It has been shown recently that border collision bifurcation in a piecewise smooth map can lead to a situation where a fixed point remains stable at both sides of the bifurcation point, and yet the orbit becomes unbounded at the point of bifurcation because the basin of attraction of the stable fixed point shrinks to zero size. Such bifurcations have been named "dangerous bifurcations". In this paper we provide explanation of this phenomenon, and develop the analytical conditions on the parameters under which such dangerous bifurcations will occur

Banerjee, Soumitro

Ganguli, Anindita

English

Publications of the IAS Fellows

Dangerous bifurcation at border collision: When does it occur?
Anindita Ganguli and Soumitro Banerjee*
Center for Theoretical Studies, Indian Institute of Technology, Kharagpur-721302, India
sReceived 27 August 2004; published 26 May 2005d
It has been shown recently that border collision bifurcation in a piecewise smooth map can lead to a situation
where a fixed point remains stable at both sides of the bifurcation point, and yet the orbit becomes unbounded
at the point of bifurcation because the basin of attraction of the stable fixed point shrinks to zero size. Such
bifurcations have been named “dangerous bifurcations.” In this paper we provide explanation of this phenom-
enon, and develop the analytical conditions on the parameters under which such dangerous bifurcations will
occur.
DOI: 10.1103/PhysRevE.71.057202 PACS numberssd: 05.45.Ac
The bifurcations occurring in piecewise smooth maps
have attracted significant research attention in recent years
because of their applicability in a wide class of systems of
practical interest f1g including switching circuits f2g, impact
oscillators f3g walking robots f4g, and cardiac dynamics f5g.
In such systems, as a parameter is varied, a fixed point can
collide with the borderline between two smooth regions, re-
sulting in an abrupt change in the Jacobian matrix. This leads
to a new class of bifurcations, known as border collision
bifurcations f6g.
It has been shown that such bifurcations can lead to atypi-
cal transitions like a period-1 orbit directly bifurcating into a
chaotic orbit, or a periodic orbit suddenly vanishing as it hits
the border. Analysis of such bifurcations have been devel-
oped f6–8g depending on the eigenvalues of the Jacobian
matrix at the two sides of the border. It has been found that
under some conditions a stable fixed point occurs at both
sides of a border collision event, and it was believed that this
situation would result in no observable change in system
behavior f7,8g. However, it has been recently shown f9g that
border collision bifurcations can also lead to a peculiar situ-
ation where the system collapses at the point of border col-
lision, even though the fixed point remains stable throughout
the range of parameter variation. The basin of attraction of
the stable fixed point shrinks as the parameter is varied to-
ward the bifurcation value, and at the bifurcation point the
basin of attraction has zero size. As a result, orbits starting
from all points other than the fixed point become unbounded
sat least from a local point of view in a neighborhood of the
bifurcating pointd. This revelation is a matter of serious con-
cern for practical systems that are modeled by piecewise
smooth maps, because the eigenvalues of the fixed point do
not give any signal of the impending collapse.
This possibility has been pointed out through numerical
exploration but the mechanism causing its occurrence is not
yet known. Moreover, in order to apply this knowledge in
practical situations, it is necessary to know the conditions
under which such bifurcations are expected to occur. The
purpose of this paper is to explain the mechanism of such
“dangerous bifurcations” and to obtain explicit conditions on
the system parameters which lead to such behavior.
It has been shown f6g that dynamical phenomena related
to border collision bifurcations sBCBd can be probed using
the piecewise linear approximation in the neighborhood of
the border crossing fixed point, expressed in the convenient
normal form:
s1d
where tL is the trace and dL is the determinant of the Jaco-
bian matrix JL of the system at a fixed point in RA“ hsx ,yd
PR2 :xł0j and close to the border and tR is the trace and dR
is the determinant of the Jacobian matrix JR of the system
evaluated at a fixed point in RB“ hsx ,ydPR2 :xø0j near the
border.
We assume that the determinants of the Jacobian matrices
at the two sides of the border satisfy the conditions 1.dL
.0 and 1.dR.0. For a specific combination of dL and dR,
the type of BCB depends on the two parameters tL and tR.
We restrict our attention to the parameter space region
bounded by
− s1 + dLd , tL , s1 + dLd , s2ad
− s1 + dRd , tR , s1 + dRd , s2bd
because, the map has a fixed point attractor for m,0 if s2ad
is true, and for m.0 if s2bd is true.
In order for an “attractor at infinity” to occur in addition
to the stable fixed point, it is necessary that there must be an
invariant subspace forming the boundary between the basins
of attraction of the two attractors. It is known that the stable
manifold of a saddle fixed point can form such a basin
boundary. Therefore we probe which fixed point can serve
this purpose.
*Also at the Department of Electrical Engineering. Electronic
address: soumitro@ee.iitkgp.ernet.in
PHYSICAL REVIEW E 71, 057202 s2005d
1539-3755/2005/71s5d/057202s4d/$23.00 ©2005 The American Physical Society057202-1
Since an unstable period-1 orbit does not occur in the
region of parameter space given by s2d, a period-1 orbit can-
not form the basin boundary. A period-2 orbit exists for m
.0 if tR,−s1+dRd and for m,0 if tL,−s1+dLd, and
therefore it cannot coexist with a stable period-1 orbit.
Next, we analyze the existence and the stability of high
periodic orbits sHPOd starting from period-3. In the follow-
ing discussion we will name particular types of orbits de-
pending on the partitions sL or Rd in which the points fall.
For example, an LRR orbit implies a period-3 orbit with one
point on the left-hand side and two points on the right-hand
side. Suppose this orbit has the points sxL1 ,yL1d, sxR1 ,yR1d,
and sxR2 ,yR2d. The conditions of existence of the LRR orbit
are given by xL1,0, xR1.0, and xR2.0. Out of these, the
first one is always satisfied, since there cannot be a high-
periodic orbit with all the points in one linear side. From the
other two conditions we get the inequalities
s1 + tL − dR + tRtL + dLdR + dLtRdm
1 + dR
2dL + tLdR + dLtR + dRtR − tLtR
2 . 0, s3d
s1 + tR − dL + tRtL + dLdR + tLdRdm
1 + dR
2dL + tLdR + dLtR + dRtR − tLtR
2 . 0. s4d
Notice that the denominator of both these inequalities is
the same. The region of existence of this orbit in the param-
eter space will be determined by the intersection of the re-
gions given by the above inequalities. This region is plotted
in Fig. 1. Since the LHS of the inequalities are expressed as
ratios of two functions, the ratio will be positive if both the
numerator and the denominator are positive, or if both are
negative. This leads to the conclusion that the contours of the
region of existence will be formed by
1 + tL − dR + tRtL + dLdR + dLtR = 0, s5d
1 + tR − dL + tRtL + dLdR + tLdR = 0, s6d
1 + dR
2dL + tLdR + dLtR + dRtR − tLtR
2
= 0. s7d
These lines are marked in Fig. 1.
Now, in order for this orbit to be stable, the eigenvalues of
the composite matrix JLJR2 must be inside the unit circle. Let
t be the trace and d be the determinant of the above matrix.
If the eigenvalues are complex conjugate, ulu2=d,1 since
the system is assumed to be dissipative at both sides. There-
fore the eigenvalues can go out of the unit circle only when
they are real, giving the stability condition
− s1 + dd , t , s1 + dd . s8d
From this, the stability conditions for the LRR orbit are
found to be
1 + dR
2dL + tLdR + dLtR + dRtR − tLtR
2 . 0, s9d
1 + dR
2dL − tLdR − dLtR − dRtR + tLtR
2 . 0. s10d
Notice that the left-hand side sLHSd in s9d is the same as the
LHS in s7d—which corresponds to the condition where the
eigenvalue becomes equal to +1.
If s9d is satisfied, then the denominator of the existence
conditions becomes positive. Therefore in the parameter re-
gion shown in Fig. 1, the stable LRR orbit must coexist with
the stable fixed point in L. These two orbits must have their
own basins of attraction, separated by the stable manifold of
a saddle-type periodic point. Which fixed point serves this
purpose?
Notice that for the LLR type period-3 orbit, the stability
conditions are
1 + dL
2dR + tRdL + dRtL + dLtL − tRtL
2 . 0, s11d
1 + dL
2dR − tRdL − dRtL − dLtL + tRtL
2 . 0. s12d
From these conditions we find that in the parameter region
under consideration, the LLR orbit is unstable—a regular
saddle. Indeed, this is the orbit whose stable manifold creates
the basin boundary.
The condition of existence of the LLR orbit is given by
xL1,0, xL2,0, and xR1.0. Ignoring the last condition
which is always satisfied, we get the inequalities
s1 + tR − dL + tLtR + dRdL + dRtLdm
1 + dL
2dR + tRdL + dRtL + dLtL − tRtL
2 , 0, s13d
s1 + tL − dR + tLtR + dRdL + tRdLdm
1 + dL
2dR + tRdL + dRtL + dLtL − tRtL
2 , 0, s14d
as the condition for its existence.
We now plot the intersection of these existence conditions
in Fig. 2. In this case also the boundaries of the region are
obtained from the numerators and denominators of s13d and
s14d, as
1 + tR − dL + tLtR + dRdL + dRtL = 0, s15d
1 + tL − dR + tLtR + dRdL + tRdL = 0, s16d
1 + dL
2dR + tRdL + dRtL + dLtL − tRtL
2
= 0. s17d
FIG. 1. Region of existence of the LRR orbit, given by the
intersection of s3d and s4d for m=−0.3 and dL=dR=0.9. The con-
tours are marked by the corresponding equations.
BRIEF REPORTS PHYSICAL REVIEW E 71, 057202 s2005d
057202-2
Now notice that s6d and s15d are identical and s5d and s16d
are also identical. Therefore, the curves marked s5d in Fig. 1
and in Fig. 2, respectively, are identical, and the curves
marked s6d in Fig. 1 and in Fig. 2 are also identical. But the
curves marked s7d in Fig. 1 representing the first stability
condition of the LRR orbit and that marked s17d in Fig. 2
representing the first stability condition of the LLR orbit are
not identical.
This observation leads to the conclusion that whenever
the stable LRR orbit exists, the unstable LLR orbit swe call it
the complementary orbitd also exists. But the reverse is not
true. In the parameter space there exists a region where the
unstable LLR orbit exists but the stable LRR orbit does not.
This region is obtained by subtracting the regions of exis-
tence of the LRR orbit from that of the LLR orbit, and is
shown in Fig. 3. When tL is reduced from zero with tR fixed
at a value lying between −1 and −1.9, if tL crosses the line
marked s17d, a single unstable period-3 orbit comes into ex-
istence. As it is reduced below the parameter value limited
by curve s7d, the stable LRR orbit begins to exist.
For a system with parameters placed in the shaded region
in Fig. 3, a stable period-1 fixed point exists in L, and a
regular saddle type period-3 fixed point also exists. Natu-
rally, the stable manifold of the saddle fixed point will divide
the phase space into two regions. One is the basin of attrac-
tion of the stable fixed point. In the absence of any other
stable orbit, the other attractor will be at infinity.
Since every distance in this map scales with the parameter
m, as the parameter is varied toward zero, the maximum
distance between the stable fixed point and the points of the
unstable period-3 orbit reduces. At the point of border colli-
sion bifurcation, i.e., at m=0 the distance becomes zero.
Therefore the area of the basin of attraction of the stable
fixed point reduces to zero.
Because of the symmetry in the map, for positive values
of m, the same phenomenon will be observed. Only, the LLR
orbit will now be stable and the LRR orbit will be unstable.
In the equations, the terms with suffix L will have to be
substituted by terms with suffix R and vice versa. For the
same values of the determinants at the two sides, these re-
gions obtained for m.0 will be identical with that obtained
for m,0 sthough the regions will be different if the deter-
minants at the two sides are unequald. For a specific param-
eter combination placed in this region, the basins of attrac-
tion and the positions of the stable and unstable fixed points
are shown in Fig. 4.
Now we take up the case of orbits of periodicity four and
above. Obviously, there can be many different types of orbits
of each periodicity. We define “regular orbits” as those orbits
having the symbols L and R occurring consecutively at least
in one cyclic permutation. We find that for m.0, two types
of regular periodic orbits occur in the parameter region s2d:
sad those with one point on the right-hand side and the rest
on the left-hand side sthe Ln−1R orbitsd, and sbd those with
two points on the left-hand side and the rest on the right-
hand side sthe L2Rn−2 orbitsd. For m,0, the regular orbits
are of LRn−1 and Ln−2R2 types. Based on numerical investi-
gation, it is our conjecture that only these regular orbits are
responsible for the dangerous BCBs.
For each stable high periodic orbit, there will be a
FIG. 2. The region of existence of the LLR orbit for m=−0.3 and
dL=dR=0.9, obtained as the intersection of the parameter regions
given by s13d and s14d.
FIG. 3. The region where only the unstable LRR orbit exists
along with the stable period-1 fixed point.
FIG. 4. The structure of the phase space for dL=dR=0.9, tL=
−0.5, tR=−1.5. sad For a negative value of the parameter m and sbd
for a positive value of m. The stable fixed point is shown with a
black dot and the unstable period-3 fixed point is shown with open
circles. The basin of attraction of the stable fixed point reduces to
zero size at m=0.
BRIEF REPORTS PHYSICAL REVIEW E 71, 057202 s2005d
057202-3
complementary orbit with the following properties:
s1d It must have the same periodicity as the observed pe-
riodic orbit.
s2d Its symbol sequence differs from that of the stable
periodic orbit by only one letter, with the exception of se-
quences consisting of only one symbol.
s3d Whenever a high periodic orbit coexists with a stable
period-1 orbit, the stable manifold of the complementary or-
bit forms the basin boundary.
In the present case, the complementary of the Ln−1R orbits
are the Ln−2R2 orbits, and the complementary of the LRn−1
orbits are the L2Rn−2 orbits. Note that the period-3 LLR orbit
belongs to both Ln−1R class and the L2Rn−2 class.
There will be regions of the parameter space where the
above regular orbits of periods 4, 5, 6, etc., will occur. There
will also be adjacent regions where these regular orbits do
not occur, but their complementary orbits occur. These rep-
resent conditions for the occurrence of dangerous border col-
lision bifurcations. To compute these, one must obtain the
composite matrices of the regular orbits and their comple-
mentary orbits, compute their trace t and determinant d, and
then substituting into the equation t=1+d one gets the con-
tours of the regions. These regions can be easily obtained
using any symbolic computation program. For example, for
m,0 there will be a region where the regular period-4 LR3
orbit will not occur, but its complementary L2R2 orbit will
occur. This region is delimited by two lines obtained by sub-
stituting the traces and determinants of the matrices
JLJRJRJR and JLJLJRJR into the equation t=1+d.
Following this procedure, we obtain the parameter regions
where the above phenomenon occurs with orbits of period 3,
4, 5, etc., i.e., where only the unstable complementary orbits
occur, but the stable high-periodic orbits do not occur. These
regions are shown in Fig. 5. In the dark areas of the region
tL.tR, the complementary orbits belonging to the class
Ln−1R exist for m,0, and those belonging to the class Ln−2R2
exist for m.0. By symmetry, in the region tL,tR the
complementary orbits belonging to the class L2Rn−2 exist for
m,0, and those belonging to the class LRn−1 exist for m
.0. Notice that the regions for the occurrence of dangerous
bifurcation thus obtained analytically exactly match those
obtained in Ref. f9g through numerical exploration.
In this paper we have shown that it is possible for a single
unstable periodic orbit to come into existence in a piecewise
linear continuous map, and that the dangerous border colli-
sion bifurcations in nonsmooth systems are caused by the
occurrence of such singleton unstable orbits. We have de-
scribed a method of obtaining the parameter space regions
where such bifurcations occur. This knowledge will help
avoid the occurrence of dangerous bifurcations in practical
systems.
This work was supported in part by the Department of
Atomic Energy, Government of India under Contract No.
2003/37/11/BRNS.
f1g H. E. Nusse, E. Ott, and J. A. Yorke, Phys. Rev. E 49, 1073
s1994d.
f2g J. H. B. Deane and D. C. Hamill, IEEE Trans. Power Electron.
5, 260 s1990d; T. Kousaka, T. Ueta, and H. Kawakami, IEEE
Trans. Circuits Syst., II: Analog Digital Signal Process. 46,
878 s1999d; G. H. Yuan, S. Banerjee, E. Ott, and J. A. Yorke,
IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 45, 707
s1998d.
f3g A. B. Nordmark, J. Sound Vib. 145, 279 s1991d; C. Budd and
F. Dux, Philos. Trans. R. Soc. London, Ser. A 347, 365 s1994d.
f4g B. Thuilot, A. Goswami, and B. Espiau, in Proceedings of the
IEEE International Conference on robotics and automation
sAlbuquerque, New Mexico, USA, 1997d; M. Garcia, A. Chat-
terjee, A. Ruina, and M. Coleman, ASME J. Biomech. Eng.
120, 281 s1998d.
f5g J. Sun, F. Amellal, L. Glass, and J. Billette, J. Theor. Biol.
173, 79, 1995; D. Chen, H. O. Wang, and W. Chin, in Pro-
ceedings of the IEEE International Symposium on circuits &
systems sMonterey, California, 1998d, Vol. 3, pp. 635–638.
f6g H. E. Nusse and J. A. Yorke, Physica D 57, 39 s1992d.
f7g S. Banerjee and C. Grebogi, Phys. Rev. E 59, 4052 s1999d.
f8g M. di Bernardo, M. I. Feigin, S. J. Hogan, and M. E. Homer,
Chaos, Solitons Fractals 10, 1881 s1999d.
f9g M. A. Hassouneh, E. H. Abed, and H. E. Nusse, Phys. Rev.
Lett. 92, 070201 s2004d.
FIG. 5. The regions of dangerous border collision bifurcation
obtained using the conditions of existence of complementary orbits
for dL=0.9, dR=0.9.
BRIEF REPORTS PHYSICAL REVIEW E 71, 057202 s2005d
057202-4


Dangerous bifurcation at border collision: when does it occur?

http://repository.ias.ac.in/21401/1/341.pdf

Dangerous bifurcation at border collision: when does it occur?

Abstract

Similar works

Full text

Available Versions

Publications of the IAS Fellows