occurring on the
border. Since this region in the parameter space always
has a periodic attractor for µ> 0, we exclude this region
from our analysis when looking for chaotic behavior. The
condition τR=−(1+ δR), results in a nongeneric situation
where all points on the line joining the points ( µ
1+δR
, 0)
and (0,− δRµ
1+δR
) are fixed points of the second iterate. We
therefore concentrate on the parameter space region
τL>(1 + δL) and τR<−(1 + δR) (4)
and investigate the property of the attractor for µ>0. We
first consider the case 1>δL≥0 and 1>δR≥0.
For (4), L∗ is a regular saddle and R∗ is a flip saddle.
Let UL and SL be the unstable and stable manifold of L
∗
and UR and SR be the unstable and stable manifold of
R∗ respectively. For (2), all intersections of the unstable
manifolds with x = 0 map to the line y = 0. Since one
linear map changes to another linear map across the x=0
line, UL and UR experience folds along the x-axis. And
all images of fold points will be fold points. By a similar
argument we conclude that SL and SR fold along the y-
axis, and all pre-images of fold points are fold points.
Let λ1L, λ2L be the eigenvalues in side L and λ1R,
λ2R be the eigenvalues at side R. For condition (4),
λ1L > λ2L > 0 and 0 > λ1R > λ2R. The stable eigenvec-
tor at R∗ has a slope m1 = (−δR/λ1R) and the unstable
eigenvector has a slope m2 = (−δR/λ2R). Since points
on an eigenvector map to points on the same eigenvector
and since points on the y-axis map to the x-axis, we con-
clude that points of UR to the left of y-axis map to points
above x-axis. From this we find that UR has an angle
m3=
δLλ2R
δR−τLλ2R
after the first fold. Under condition (4) we
have m1 >m2 > 0 and m3 < 0. Therefore there must be
a transverse homoclinic intersection in R. This implies an
infinity of homoclinic intersections and the existence of a
chaotic orbit.
We now investigate the stability of this orbit. The basin
boundary is formed by SL. SL folds at the y-axis and
intersects the x-axis at point C. The portion of UL to
the left of L∗ goes to infinity and the portion to the right
of L∗ leads to the chaotic orbit. UL meets the x-axis at
point D, and then undergoes repeated foldings leading to
an intricately folded compact structure as shown in Fig.3.
The unstable eigenvector at L∗ has a negative slope
given by (−δL/λ1L). Therefore it must have a hetero-
clinic intersection with SR. Since both UL and UR have
transverse intersections with SR, by the Lambda Lemma
[9] we conclude that for each point q on UR and for each
ǫ-neighborhood Nǫ(q), there exist points of UL in Nǫ(q).
SinceUL comes arbitrarily close toUR, the attractor must
span UL in one side of the heteroclinic point.
Since all initial conditions in L converge on UL and all
initial conditions in R converge on UR, and since there
Figure 3: The stable and unstable manifolds of L∗ for
τL = 1.7, δL = 0.5, τR =−1.7, δR = 0.5. R∗ is marked by
the small cross inside the attractor.
are points of UL in every neighborhood of UR, we con-
clude that the attractor is unique. This chaotic attractor
can not be destroyed by small changes in the parameters.
Since small changes in the parameters can only cause small
changes in the Lyapunov exponents, where the chaotic at-
tractor is stable, it is also robust.
It is clear from the above geometrical structure that no
point of the attractor can be to the right of point D. If
D lies towards the left of C, the chaotic orbit is stable. If
D falls outside the basin of attraction, it is an unstable
chaotic orbit or chaotic saddle. From this, the condition
of stability of the chaotic attractor is obtained as
δLτRλ1L−δRλ1Lλ2L+δRλ2L−δLτR+τLδL−δ2L−λ2LδL > 0
(5)
If δL=δR=δ this condition reduces to τRλ1L − λ1Lλ2L +
τL − τR − δ>0.
The robust chaotic orbit continues to exist as τL is re-
duced below (1+δL). With τL slightly below (1+δL), there
is no fixed point in L for µ>0 but the invariant manifolds
suffer only slight change. The invariant manifold of L as-
sociated with λ1L still forms the attractor. The invariant
manifolds in L, however, cease to exist for τL<2
√
δL since
the eigenvalues become complex. As τL is reduced below
2
√
δL there is a sudden reduction in the size of the attrac-
tor as it spans only UR. So long as UL exists, multiple
attractors can not exist and therefore if the main attractor
is chaotic, it is also robust.
Therefore we see that for 1 > δL > 0, 1 > δR > 0, the
normal form (2) exhibits robust chaos in a portion of pa-
rameter space bounded by the conditions τR=−(1 + δR),
τL>2
√
δL and (5), as shown in Fig.4. There is a symmet-
ric region of the parameter space with the roles of R and
L interchanged, where the same phenomena are observed
for µ<0.
3
Figure 4: Schematic diagram of the parameter space re-
gion of the normal form (2) where robust chaos is observed
for 1>δL>0, 1>δR>0 and µ>0.
At very low values of the determinant, i.e., when the
system is very close to being one-dimensional, the main
attractor may not remain chaotic even for τL > 2
√
δL, as
periodic orbits become stable. The conditions for emer-
gence of periodic windows for the one-dimensional case
have been derived in [8, 10]. Therefore for one dimen-
sional systems, the parameter range for robust chaos is
bounded by τR=1, τR > − τLτL−1 , and the lower limit of τL
is given by the conditions of existence of various periodic
windows. Here the τ ’s are to be interpreted as the slopes
of the piecewise linear function in the two halves separated
by x= 0. On the other hand, if the determinants in the
two sides are unity, the region in the τL− τR space for
robust chaos shrinks to zero area.
The cases with negative determinant are investigated
following the same method. For the sake of brevity, we
present the results without much explanation.
For −1< δR < 0, we have 1 > λ1R > 0, λ2R < −1, and
R∗ is located above the x-axis. A positive value of λ1R
implies that UL converges on UR from one side. If
λ1L − 1
τL − 1− δL
>
λ2R − 1
τR − 1− δR
(6)
then the intersection of UL with the x-axis remains the
rightmost point of the attractor and (5) still gives the pa-
rameter range for boundary crisis. But if (6) is not satis-
fied, the intersection of UR with the x-axis becomes the
rightmost point of the attractor, and the condition of ex-
istence of the chaotic attractor changes to
λ2R − 1
τR − 1− δR
<
δL (τL − δL − λ2L)
(τL − 1− δL) (δRλ2L − δLτR)
(7)
For δL < 0 and δR < 0, L
∗ is below the x-axis and the
same logic as above applies. But if δL < 0 and δR > 0,
the stable manifold of R∗ has a negative eigenvalue and
hence UL does not approachUR from one side. Therefore,
if (6) is not satisfied, there is no analytic condition for
boundary crisis — it has to be determined numerically.
For δL<0, the invariant manifoldsUL and SL always exist
as the eigenvalues are real for all τL. Therefore multiple
attractors can not exist for δL<0.
Since (2) is a normal form of the piecewise smooth map
(1), it is expected that robust chaos would be observable
in many piecewise smooth maps in the neighborhood of
border collision bifurcations, provided that there are no
more than one period-1 fixed point in Ra and Rb, there
exist homoclinic as well as heteroclinic intersections of the
invariant manifolds associated with these fixed points, and
the trace and determinant at the two sides of the border-
line satisfy the above conditions. The example of the boost
converter is a case in point.
A major conclusion of this Letter is that one should
use piecewise smooth systems in applications that require
reliable operation under chaos.
∗ Electronic address: soumitro@ee.iitkgp.ernet.in
† Electronic address: yorke@ipst.umd.edu
‡ Also with the Institute for Plasma Research and De-
partment of Mathematics, University of Maryland, College
Park, USA. Electronic address: grebogi@chaos.umd.edu
References
[1] S. Hayes, C. Grebogi and E. Ott, Phys. Rev. Lett. 70,
3031 (1993).
[2] J. M. Ottino, The Kinematics of Mixing: Stretching,
Chaos and Transport, (Cambridge Univ. Press, 1989).
[3] S. H. Isabelle, G. C. Verghese, and S. Venkataraman,
To appear in IEEE Trans. Circuits & Systems–I.
[4] J. H. B. Deane and D. C. Hamill, Electronics Letters,
32, 1045 (1996).
[5] J. Graczyk and G. S˜wica˜tek, Annals Math. 146,1
(1997).
[6] E. H. Nusse and J. A. Yorke, Physica D, 57, 39 (1992).
[7] E. H. Nusse, E. Ott, and J. A. Yorke, Phys. Rev. E,
49, 1073 (1994).
[8] E. H. Nusse and J. A. Yorke, Int. J. Bifurcation chaos
Appl. Sci. Eng., 5, 189 (1995).
[9] K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos:
An Introduction to Dynamical Systems, (Springer,
1996).
[10] Y. L. Maistrenko, V. L. Maistrenko, and L. O. Chua,
Int. J. Bifurcation Chaos Appl. Sci. Eng., 3, 1573
(1993).
4


Robust chaos

Indian Academy of Sciences

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Robust Chaos
Soumitro Banerjee 1,∗, James A. Yorke 2,† and Celso Grebogi 2,‡
1 Department of Electrical Engineering, Indian Institute of Technology, Kharagpur–721302, India
2 Institute of Physical Science & Technology, University of Maryland, College Park, MD 20742, USA
Practical applications of chaos require the chaotic orbit to be robust, defined by the absence of periodic
windows and coexisting attractors in some neighborhood of the parameter space. We show that robust
chaos can occur in piecewise smooth systems and obtain the conditions of its occurrence. We illustrate this
phenomenon with a practical example from electrical engineering.
It has been proposed to make practical use of chaos in
communication [1], in enhancing mixing in chemical pro-
cesses [2] and in spreading the spectrum of switch-mode
power suppies to avoid electromagnetic interference [3, 4].
In such applications it will be necessary to obtain reliable
operation in the chaotic mode.
It is known that for most smooth chaotic systems (take
the logistic map [5] for example), there is a dense set of pe-
riodic windows for any range of parameter values. There-
fore in practical systems working in chaotic mode, slight
inadvertent fluctuation of a parameter may take the sys-
tem out of chaos. The question is, how to guarantee that
there is no periodic window for a given range of param-
eter values and the maximal Lyapunov exponent remains
positive throughout the range? In this Letter, we address
this problem.
We say a chaotic attractor is robust if, for its param-
eter values there exists a neighborhood in the parameter
space with no periodic attractor and the chaotic attrac-
tor is unique in that neighborhood. It is known that ro-
bust chaos cannot occur in smooth systems. In this Let-
ter we show that such situations can occur in piecewise
smooth maps and obtain the conditions of existence of ro-
bust chaos.
We first give a practical example from electrical engi-
neering to demonstrate robust chaos. The circuit shown in
Fig.1 is known as the boost converter. It consists of a con-
trolled switch S, an uncontrolled switch D, an inductor L,
a capacitor C and a load resistor R. When the controlled
switch is turned on, the current in the inductor increases
and energy is stored in it. When the controlled switch is
turned off, the stored energy in the inductor drops and the
polarity of the inductor voltage changes so that it adds to
the input voltage. The voltage across the inductor and
the input voltage together “boosts” the output voltage to
a value higher than the input voltage. Such circuits are
widely used in regulated dc switch-mode power supplies.
Regulation of the output current is achieved by con-
trolling the switching by current feedback — known as
Figure 1: The current mode controlled boost converter
“current-mode control”. In this control logic, the switch
is turned on by clock pulses that are spaced T seconds
apart. When the switch is closed, the inductor current
increases till it reaches the specified reference value Iref .
The switch opens when i = Iref . Any clock pulse arriv-
ing during the on period is ignored. Once the switch has
opened, the next clock pulse causes it to close.
We obtain a discrete-time model by observing the state
variables at every clock instant. There are two ways in
which a state can evolve from one clock instant to the next.
If the on-time Ton = L(Iref − in)/Vin is less than T , the
evolution between observation instants includes one on pe-
riod and one off period. Since the clock period typically is
much smaller than the characteristic time of the LCR cir-
cuit, we assume the waveforms to be linear between clock
instants. By neglecting the higher order Taylor terms, the
two dimensional map for Ton <T is derived as:
in+1 = Iref +
1
L
(
Vin − vn +
vnTon
CR
)
(T − Ton)
vn+1 = vn−
vnTon
CR
+
(
Iref
C
−
vn
CR
+
vnTon
C2R2
)
(T−Ton)
On the other hand, if the clock pulse arrives while
i < Iref , the switch remains on between the observation
1
instants. If Ton ≥ T , then the map takes the form
in+1 = in +
Vin
L
T
vn+1 = vn −
vn
CR
T
The borderline between the two cases is given by the
case where the current reaches Iref exactly at the arrival
of the next clock pulse, i.e., Iborder = Iref − VinT/L. The
resulting map, therefore, is piecewise smooth.
Figure 2: The bifurcation diagram and the Lyapunov spec-
trum of the boost converter. The parameter values are:
c = 220µF , Iref = 0.5A, Vin = 30V , T = 400µs, L = 0.1H
The bifurcation diagram and the Lyapunov spectrum of
the boost converter is presented in Fig.2. It may be noted
that there is no periodic window or coexisting attractor in
the parameter range R= [241, 500]Ω. The chaotic attrac-
tor therefore satisfies the conditions of robustness.
We now obtain the general conditions of occurrence of
robust chaos. Let f(x̂, ŷ; ρ) be a two-dimensional piece-
wise smooth map which depends on a single parameter ρ.
Let Γρ, given by x̂ = h(ŷ, ρ) denote a smooth curve that
divides the phase plane into two regions Ra and Rb. The
map is given by
f(x̂, ŷ; ρ) =
{
f1(x̂, ŷ; ρ) for x̂, ŷ ∈ Ra,
f2(x̂, ŷ; ρ) for x̂, ŷ ∈ Rb
(1)
It is assumed that the functions f1 and f2 are both con-
tinuous and have continuous derivatives. The map f is
continuous but its derivative is discontinuous at the line
Γρ, called the “border”. It is further assumed that the
one-sided partial derivatives at the border are finite. We
study the bifurcations of this system as the parameter ρ
is varied.
If a bifurcation occurs when the fixed point of the map is
in one of the smooth regions Ra or Rb, it will be one of the
“standard” types, namely period doubling, saddle-node or
Hopf bifurcation. But if the bifurcation occurs when the
fixed point is on the border, there is a discontinuous change
in the elements of the Jacobian matrix as ρ is varied. A
rich variety of bifurcations have been reported [6, 7, 8]
in this situation, which have been called border collision
bifurcation. We show that under certain conditions border
collision bifurcation results in robust chaos.
It has been shown [6] that by a change of coordinates,
any piecewise smooth map can be reduced to the normal
form (2) in some small neighborhood of the fixed point
undergoing border collision bifurcation.
Gµ =







(
τL 1
−δL 0
) (
x
y
)
+ µ
(
1
0
)
, for x ≤ 0,
(
τR 1
−δR 0
) (
x
y
)
+ µ
(
1
0
)
, for x > 0,
(2)
where x and y are the new coordinates for which the border
is along the line x = 0, dividing the phase space into two
halves L and R. µ is the new parameter, which is obtained
by scaling ρ. τL and δL are the trace and determinant of
the Jacobian matrix in side L. τR and δR are the corre-
sponding values in side R. Since the trace and determinant
are invariant under change of coordinates, these quantities
are the same as in map f , calculated in the neighborhood
of the point where border collision occurs. As the parame-
ter µ is varied through zero, local bifurcations depend only
on the values of τL, δL, τR, and δR appearing in (2) and
therefore it suffices to study the bifurcations in the normal
form (2) in exploring the border collision bifurcations in
the piecewise smooth map (1).
The fixed points of the system in the two sides are given
by
L∗ =
(
µ
1 − τL + δL
,
−δLµ
1 − τL + δL
)
R∗ =
(
µ
1 − τR + δR
,
−δRµ
1 − τR + δR
)
and their stability is determined by the eigenvalues λ1,2 =
1
2
(
τ ±
√
τ2 − 4δ
)
.
It may be noted that if
τL >(1 + δL) and τR <(1 + δR) (3)
then there is no fixed point for µ<0 and there are two fixed
points, one each in L and R, for µ>0. The two fixed points
are born on the border at µ = 0. We call this a border
collision pair bifurcation. An analogous situation occurs
if τL <(1 + δL) and τR >(1 + δR) as µ is reduced through
zero. Due to symmetry of the two cases, we consider only
the parameter region (3).
2
If (1 + δR) > τR > −(1 + δR) then for µ > 0, the fixed
point in L is a regular saddle and the one in R is an attrac-
tor. This is like a saddle-node bifurcation occurring on the
border. Since this region in the parameter space always
has a periodic attractor for µ > 0, we exclude this region
from our analysis when looking for chaotic behavior. The
condition τR =−(1+ δR), results in a nongeneric situation
where all points on the line joining the points ( µ
1+δR
, 0)
and (0,− δRµ
1+δR
) are fixed points of the second iterate. We
therefore concentrate on the parameter space region
τL >(1 + δL) and τR <−(1 + δR) (4)
and investigate the property of the attractor for µ>0. We
first consider the case 1>δL≥0 and 1>δR≥0.
For (4), L∗ is a regular saddle and R∗ is a flip saddle.
Let UL and SL be the unstable and stable manifold of L
∗
and UR and SR be the unstable and stable manifold of
R∗ respectively. For (2), all intersections of the unstable
manifolds with x = 0 map to the line y = 0. Since one
linear map changes to another linear map across the x=0
line, UL and UR experience folds along the x-axis. And
all images of fold points will be fold points. By a similar
argument we conclude that SL and SR fold along the y-
axis, and all pre-images of fold points are fold points.
Let λ1L, λ2L be the eigenvalues in side L and λ1R,
λ2R be the eigenvalues at side R. For condition (4),
λ1L > λ2L > 0 and 0 > λ1R > λ2R. The stable eigenvec-
tor at R∗ has a slope m1 = (−δR/λ1R) and the unstable
eigenvector has a slope m2 = (−δR/λ2R). Since points
on an eigenvector map to points on the same eigenvector
and since points on the y-axis map to the x-axis, we con-
clude that points of UR to the left of y-axis map to points
above x-axis. From this we find that UR has an angle
m3 =
δLλ2R
δR−τLλ2R
after the first fold. Under condition (4) we
have m1 > m2 > 0 and m3 < 0. Therefore there must be
a transverse homoclinic intersection in R. This implies an
infinity of homoclinic intersections and the existence of a
chaotic orbit.
We now investigate the stability of this orbit. The basin
boundary is formed by SL. SL folds at the y-axis and
intersects the x-axis at point C. The portion of UL to
the left of L∗ goes to infinity and the portion to the right
of L∗ leads to the chaotic orbit. UL meets the x-axis at
point D, and then undergoes repeated foldings leading to
an intricately folded compact structure as shown in Fig.3.
The unstable eigenvector at L∗ has a negative slope
given by (−δL/λ1L). Therefore it must have a hetero-
clinic intersection with SR. Since both UL and UR have
transverse intersections with SR, by the Lambda Lemma
[9] we conclude that for each point q on UR and for each
ǫ-neighborhood Nǫ(q), there exist points of UL in Nǫ(q).
Since UL comes arbitrarily close to UR, the attractor must
span UL in one side of the heteroclinic point.
Since all initial conditions in L converge on UL and all
initial conditions in R converge on UR, and since there
Figure 3: The stable and unstable manifolds of L∗ for
τL = 1.7, δL = 0.5, τR =−1.7, δR = 0.5. R∗ is marked by
the small cross inside the attractor.
are points of UL in every neighborhood of UR, we con-
clude that the attractor is unique. This chaotic attractor
can not be destroyed by small changes in the parameters.
Since small changes in the parameters can only cause small
changes in the Lyapunov exponents, where the chaotic at-
tractor is stable, it is also robust.
It is clear from the above geometrical structure that no
point of the attractor can be to the right of point D. If
D lies towards the left of C, the chaotic orbit is stable. If
D falls outside the basin of attraction, it is an unstable
chaotic orbit or chaotic saddle. From this, the condition
of stability of the chaotic attractor is obtained as
δLτRλ1L−δRλ1Lλ2L+δRλ2L−δLτR+τLδL−δ2L−λ2LδL > 0
(5)
If δL =δR =δ this condition reduces to τRλ1L − λ1Lλ2L +
τL − τR − δ>0.
The robust chaotic orbit continues to exist as τL is re-
duced below (1+δL). With τL slightly below (1+δL), there
is no fixed point in L for µ>0 but the invariant manifolds
suffer only slight change. The invariant manifold of L as-
sociated with λ1L still forms the attractor. The invariant
manifolds in L, however, cease to exist for τL <2
√
δL since
the eigenvalues become complex. As τL is reduced below
2
√
δL there is a sudden reduction in the size of the attrac-
tor as it spans only UR. So long as UL exists, multiple
attractors can not exist and therefore if the main attractor
is chaotic, it is also robust.
Therefore we see that for 1 > δL > 0, 1 > δR > 0, the
normal form (2) exhibits robust chaos in a portion of pa-
rameter space bounded by the conditions τR =−(1 + δR),
τL >2
√
δL and (5), as shown in Fig.4. There is a symmet-
ric region of the parameter space with the roles of R and
L interchanged, where the same phenomena are observed
for µ<0.
3
Figure 4: Schematic diagram of the parameter space re-
gion of the normal form (2) where robust chaos is observed
for 1>δL >0, 1>δR >0 and µ>0.
At very low values of the determinant, i.e., when the
system is very close to being one-dimensional, the main
attractor may not remain chaotic even for τL > 2
√
δL, as
periodic orbits become stable. The conditions for emer-
gence of periodic windows for the one-dimensional case
have been derived in [8, 10]. Therefore for one dimen-
sional systems, the parameter range for robust chaos is
bounded by τR =1, τR > − τLτL−1 , and the lower limit of τL
is given by the conditions of existence of various periodic
windows. Here the τ ’s are to be interpreted as the slopes
of the piecewise linear function in the two halves separated
by x = 0. On the other hand, if the determinants in the
two sides are unity, the region in the τL − τR space for
robust chaos shrinks to zero area.
The cases with negative determinant are investigated
following the same method. For the sake of brevity, we
present the results without much explanation.
For −1 < δR < 0, we have 1 > λ1R > 0, λ2R < −1, and
R∗ is located above the x-axis. A positive value of λ1R
implies that UL converges on UR from one side. If
λ1L − 1
τL − 1 − δL
>
λ2R − 1
τR − 1 − δR
(6)
then the intersection of UL with the x-axis remains the
rightmost point of the attractor and (5) still gives the pa-
rameter range for boundary crisis. But if (6) is not satis-
fied, the intersection of UR with the x-axis becomes the
rightmost point of the attractor, and the condition of ex-
istence of the chaotic attractor changes to
λ2R − 1
τR − 1 − δR
<
δL (τL − δL − λ2L)
(τL − 1 − δL) (δRλ2L − δLτR)
(7)
For δL < 0 and δR < 0, L
∗ is below the x-axis and the
same logic as above applies. But if δL < 0 and δR > 0,
the stable manifold of R∗ has a negative eigenvalue and
hence UL does not approach UR from one side. Therefore,
if (6) is not satisfied, there is no analytic condition for
boundary crisis — it has to be determined numerically.
For δL <0, the invariant manifolds UL and SL always exist
as the eigenvalues are real for all τL. Therefore multiple
attractors can not exist for δL <0.
Since (2) is a normal form of the piecewise smooth map
(1), it is expected that robust chaos would be observable
in many piecewise smooth maps in the neighborhood of
border collision bifurcations, provided that there are no
more than one period-1 fixed point in Ra and Rb, there
exist homoclinic as well as heteroclinic intersections of the
invariant manifolds associated with these fixed points, and
the trace and determinant at the two sides of the border-
line satisfy the above conditions. The example of the boost
converter is a case in point.
A major conclusion of this Letter is that one should
use piecewise smooth systems in applications that require
reliable operation under chaos.
∗ Electronic address: soumitro@ee.iitkgp.ernet.in
† Electronic address: yorke@ipst.umd.edu
‡ Also with the Institute for Plasma Research and De-
partment of Mathematics, University of Maryland, College
Park, USA. Electronic address: grebogi@chaos.umd.edu
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4


Robust Chaos

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