Border-collision bifurcations including "period two to period three" for piecewise smooth systems*
H e l e n a E. Nusse a'b and J ames A. Yo rk e a'c alnstitute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA bVakgroep Econometrie, F.E.W., Rijksuniversiteit Groningen, Postbus 800, 9700 AV Groningen, The Netherlands CDepartment of Mathematics, University of Maryland, College Park, MD 20742, USA
Received 15 March 1991 Revised manuscript received 5 February 1992 Accepted 16 February 1992 Communica ted by J. Guckenheimer
We examine bifurcation phenomena for maps that are piecewise smooth and depend continuously on a parameter /~ . In the simplest case there is a surface F in phase space along which the map has no derivative (or has two one-sided derivatives). F is the border of two regions in which the map is smooth. As the parameter /z is varied, a fixed point E~, may collide with the border F , and we may assume that this collision occurs a t / z = 0. A variety of bifurcations occur frequently in such situations, but never or almost never occur in smooth systems. In particular E~, may cross the border and so will exist for/z < 0 and for g > 0 but it may be a saddle in one case, say/~ < 0, and it may be a repellor for/~ > 0. For/.~ < 0 there can be a stable period two orbit which shrinks to the point E 0 as /~ ~ 0, and for /~ > 0 there may be a stable period 3 orbit which similarly shrinks to E 0 as g ---, 0. Hence one observes the following stable periodic orbits: a stable period 2 orbit collapses to a point and is reborn as a stable period 3 orbit. We also see analogously "stable period 2 to stable period p orbit bifurcations", with p = 5,11,52, or period 2 to quasi-periodic or even to a chaotic attractor. We believe this phenomenon will be seen in many applications.
1. Introduction
Certain bifurcation phenomena have been re- ported repeatedly in numerous studies of low dimensional dynamical systems, that depend on one parameter. The rather familiar local bifurca- tion phenomena describing the evolution of at- tractors as a parameter is varied include the saddle node bifurcation, the period doubling (or halving) bifurcation, and the Hopf bifurcation. In the literature dealing with bifurcation theory, it is frequently assumed that the map corresponding to the dynamical system is differentiable; see for example [2, 6, 10, 11]. To remind the reader so
* Research in part supported by the Depar tment of Energy (Scientific Comput ing Staff Office of Energy Research), and by D A R P A / O N R .
that we may draw contrasts, the well known bifur- cation diagram of the quadratic map Q,(x)=
- x z is given in fig. 1 (1 </z < 1.5). All the computer assisted pictures were made by using the DYNAMICS program [12].
We say a map is smooth if the map has a continuous derivative. A region is a closed, con- nected subset in phase space. We examine con- tinuous maps which are piecewise smooth. We restrict attention to those which are smooth on two regions of the plane with the border between these regions being a smooth curve. From now on we assume that there is a smooth curve F which separates the plane into two regions denoted by R A and R B. We say, a map F from the phase space •2 to itself is piecewise smooth if (1) F is continuous, and (2) F is smooth on both the regions R A and R B. Note that on the border F
40 H.E. Nusse, J.A. Yorke /Border-collision bifurcations for piecewise smooth systems
p.
/ I
1.5 -1 x 2
Fig. 1. Bifurcation diagram of the quadratic map Q~,(x)= hi, --X 2.
-0.1
~t
0.2 -1 X
Fig. 2. Bifurcation diagram exhibiting the "period two to period three" bifurcation of the map
between the regions, the mappings must be equal since F, is assumed to be continuous. A special case that we shall refer to frequently is the fol- lowing prototype example, a piecewise linear map into which other generic piecewise linear maps in the plane can be transformed by changes in coor- dinates.
Let u and w be vectors in the plane. Let x and y be the phase space coordinates and st is a scalar parameter. Let P , be the map defined by
P A x , y) =xu + Ixlw + ( y + st)(1,0)
and we investigate trajectories (Xn+~,yn+l)= P~,(X n, yn). The regions R A and Ra are the left and right half plane separated by F, the y-axis. Notice that if we set u = (0, b), w -- (a, 0), st = 1, then the Lozi map (L) is retrieved.
To illustrate the "per iod two to period three" border-collision bifurcation phenomenon, consider the one-parameter family of maps f~ ( - oo < st < oo) from the plane to itself, defined by
[ ( - 1 . 4 x + y , - 0 . 1 x ) + s t ( l , 0 ) if x < 0 ,
f~ , (x ,y) / ( - 3 x + y , - 4 x ) + s t ( l , 0 )
if x > 0 .
l - l . 4 x + y + / x , - 0 . 1 x ) if x < 0 , f~(x,y)= - 3 x + y + ~ , - 4 x ) if x>0.
Notice that the map is smooth in each of the half planes x < 0 and x > 0 , and the y-axis is the border which is a smooth curve. Note that to write f~ in the form of P, , let u = ( - 2.2, -2.05) , and w = ( - 0 . 8 , - 1.95). The bifurcation diagram exhibiting the "period two to period three" bifur- cation, is presented in fig. 2 ( - 0.1 < st < 0.2). All the bifurcation diagrams in this paper show a projection of the attractor, projecting (x, y) onto the X-axis, which is plotted vertically; the hori- zontal coordinate is st.
The purpose of this paper is to study the occur- rence of such a new bifurcation phenomenon for continuous, piecewise smooth maps. These systems include, for example, two-dimensional continuous, piecewise-linear maps. In [5] the dy- namics of a simple economic model was studied, and a "per iod three to period two" bifurcation was observed numerically, and was established rigorously in [4] for a degenerate piecewise-linear situation. The "border-collision bifurcation" phe- nomena is a much richer class of bifurcation phenomena than just a "period two to period three" bifurcation and occur for generic piece-
H.E. Nusse, J.A. Yorke /Border-collision bifurcations for piecewise smooth systems 41
wise smooth maps. We present phenomena that occur when the nature of an unstable fixed point of a piecewise smooth map is changed while the fixed point collides with the border between two regions in which the map is smooth. Since the fixed point is unstable before and after collision, it is not shown in the bifurcation diagram in fig. 2. While we consider maps in the plane, higher dimensional analogues exist. We know of no phenomena that can occur only in higher dimen- sional cases. There is also no difficulty in chang- ing the notation to that there are more than two regions on which the map is smooth. We could also allow F to depend on /z, but coordinates could be chosen so that it remains fixed, so our case in practice includes moving boundaries. With moving boundaries the map would be piecewise smooth in /z .
We say a fixed point Ez is a border crossing fired point if it crosses the border F between two regions in which the map is smooth. We will assume that the crossing occurs at /~ = 0. The fixed point Eg is called a flip saddle if the eigen- values a and v of the Jacobian matrix DF~(E~) if h < - 1 < v < 1. Assume that there exists a one- parameter family of piecewise smooth maps and assume that there is a border crossing fixed point (or periodic point) Ez, we emphasize the case when Ez crosses the border F it changes from being a flip saddle to a repellor with complex eigenvalues. The above example has this behav- ior.
In section 2, we discuss why the border-colli- sion bifurcation phenomenon occurs when the nature of an unstable equilibrium changes when it crosses the border of two regions. To be some- what more specific, assume that a border crossing fixed point (or periodic point) Ez of a one- parameter family of piecewise smooth maps changes from being a flip saddle to a repellor with complex eigenvalues when it crosses the border F. Then at ~ -- 0, a border-collision bifur- cation occurs at this fixed point Ez on the border.
In section 3, we mainly deal with two piecewise smooth systems of the plane, one piecewise linear
and one piecewise nonlinear. The first system is the map P, (derived in section 2) that corre- sponds with a generic piecewise smooth nonlinear map, and the other system is based on the Henon map. For the piecewise linear map Pt, we present several examples including "per iod 2 to period p " (p = 5, 11, and 52), "per iod 2 to quasi-peri- odic" and "period 2 to chaotic" bifurcation. We also present an example of a border-collision bifurcation for the map Pz in which no attractors but chaotic saddles are involved. The system of the plane involving the H6non map at the left side and a linear map at the right side of the border, different border-collision bifurcations are observed. We present a variety of examples. Al- though we do not have an exhaustive list of types of border-collision bifurcation of one-parameter families of maps under consideration, we point out that several other types of bifurcation occur. We believe this phenomenon will be seen in many applications.
In section 4 we prove that certain one-parame- ter families of piecewise smooth maps exhibit a "period 2 to period 3" border-collision bifurca- tion. This phenomenon persists under small per- turbations of the involved maps.
In section 5, we discuss the state of the art, and pose several questions which remained unre- solved. This paper does not give a complete theory, but can be considered as initiating a bifur- cation theory of piecewise smooth maps.
2. The border-collision bifurcation phenomenon
In the bifurcation theory for maps, attention is focused on differentiable maps when one or more eigenvalues of a fixed point (or periodic point) cross the unit circle. When this occurs, the nature of the fixed point changes. For example, a fixed point attractor becomes a saddle (possibly a flip saddle) or a repellor. For border crossing fixed points, the Jacobian matrix of the fixed point generally changes discontinuously, and the fixed
42 H.E. Nusse, J.A. Yorke /Border-collision bifurcations for piecewise smooth systems
point can for example change from being a repel- lor to a saddle as U crosses zero.
Let F ( . , ~ ) = F. be a one-parameter family of piecewise smooth maps from the phase space •2
to itself, depending smoothly on the parameter /~ , and where /x varies in a certain interval on the real line. Let E . denote a fixed point of F~ defined on - e < ~ < e and depending con- tinuously on /z, for some e > 0. For a general approach (which is given below) we need the concept of the "orbit index" of a periodic orbit [8]. The orbit index is a number associated with a periodic orbit, and this number is useful in un- derstanding the patterns of bifurcations the orbit undergoes. We say an orbit of period p is typical if its Jacobian matrix exists (that is, the Jacobian matrix of the p th iterate of the map at a point of the orbit) and neither + 1 nor - 1 is an eigen- value (of the Jacobian matrix). For typical orbits, the orbit index is - 1, 0, or + 1. The orbit index is a bifurcation invariant in the sense that if one examines the periodic orbits that collapse to the fixed point E~, as /z ~ 0, and adds the orbit indexes of the periodic orbits that exist just before a bifurcation, then that sum equals the corresponding sum just after that bifurcation. Suppose a typical periodic orbit PO of a map F has (minimum) period p. The orbit index of that orbit depends on the eigenvalues of the Jacobian matrix Ap of the map F p at one of the points in PO. Now we define the orbit index Ipo of PO. Let m be the number of real eigenvalues of Ap smaller than - 1, and let n be the number of real eigenvalues of Ap greater than + 1. The orbit index Ipo of PO is defined by
Ipo = 0
Iao = -- 1
Ipo = + 1
if m is odd,
if m is even and n is odd,
if both m and n are even.
If the orbit index = - 1, then the orbit is called a regular saddle. The typical orbits with orbit index + 1 in the plane are repellors and attractors and fixed points with non-real eigenvalues. The deft-
nition of orbit index is technical when a point of the orbit lies on the boundary and so does not have a Jacobian matrix, and the definition is unnecessary since we consider orbits for/~ ~ 0.
For a moment, assume that E , is in the inte-
rior of the region RA (or the region RB), and write A and v for the eigenvalues of DF~(E,) . If
neither of the two eigenvalues A and v is on the unit circle, then the fixed point E , is called a flip saddle (and has index 0) if A < - 1 < v < 1; E , is a regular saddle (and has index - 1) if - 1 < v < 1 < A; E~ is a repellor (and has index + 1) if both LAI > 1 and Ivl > 1; and E , is an attractor (and has index +1) if both I~1 < 1 and Ivl < 1. Note that Eu has orbit index + 1 if the eigenvalues are not real. Hence, a typical fixed point is a flip saddle, a regular saddle, a repellor or an attrac- tor. Similarly, the nature of periodic points is defined.
Now we are able to provide a definition of the notion "border-collision bifurcation". Let the re- gions R A and R a, the map F, and the fixed point (periodic point) E~, be as above. Assume there exists a number e > 0 such that (1) E 0 is on the border of the two regions R g and RB, (2) for --e </~ < 0 the fixed point E , is in the region
RA, and its index is IA, and (3) for 0 < ~ < e the fixed point E~, is in the region R B, and its index is I B. If I A and I s are different, then (as stated below) some bifurcation must occur at E 0, since the orbit index of E~, is changing from I A to Ia , while the parameter /~ is increasing from - e
to +e. We say a periodic orbit PO is an isolated
border crossing orbit if (1) PO includes a point that is a border crossing fixed point under some iterate of the map, and (2) the orbit PO is iso- lated in phase space when tz = 0, that is, in the plane there exist a neighborhood U of the orbit PO such that PO is the only periodic orbit in U when /x = 0. From the topological degree theory as described in [8] (see also [1] for the two di- mensional case), the following "border-collision bifurcation" result follows after some minor mod- ifications.
H.E. Nusse, J.A. Yorke / Border-collision bifurcations for piecewise smooth systems 43
Border-collision bifurcation theorem. For each two-dimensional piecewise smooth map and de- pending smoothly on a paramete r IZ, if the index of an isolated border crossing orbit changes as tz crosses 0, then at Ix = 0 a bifurcation occurs at this point, a bifurcation involving at least one additional periodic orbit.
This result says that additional fixed points or
periodic points must bifurcate from E 0 at /x = 0. These bifurcating orbits need not to be stable. An example of the preservation of orbit index occurs with a period doubling bifurcation. If for ~ < 0 there is an attracting fixed point (and no other entering orbits), the total index is + 1. Then for /~ > 0 we can have a flip saddle (orbit index 0) and a period 2 attractor (orbit index + 1). Hence, the sum of the orbit indices before and after / x = 0 is +1. Note that the two points of the period 2 orbit are collectively assigned + 1, not individually, since that orbit has index + 1. Since this bifurcation occurs while the fixed point (or periodic point) collides with the border of the regions R A and RB, we call it a border-collision bifurcation. In other words, a border-collision bi- furcation is a bifurcation at a fixed point (or periodic point) on the border of two regions when the orbit index of the fixed point (or periodic point) before the collision with the border is different from the orbit index of the fixed point
after the collision. In addition, if h I and A 2 are attractors for tt < 0 and t z > 0, respectively, which are involved in a border-collision bifurcation at a fixed point (or periodic point) and shrink to the fixed point (or periodic point) when IZ ---> 0, we also say there is a border-collision bifurcation from attractor A~ to attractor A 2. Moreover, if these attractors are periodic orbits with period P1 and P2, respectively, we will also refer to such a bifurcation as period P1 to period Pz border-colli- sion bifurcation. For example, if A 1 is a period 2 attractor and A 2 is a period 5 attractor, we say there is a border-collision bifurcation from a pe- riod 2 attractor to a period 5 attractor (or a period 2 to period 5 border-collision bifurcation);
see e.g. example 1 in section 3. Similarly, if A, is a period 2 attractor and A 2 is a chaotic attractor, we say there is a border-collision bifurcation from a period 2 attractor to a chaotic attractor; see e.g. example 2 in section 3. We would like to point out that if A, is an attractor for F, when /~ < 0, then besides this attractor A~, the map F, may have other attractors. These other attractors may
be far away from the location(s) at which the border-collision bifurcation occurs. On the other hand, there may be more than one attractor
involved in the border-collision bifurcation for, say/z > 0. The figs 9a and 9b illustrate this phe- nomenon: a border-collision from a 5-piece chaotic attractor (~ < 0) to both a period 4 at- tractor and a 1-piece chaotic attractor (/z > 0). To illustrate that border-collision bifurcations may even be more complicated, we provide an exam- ple of a border-collision bifurcation in which the invariant sets being involved in this bifurcation are not attractors, but chaotic saddles; see exam- ple 3 in section 3 for more details.
We derive the map P, that was introduced in section 1, from nonlinear piecewise smooth maps. We assume coordinates are chosen so that the curve F is a straight line. Let z denote any vector in the plane, and write Fg(z) = F(z; Iz), and write z 0 = E 0. From the assumption Fg is piecewise smooth, we have that on each of the regions R A
and R B
F(z;Iz) =F(zo;O ) + D~F(zo,O)(z -Zo)
+ D~,F(zo,0)/z + H O T ,
where H O T stands for higher order terms. Hence, there exist matrices M A and M B and vectors v A and v a such that if z is in the region R a then
F ( z ; / . Q = F ( z o ; O ) + M a ( z - z 0 ) + VA/~ + H O T
and if z is in the region R a then
F ( z ; / z ) = F ( z o ; 0 ) + M B ( z - z o ) + vatz + HOT.
44 H.E. Nusse, J.A. Yorke ~Border-collision bifurcations for piecewise smooth systems
Let e~ be the unit vector tangent to F at z 0. The assumption F~, is piecewise smooth and de- pends smoothly on ~ implies MAe ~ = MBe 1 = e 2
and v A -- v B = v. Choose coordinates so that z 0 = 0, so F(zo,O) -- 0. Assume that e 2 is independent of e~, so we may use e 1 and e 2 as basis vectors. We let e~ and e 2 be the basis vectors of the plane. We assume that e 2 is independent of v and that v is not parallel with e~ - e 2. We claim that by a change of variables and by rescaling /z we may assume that v = e r Write v = (vx ,vy) .
We now assume that vx # O. We can make vr = 0 after a change of variables, and v~ = 1 by rescal- ing of /z. If Vy is not 0 then we can change variables, setting ~ = y - vfl.~ (where x is un- changed), and the new vector v for the (x, Y) system will have its second coordinate O. By rescaling ~, that is, by introducing ~ = ~v~, we can change the system so that the new vector v is (1,0), when ~ is the parameter. Therefore, we
may write M A - - ( b ~ ) , M B = ( ~ 1 ) , a n d v =
(1, 0). Since all these assumptions are generic, we say the prototype piecewise linear f o rm of F~ for/z
small is defined by F ( z ; / x ) = (~ ~tz + ~ ( 1 , 0 ) i f k - -
1 }z +/1(1, 0) is in the region RA, F(z ; /x) = (~ z 0 /
if z is in the region R B. To write the prototype piecewise linear form of
F~, in the form of the map P~,, let u = (½(a + c), ½(b + d)), and w = (½(a - c), ½(b - d)).
We observe the following fact. Assume that the fixed point E , is a flip saddle (orbit index 0) in region R A and a repellor with complex eigenval- ues (orbit index + 1) in region R B. If there exists a stable periodic orbit with period 2 in R A that converges to E 0 when u approaches 0, then the total degree in R A is + 1. Hence, if there exists a stable periodic orbit in R B that converges to E 0 when/~ goes to 0, then there must exist a regular saddle periodic orbit of the same period (orbit index - 1 ) in R B that converges to E 0 when goes to 0, since the total orbit index is a bifurca- tion invariant. Consequently, for the family of maps f~ in the section 1 exhibiting a "per iod two to period three" bifurcation in figure 2, there
must also exist a regular saddle periodic orbit with period 3.
Period two to period three border-collision bifurca-
tion theorem. Le t F~, be a one-parameter family of piecewise smooth maps which has a prototype piecewise linear form at p . - -0 , and assume that (1) a < - 1 , d < c < - 1 ; ( 2 ) c 2 + 4 d < O < c 2 + d ;
and (3) 0 < a(ac + d) < 1. Then, there exists e > 0 such that if [hi < e, then the family F~, has a "period two to period three" border-collision bi- furcation at (0, 0).
We point out that the border-collision bifurca- tions persist under small perturbations. The proof follows of the theorem from the result obtained in section 4. The proof given in section 4, might give insight why other bifurcations (for example, period 5 to period 2 bifurcation) may occur in piecewise smooth systems. Presumably, the method of proof only works if one of the two maps involved has a small Jacobian. Hence, when the piecewise smooth map consists of maps that all have Jacobian bounded (far) away from zero, new techniques have to be developed to obtain rigorous border-collision bifurcation results.
3. A variety of border-collision bifurcations
In this section we present a variety of numeri- cal examples exhibiting a border-collision bifurca- tion. The first series of examples is from the piecewise linear map P~,, and the second series is based on the H6non map. We will present exam- ples showing that in a border-collision bifurcation not only attracting periodic orbits are involved, but also chaotic saddles may play a role. There- fore, in order to describe the qualitatively differ- ent border-collision bifurcations in a consistent manner, we refer to the invariant sets that are involved in the border-collision bifurcation. A chaotic saddle is a compact, invariant set that is not an attractor which contains a chaotic trajec- tory. If an attractor A of a map F is an attracting
H. E. Nusse, LA. Yorke /Border-collision bifurcations for piecewise smooth systems 45
periodic orbit with period p, then we call A a period p attractor, and we say instead of “period two to period three” bifurcation a bifurcation from a period 2 attractor to a period 3 attractor.
The bifurcation diagrams below show the long term behavior of the coordinate x for p between -0.1 and 0.2. The diagrams have been con- structed as follows. For the minimum value - 0.1 of II, and initial value (0, O), calculate the first 200 points (transient time 200) of the orbit and plot the next 1000 points of the orbit. Increase Jo slightly, say by 0.001, take for the initial value the last point which was plotted, calculate 200 points of this orbit and plot the next 1000 points. In- crease p again, and continue increasing until p achieves the maximum value 0.2. Hence, once the orbit is close to an attractor, as the parameter is increased, this attractor is “followed” as long as it exists. In the diagrams, the x-coordinate is plot- ted horizontally, and the parameter p is plotted vertically.
Define the map GL, from the plane to itself to be the prototype piecewise linear form of F,, that is,
GL,(x,y) =(ax+y,bx) +&l,O) if x50,
GL,(x,y)=(cx+y,du)+~(l,O) ifxr:O.
Recall that the map GL, is equivalent to the map P,, since to write the map GL, in the form of the map P,, let u = <;<a + c), t(b + d)), and w = <+<a -cl, :(b -d)). We present a few nu- merical examples for this map GL, exhibiting a border-collision bifurcation. In all these exam- ples, the fixed point is a flip saddle for p < 0 and a repellor with complex eigenvalues for I_L > 0.
Example 1. The presumably simplest border-col- lision bifurcation is from a period 2 attractor to a period 3 attractor presented in fig. 1. We present parameter values for which the map GL, shows a bifurcation from a period 2 attractor to a period p attractor for a variety of period p.
For a = -1.25, b = -.035, c = -2, d = -1.75, the bifurcation diagram in fig. 3a exhibits a bifur-
-0.1
v
0.2
-O-l l--T---
-0.3 X 0.3
Fig. 3. (a) Bifurcation diagram of
GL,(x, Y) = (-1.25x+y+p,-0.035x) if xi;O,
(-2x+y+p,-1.75x) ifxr:O
exhibits at CL,, = 0 a border-coWion bifurcation from a period 2 attractor to a period 5 attractor. (b) Bifurcation diagram of
(-1.25x+y+p,-0.0435x) ifxS0,
(-2x+y+j~,-2.175~) ifxz0
exhibits at cl0 = 0 a border-collision bifurcation from a period 2 attractor to a period 11 attractor.
46 H.E. Nusse, J.A. Yorke /Border-collision bifurcations for piecewise smooth systems
cation from a period 2 attractor to a period 5 attractor.
For a = -1.25, 6 = -0.0435, c = -2, d = - 2.175, the bifurcation diagram in fig. 3b exhibits a bifurcation from a period 2 attractor to a period 11 attractor.
For other choices for a, b, c, and d we have found bifurcations from a period 2 attractor to a period p attractor, where p = 6, 7, 8, 9, 10, 11, 13, 19, 21, 23, 29, 31, 37, 41, 52, etc.
Example 2. The simplest border-collision bifurca- tion in which chaotic attractors are involved is presumably the bifurcation from a period 2 at- tractor to a (l-piece) chaotic attractor. Fre- quently, the border-collision bifurcation from a period 2 attractor to a p-piece chaotic attractor is observed.
For a = -1.25, b = -0.042, c = -2, and d = - 2.1, the bifurcation diagram in fig. 4a exhibits a bifurcation from a period 2 attractor to a l-piece chaotic attractor.
For a = -1.36, b = -0.12, c = -2, and d = -2, the bifurcation diagram in fig. 4b seems to exhibit a bifurcation from a period 2 attractor to a 1Zpiece chaotic attractor, but using the phase space it turns out that the bifurcation is from a period 2 attractor to a 18-piece chaotic attractor.
We have observed many other values of p, the map GL, shows a bifurcation from period 2 attractor to p-piece chaotic attractor.
For the selection a = - 1.25, b = -0.03865, c = - 2, and d = - 1.9325, we obtain a bifurca- tion diagram similar to fig. 4a, but in this case the border-collision bifurcation is from a period 2 attractor to what appears to be a quasi-periodic attractor.
Example 3. A border-collision bifurcation in which chaotic saddles (rather than attractors) are involved, will not be exhibited by bifurcation dia- grams. Therefore, some other numerical method is needed to detect these sets. We use the saddle straddle trajectory (SST) method introduced in [9] to detect such sets.
-0.1
P
0.2 -0.3 X 0.3
-O.l I P
0.2 -0.3 X
Fig. 4. (a) Bifurcation diagram of
0.3
(-1.25x+y+w,-0.042x) ifx<O,
(-2x+y+~,-2.13-) ifxt0
exhibits at CL,, = 0 a border-collision bifurcation from a period
2 attractor to a l-piece chaotic attractor. (b) Bifurcation
diagram of
(-1.36x+y+~, -0.12~) if ~50,
(-2x+y+/.L-2X) ifxz0
exhibits at pLo = 0 a border-collision bifurcation from a period
2 attractor to a 18-piece chaotic attractor.
H.E. Nusse, J.A. Yorke ~Border-collision bifurcations for piecewise smooth systems
We select a = - 1.25, b = 0.18, c = 2, and d = -3 . For/z = -0 .05 the invariant set (obtained by the SST method) is presented in fig. 5a, and the
invariant set for ~ = 0.05 is in fig. 5b. Presumably,
it is correct to say that the border-collision bifur- cation is a bifurcation from a chaotic saddle to
another chaotic saddle.
Now we present a few examples based on the
H6non map. In fact, in these examples we have a
moving border. Define the map H from the plane
to itself by
n ( x , y) = (A - x 2 + By, x)
and define the map L , ( -oo < ~ < oo) from the
plane to itself by
0.02
-0.25
I (a),
tit V
-0.2 x 0.1
L , ( x , y) = (.4 + Cx + B y - + C ) u ,
Dx + (1 - D ) ~ ) .
The regions R A and R B are the half planes to the left and the right of the straight line x =/x. The map we are investigating is defined being the
H6non map on R A and the "linear" map Lu on R B. Define the one-parameter family of maps F, from the plane to itself by
H(x,y) i f x < ~ , F g ( x , y ) = L ~ ( x , y ) ifx>__/z.
Notice that the map is smooth in each of the h a l f
planes x </z and x >/z, and the line x = g is the
border which is a smooth curve. Since the map F, is continuous, it is a piecewise smooth map. Note
that for this family F, border-collision bifurca-
tions occur presumably for values go different from zero.
Example 4. Simple border-collision bifurcations are bifurcations from a period p attractor to a period q attractor.
For A = 1 . 4 , B = 0 . 3 , C = 0 . 9 , and D = - 5 , the bifurcation diagram in fig. 6a exhibits a bifur- cation from a period 3 attractor to a period 4
0.2
-1.8
<b)l
-1 x
Fig. 5. (a) Chaotic saddle of
0.6
(-1.25x+y+lz,O.18x) if x_<0, GL~,(x, y) = ~ (2x +y +/z , -3x) if x_>0
when g = -0.05. (b) Chaotic saddle of
[(-1.25x+y+lz,O.18x) if x_<O, OL~,(x, y) = ~ (2x +y +/.~, "3x) if x > 0
when /z = 0.05.
47
48 H.E. Nusse, J.A. Yorke ~Border-collision bifurcations for piecewise smooth systems
attractor, where tz (plotted vertically) varies f rom 0.89
0.89 to 0.87. In the region R A the fixed point is a
flip saddle and in the region R B the fixed point is
a repellor. The border-coll ision bifurcation oc-
curs at Ix = ~0 = 0.884. For /x >/~0 (the side of the period 3 at t ractor which has orbit index + 1) g the fixed point is a flip saddle (orbit index 0) and
we find no o ther periodic orbits on this side of
the bifurcation. F o r / x </x 0 (the side of the pe-
riod 4 at tractor which has orbit index + 1) the
fixed point is a repellor (orbit index + 1); there also exists a period 4 regular saddle (orbit index
- 1 ) . The regular saddle also shrinks to a point (the fixed point) as /1,---,/z 0. Hence, the orbit 0.87
index is + 1 on both sides o f / x 0.
For A = 1 . 4 , B = 0 . 3 , C = l , and D = - 5 , the 1.05
bifurcation diagram in fig. 6b exhibits a bifurca-
tion f rom a period 6 at t ractor to a period 4
attractor, w h e r e / x (plotted vertically) varies f rom
1.05 to 0.8. In the figure one might first notice a
bifurcation f rom a 6-piece chaotic a t t ractor to a
period 4 attractor, but closer examination gives g the above ment ioned bifurcation f rom a period 6
at t ractor to a per iod 4 attractor. Similarly as
above, the periodic orbits involved in the border- col l is ion b i fu rca t ion tha t occurs at /z =
/x o = 0.884 are the following. F o r / z > /x 0 there is period 6 at t ractor and the fixed point is a flip saddle, and f o r / z < / z o the fixed point is a repel-
lor and there is a per iod 4 at t ractor a period 4
regular saddle. Hence , the orbit index is + 1 on
both sides o f / x 0.
Example 5. In this example we present two cases
of a border-coll ision bifurcation f rom a per iod p
at t ractor to a q-piece chaotic attractor. For A = 1.4, B = 0.3, C = 1.1, and D = - 5 , the bifurca-
tion diagram in fig. 7a exhibits a bifurcation f rom a 1-piece chaotic at t ractor to a period 4 attractor,
where p~ (plotted vertically) varies f rom 1.05 to 0.8. The border-coll ision bifurcation occurs at
/x = /x 0 ~ 0.885. For /x > /x o (the side with the chaotic at tractor) we do not know the (total) orbit
index since the chaotic a t t ractor contains a lot o f periodic orbits. For/~ > /x 0 the fixed point is a flip
exhibits at /~0 ~- 0.884 a border-collision bifurcation from a period 6 attractor to a period 4 attractor.
H.E. Nusse, J.A, Yorke / Border-collision bifurcations for piecewise smooth systems 49
saddle (orbit index 0). For/~ </z 0 (the side of the 1.05 period 4 attractor which has orbit index + 1) the fixed point is a repellor (orbit index + 1) there also exists a period 4 regular saddle (orbit index - 1). The regular saddle also shrinks to the fixed
point as ~ ~/~0. Hence, presumably we have a P
border-collision bifurcation from a period 4 at- tractor to a 1-piece chaotic attractor.
For A = 1 . 4 , B = 0 . 3 , C = 1 . 5 , and D = - 4 , the bifurcation diagram in fig. 7b exhibits a bifur- cation from an 8-piece chaotic attractor to a period 5 attractor, where ~ (plotted vertically) varies from 0.91 to 0.86. The border-collision
0.8 bifurcation occurs at ~ = tz 0 = 0.884. For ~ >/z 0 (the side of the 8-piece chaotic attractor) we do not know the (total) orbit index since the chaotic
0.91 attractor contains a lot of periodic orbits, and the fixed point is a flip saddle (orbit index 0). For
< ~0 (the side of the period 5 attractor which has orbit index + 1) the fixed point is a repellor (orbit index + 1); there also exists a period 5 regular saddle (orbit index - 1). The regular sad- p dle also shrinks to the fixed point as tz ~ ~0. In the figure one might first notice a bifurcation from an 5-piece chaotic at tractor to a period 5 attractor, but closer examination in the phase space gives the above mentioned bifurcation from an 8-piece chaotic attractor to a period 5 attrac-
tor. Hence, presumably we have a border-colli- 0.86 sion bifurcation from a period 5 attractor to an 8-piece chaotic attractor.
/ / ........... i ................. -0.5 x 2.0
I
0 x 2.0
Fig. 7. (a) Bifurcation diagram of
Example 6. Border-collision bifurcation from a
p-piece chaotic at tractor to a q-piece chaotic at- tractor. We present just one example, namely p = q = l .
For A = 1 . 4 , B = 0 . 3 , C = 1 . 2 , and D = - 4 , the bifurcation diagram in fig. 8 exhibits a bifur- cation from a 1-piece chaotic at tractor to a 1-piece chaotic attractor, where /.t (plotted vertically) varies from 0.95 to 0.85. The border-collision bifurcation occurs at ~ = lz 0 -- 0.884 and we only can say that on both sides infinitely many periodic orbits are involved in the border-collision bifurca- tion, since the attractors are chaotic. Hence, pre-
( 1 . 4 - x 2 + 0 . 3 y , x ) if x_</~,
f , (x , y ) = ~ (1 .4 + 1.1x + 0.3y - (/~ + 1.1)/.~, - 5 x + 6p,)
L if x >p ,
exhibits at /.% ~ 0.885 a border-collision bifurcation from a I-piece chaotic attractor to a period 4 attractor. (b) Bifurca- tion diagram of
F..(x, Y)= { (1 .4-x2 +O.3y, x) if x_</~,
(1.4 + 1.5x + 0.3y - (/,L + 1.5)p,, - 4 x + 5/~)
if x > ~ .
exhibits at /,% = 0.884 a border-collision bifurcation from an 8-piece chaotic attractor to a period 5 attractor.
50 H.E. Nusse, J.A. Yorke /Border-collision bifurcations for piecewise smooth systems
exhibits at /z 0 = 0.884 a border-collision bifurcation from a 5-piece chaotic attractor to a period 4 attractor.
H.E. Nusse, J.A. Yorke / Border-collision bifurcations for piecewise smooth systems 51
5-piece chaotic a t t ractor to a period 4 attractor,
where /z (plotted vertically) varies f rom 0.874 to 0.895. Hence, we may have a border-coll ision
bifurcation f rom a 5-piece chaotic at t ractor to a
coexisting 1-piece chaotic a t t ractor and a period 4
attractor.
Example 8. Anot he r simple border-collision bi-
furcation for the piecewise linear map GL~, is a
bifurcation f rom a fixed point a t t ractor to a
chaotic attractor. For a = 1.4, b = - 0 . 9 5 , c = - 1.04, d = 0.9, the bifurcation diagram in fig. 10a
exhibits a bifurcation f rom a fixed point at t ractor
to a chaotic attractor. The chaotic s trange attrac-
tor fo r / z = 0.05 is given in fig. 10b.
Now we consider an example in which the curve F , is the straight line y = - x + /z . In this
example we have a moving border. Let the map
H f rom the plane to itself be defined as above, that is, H(x , y) = (A - x z + By, x), and define the
map Gg where -oo < /z < oo f rom the plane to
itself by
Gg( x, y) = ( A - / z C - x 2 + Cx + ( B + C ) y ,
( l + D ) x + D y - l z D ) .
The regions R A and R B are the half planes to the left and the right of the curve F t . The map
we are investigating is defined being the H6non
map on R A and the map G~, on R B. Define the one -pa rame te r family of maps F~ from the plane
to itself by
H(x,y) i f x < - y + / z ,
F ~ , ( x , y ) = G g ( x , y ) i f x > _ - y + / z .
Notice that the map Fg is a piecewise smooth map. We present an example for which the map F , has a border-coll ision bifurcation f rom a pe- riod 8 a t t ractor to an 8-piece chaotic attractor. For A = 0.7, B = 0.3, C = 0.6, and D = - 2 , the bifurcation diagram in fig. 11 exhibits a bifurca- tion f rom a per iod 8 at t ractor to an 8-piece
-0.1
0.2 -0.4 x 0.6
0.13
-0.02
(b)
-0.1 x
Fig. 10. (a) Bifurcation diagram of
0.15
~ ( l .4x + y +/x, -0 .95x) if x<0 , GL~, (x ,y)=~(_ l .04x+y+/z ,0 .9x) if x>_0
exhibits at /z 0 = 0 a border-collision bifurcation from a fixed point attractor to a 1-piece chaotic attractor. (b) Chaotic attractor of
~(1.4x+y+lz,-O.95x) if x<0 , GL~,(x,y)= ~ ( - 1 . 0 4 x + y + # , 0 . 9 x ) if x > 0
when/z = 0.05.
52
0.53
v
0.5
H.E. Nusse, J.A. Yorke /Border-collision bifurcations for piecewise smooth systems
-1.0 X
Fig. 11. Bifurcation diagram of
f,,(x, y) =
1
(0.7-xz+0.3y,x) if x15 -y+p,
(0.7 - 0.6~ - x2 + 0.6x + 0.9y,-x - 2y + 2~)
if xr -y+p
exhibits at pa = 0.602 a border-collision bifurcation from a period 8 attractor to an 8-piece chaotic attractor.
chaotic attractor, where ~1 (plotted vertically) varies from 0.53 to 0.5. The border-collision bi- furcation occurs at w = Z+ = 0.519. Hence, we may have a border-collision bifurcation from a period 8 attractor to an &piece chaotic attractor, and if we just consider the 8-th iterate of the map at one of those 8 periodic points, we have a border-collision bifurcation from a fixed point attractor to a l-piece chaotic attractor. This latter case is similar to the bifurcation for the piecewise linear map mentioned above.
4. “Period two to period three” border-collision
biircation
In this section we explain why “period two to period three” border-collision bifurcations occur for two-dimensional piecewise smooth maps. Let a, b, c, and d denote real numbers. Define the one-parameter family GL, from the plane to
itself, by
GL,(x,y) = (ax+y,bx) +&l,O) if x50,
GL,(x,y) =(cx+y,dx) +k(l,O) if x20,
where p is in an open interval Z including zero. Recall that this family GL, is equivalent with the piecewise linear map P@.
Let FM be a one-parameter family of piecewise smooth maps which has a prototype piecewise linear form at p = 0, and assume that
-a > 1, -d2 -c> 1, (Al)
c2+4d<Osc2+d, (M)
O<a(ac+d) <l. (A3)
We want to show that there exists E > 0 such that if I bl < E, then the family F, has a “period two to period three” border-collision bifurcation at (0,0X First, we show that for b = 0, the family GL, has a border-collision bifurcation from a period 2 attractor to a period 3 attractor. We write C for the set of all one-parameter families of maps GL, defined above such that b = 0.
Proposition. At p = 0, every family GL, in C has a “period two to period three” border-collision bifurcation at (0,O).
Proof of the theorem. Assume that the proposi- tion has been proved. Apply the proposition and it follows immediately from perturbation results.
The proof of the proposition (given below) might give insight why other bifurcations (for example, period 5 to period 2 bifurcation) may occur in piecewise smooth systems. Presumably, the method of proof only works if one of the two maps involved has a zero Jacobian. We first show that a border-collision bifurcation occurs at p = 0, and we present an example to give an idea of the proof.
Let GL, be in C. The fixed point E, of F, is given by E, = (p/(1 - a),O> if p I 0 and E, =
H.E. Nusse, J.A. Yorke ~Border-collision bifurcations for piecewise smooth systems 53
(/z/(1 - c - d), dix / (1 - c - d)) if Ix > 0. In the notation of section 2, define the matrices M A and
M s by M A = ( g 1 ) , M B = ( t ~ 1). Theeigenval-
ues of M A are 0 and a, so if/x < 0 then the fixed point E~, is unstable since - a > 1. In particular, E~, is a flip saddle if Ix < 0. The eigenvalues of
M B are 0.5c + 0.5~T-+ 4d and are complex, since c 2 + 4d < 0. For Ix > 0 the fixed point E~, is unstable (repelling), since the product - d of the eigenvalues of M B exceeds 1. The nature of the fixed point E~, is changing from being a flip saddle (in region R A which is the left half plane) to a repellor with complex eigenvalues (in region R B) when the parameter IX is varied from say - 0 . 1 to 0.1. We conclude that a border-collision bifurcation occurs at Ix = 0 when Ix is continu- ously varied from some negative value to a posi- tive value, since the orbit index of E~, changes from 0 to + 1. For simplicity of the explanation of this border-collision bifurcation phenomenon, we offer the following example.
Example. Consider the one-parameter family g~, from the plane to itself, defined by
g ~ , ( x , y ) - - - ( 5 ~ x + y , 0 ) + i x . ( 1 , 0 ) if x<0,_
g~,(x, y ) = ( - 2 x + y , - ~ x ) + tx "(1,0)
if x_>0.
The bifurcation diagram exhibiting the "period two to period three" bifurcation, is similar to the diagram in fig. 1. The family of maps g~, is in the class C, so it is an example for which the result above applies. The idea why a "period two to period three" border-collision bifurcation occurs for the family g,,, is the following.
For IX < 0, write W~, for the interval [ - i x / a , oo) ~ix, oo) on the x-axis. We have (1) the image
g , ( p ) of each point p on the x-axis but not in W~, is in W,,, and (2) each point p in W~, is mapped to a point p* on the x-axis after two iterates, so
Fig. 12. The map g~, is defined by g~,(x, y) = ( - 1.25x + y + /z,0)if x _< 0, and g~,(x,y)=(-2x+y+tz,-2.625x)if x > 0. For /z <0 , the return map G defined by G(x)=g~(x,O), maps the interval [0 .8- /~,~)on the x-axis into the x-axis. The map G has an unstable fixed point Pu = ~ < 0 and a stable fixed point Ps = - ~/z > 0.
g~(p) =p*. In fig. 12, the graph of the corre- sponding return map G on W~, which is defined by G ( x ) = g2(x, 0), is given. To be more specific, G ( x ) = -~x - 1 4 ~IX for ~-ix_<x_<0 and G ( x ) =
1 1 - ~ x - zix for x > 0 . 4 The map G has two fixed points Pu = ~IX < 0
2 and Ps = - ~IX > 0. The fixed point Pu is unsta- ble since the slope of G in Pu is 25 ~ , and the fixed
1 point Ps is stable since the slope of G at Ps is a. The properties (1) _4IX <pu 4 5 = ~IX <0, (2) G has slope ~6 at x for 4 1 ~IX < x < 0, (3) G has slope - for x > 0, and (4) G(0)-- 1 - xix > 0, imply that g~, has a period 2 attractor consisting of the two points Pt -- ( - gix, 0) and P2 = g~,(P1) -- (~ix, 7ix). Notice that the norms of both these points converge to zero as IX goes to zero, that is, both IJPllJ~0 and ]lP2ll~0 as Ix ~ 0 . In other words, the period 2 attractor shrinks to a point as IX goes to zero; this point to which the period 2 attractor converges is the fixed point of g~, at IX = 0. For IX > 0, each point p on the X-axis is mapped to a point p* on the X-axis after three iterates, so g 3 ( p ) = p . . The graph of the corre- sponding return map H, defined by H ( x ) = g3(x, 0), is given in fig. 13. In particular, H ( x ) =
54 H.E. Nusse, J.A. Yorke ~Border-collision bifurcations for piecewise smooth systems
LL i .
Fig. 13. The map g~, is defined by g,~(x, y) = ( - 1.25x + y + p,,0) if x_< 0, and g~,(x,y)=(-2x+y+l.~,-2.625x) if x > 0. For ~ > 0, the return map H defined by H(x)=g~(x,O), maps the x-axis into the x-axis. The map H has one unstable fixed point Pu = 4/x > 0 and two stable fixed points qs =
- ~/z < 0 and P~ = ~/x > 0.
3"~X - - 3 13_~X 3 g~ for x < 0 , H ( x ) = - - - for 0 < g/z 1 21 1 x < ~ , and H(x) = ~x + -cgl~ for x > ~tz.
The map H has an unstable fixed point 4 Pu = ~/x > 0 and two stable fixed points qs =
4 - ~/x < 0 and Ps = ~/x > 0. Fur thermore , for all x with x <Pu we have l im, ,~=H"(x)=qs , and for all x with x > p , we have l im,_.= H"(x ) =ps. The propert ies (1) H has slope between 0 and 1
for x < 0 , (2) H has slope bigger than 1 for 1 0 < x < ~/~, (3) H has slope be tween 0 and 1 for
1 3 x > ~Iz, and (4) H ( 0 ) = - ~tz < 0 and H(½p. )= 1 -~/z > ~ , imply (using the formulas for &,)
g~, has a per iod 3 at t ractor consisting of the points $ 1 = ( - 4 = = ~ , 0 ) , S 2 (~t~,0) , and S 3
( - ~t~, ~ - ~ ) . Notice that the norms of all three points converge to zero as ~ goes to zero, that is,
all three IISlll--* 0, IIS211 ~ 0, and IIS311---> 0 as /z ~ 0. In o ther words, the per iod 3 at t ractor shrinks to a point as ~ goes to zero; this point to which the period 3 at t ractor converges is the fixed point of &, at ~ = 0. The point (4 /z , 0) is a point
of a per iod 3 orbit which is a regular saddle of the map g~,.
Conclusion: at Iz = 0, there is a "pe r iod two to period th ree" border-coll ision bifurcation.
Proof of the proposition. Let GL~, be a one-
pa ramete r family in the class C, where tz is in
some interval I. We write Po = (Xo, Yo) for an initial condit ion and Pn = (x , , y , ) for its n th i~er-
ate, that is, p , = GL~,(po), for each /z. For the
part icular initial value (0, 0), we write A 0 = (0, 0),
A 1 = GL~,(Ao) , A2 = G L ~ ( A l ) , A3 = GL~,(A2) , and A 4 = GLu(A3).
For each initial value Po = (Xo, Yo) we observe the following fact. I f x 0 < 0 then Yl = 0, and if
x 0 > 0 then y~ = dx 0 < 0. Hence, it is sufficient to consider initial values in the lower half plane.
Hence, f rom now on, we assume that Yo < 0. Assume first,/x < 0. Recall that the fixed point
E u = ( /~/(1 - a ) , 0) is unstable, and is a flip sad-
dle, since - a > 1. Assume that Po = (Xo, Yo) is any initial value with Yo -< 0. Then, if x 0 < 0 then
y l = 0 , and if X o > 0 , then x, =CXo+Yo+/ .~ < 0 and so Y2 = 0. Therefore , it is sufficient to con- sider points on the x-axis, and we will do so.
Consider the initial value p o = ( 0 , 0 ) = A o . Computa t ion of the first four iterates of A 0 yields
A 1 = ( / . L , 0 ) , A 2 = ((a + 1)tz,0), A 3 = ((c(a + 1) + 1 ) ~ , d ( a + 1)/z), and A 4 = ( ( a + 1 ) ( a c + d + 1)/x, 0). The assumptions 0 < a(ac + d) < 1 and
- a > 1 imply - 1 <ac + d < 0 yielding 0 < x 4 <
x 2. F rom - 1 < ac + d < 0, and the assumptions, - a > 1, and - c > 1 follows that c(a + 1) > 0 and
d < c; therefore Ix31 > iY3[. Hence, A 1 is on the x-axis to the left of A o, A 3 is under and to the
left of A 1, and both A 2 and .44 are on the x-axis to the right of A 0 and .44 is be tween A 0
and A z. First we consider the image of the x-axis. Let
Po = (Xo, Yo) be any point on the x-axis. The image of the right half of the x-axis with end
point A 0 is the half line th rough .43 with end
point A 1 --- GL~,(Ao), since Pl = (CXo +/~, dxo) for x 0 > 0. The image of the left half of the x-axis with end point A 0 is the half line on the x-axis to
the right of A 1 with end point A 1, since Pl =
(ax 0 +/~, 0) for Xo < 0. Define Q = ( - l ~ / a , 0 ) = ( X a , 0) and R =
(aZt~/(1 - a)(ac + d) ,0) = (XR,0). The point Q is mapped to .40 iterating G L , once, that is,
H..E. Nusse, J.A. Yorke / Border-collision bifurcations for piecewise smooth systems 55
GL~.(Q) = A o, and Q is on the x-axis between A 1 and E . since A 1 = (/Z, 0), E . = (/Z/(1 - a) ,0) and - a > 1. The point R is on the X-axis to the right of A o, and R is mapped to E . iterating G L . twice, that is, G L 2 ( R ) = E~.
Let P0 = (Xo,0) be any point. Straightforward computation gives the following. I f x 0 > 0 (that is,
Po is on the x-axis to the right of A o) then
Pl = (CXo +/Z, dxo) and P2 = ((ac + d )x o + (1 + a ) / z , 0 ) , SO P2 is on the x-axis. If x 0 = 0 (that is,
p o = A o ) then Pl = ( / z , 0 ) and P2 = ( ( 1 ÷ a ) / z , 0 ) ,
so P2 is on the x-axis to the right of A o. If - / z / a _<Xo<0 (that is, Po is on the x-axis be- tween Q and A o) then p ~ = ( a X o + / z , 0 ) and
P2 = (a(ax0 +/Z) +/Z,0), so P2 is on the x-axis. If x 0 < - / z / a (that is, P0 is to the left of Q)
then pl =(aXo + /z,O) and p2 =(c(axo + /z )+ /z,
d(ax 0 +/z)) and P3 = (( a2c + ad)xo + (ad + a + d + 1)/z, 0), and so P3 is on the x-axis while P2 is not. Summarizing, for each point P0 on the x-axis to the right of Q we have P2 = GL2(p0 ) is on the x-axis. Therefore, we have a return map on the interval consisting of the points on the x-axis to the right of Q.
Let G denote the return map of GL~, on [Q,~) , so G ( x ) = GL2(x ,0) for each x > XQ. The
above results imply G ( x ) = aZx + (1 + a)/z for
- / z / a < x < O, and G ( x ) = (ac + d )x + (1 + a)/z for x > 0. The graph of G is similar to fig. 12. The map G has two fixed points, namely Pu =
/z/(1 - a), and Ps = (a + 1)/Z/(1 - ac - d), and Pu < 0 <Ps. The fixed point Pu is unstable since the slope of G in PH is a 2 > 1, and the fixed point ps is stable since the slope of G at Ps is ac + d for which - 1 < ac + d < 0. Furthermore, for all
x with Pu < X < X R we have l imn_~=Gn(x)=ps. The propert ies (1) XQ </Z/ (1 - a) < 0, (2) G has slope a 2 > 1 if XQ < x < 0, (3) G has slope - l < a c + d < O for x > 0 , a n d ( 4 ) G ( 0 ) > 0 , im- ply that GL~, has a period 2 attractor consisting of the points P l = ( ( a + l ) / z / ( 1 - a c - d ) , 0 ) and P2 = GL~,(P1) = (c - d + 1)/Z/(1 - ac + d), (a + 1 ) d / z / ( 1 - a c - d ) ) . Notice that the norms of both these points converge to zero as/Z goes to
zero, that is, both [[PIII---> 0 and [IP2ll ~ 0 a s / z
0. Hence, the period 2 attractor shrinks to a point as /z goes to zero; this point to which the period 2 attractor converges is the fixed point of G L , at
/ Z = 0 . Now assume /Z = 0 . Assume Po-- (Xo, Yo) is
any initial value with Yo -< 0, then x o _< 0 implies Yl = 0, and x o > 0 implies xl = cx o yielding Y2 ---- 0. Hence, it is sufficient to consider points on the
x-axis. Let Po = (Xo,0) be given. I f x o < 0 then Pl = (aXo, 0) which is on the positive x-axis. If x o = 0 then Pl = (aXo,0) and so Po is the fixed point of GL o. I f x o > 0, then Pl - (CXo, dxo), a n d P 2 = ((ac ÷ d)xo , O). Consequently the point A o -- (0, 0) is a globally stable fixed point of GL o, since - 1 < ac + d < O.
Now assume /Z > 0 . The fixed point E , = (/Z/(1 - c - d), d/z / (1 - c - d)) is unstable with complex eigenvalues since it was assumed - d > 1
and c 2 + 4d < 0. Assume Po = (Xo, Yo) is any ini- tial value. Then x o _< 0 implies y~ = 0. Now as- sume that x o > 0. Since the fixed point E , is repelling with complex eigenvalues, there exists a smallest positive integer N such that x N _< 0. Hence, YN+~ = 0. Therefore, it is sufficient to consider points on the x-axis.
Let Po = (Xo, Yo) = (Xo,0) be any point on the x-axis. I f x o _< 0 then Pl -- GL~(po) = (aXo +/z , 0) = (X1, Yl) , SO X 1 > 0. Every point qo = (Wo, 0) such that w o < x 0 _< 0 satisfies ql = GL, (qo) = (aWo + /Z,0) = (w 1, zl), so w~ >x~ > 0. The conclusion is that points on the x-axis to the left of A o = (0,0) are mapped monotonically into the x-axis to the right of (/Z, 0).
Let Po = (0,0). A simple computat ion shows
P l = (/Z, 0) , P 2 ----" ((1 + c)/z, d/z), P 3 = ((ac + a + d + 1)/Z, 0), and P4 = (axa +/Z, 0). Notice x 3 < 0,
hence x 4 >/z = x 1. Recall that Po =Ao, Pl =A1,
P2 =A2 , P3 =A3, and P4 =A4. The conclusion is that A o, A1, A3, and A 4 are on the x-axis, and A 3 is to the left of A o, and both A 1 and A 4 are to the right of A 0 with A 1 between A 0 and A 4.
Let P0 = (x0, 0) be any point on the x-axis for which x 0 > 0. Then Pl = (CXo +/Z, dx0). Notice that if x o = - / z / c then x I = 0 and Yl = - d / z / c . Write Bo = ( - /z / c , O) , B 1 = GL~,(B0) , B 2 =
56 H.E. Nusse, J.A. Yorke /Border-collision bifurcations for piecewise smooth systems
GL~,(B1), and B 3 = GL~,(B2). Then B 1 = (0, - d l ~ / c ) , B 2 --- ((1 - d/c ) l z , 0), and n 3 = ([a(1 - d / c ) + 1]/~, 0). Notice that B 1 denotes the point on the y-axis at which the line segment [A~,A 2] intersects the y-axis, and that B 2 is a point on the x-axis to the left of A 0. The assump- tions - a > 1 and 0 < a(ac + d) < 1 imply ac + d
< 0 and we obtain that the point A 3 --- ((ac + a +
d + 1)~, 0) is on the x-axis to the left of B 2. The image of the half line [A1, oo) through A 2
under the map GL~ is the kinked half line [h2, B 2] U [B2,oo) through h 3. The image of this kinked half line is on the x-axis. In particular, the image of the half line [B2,oo) through h 3 is [B3,oo) on the x-axis to the right of A 1 = (~,0), and the image of the line segment [Az, B 2] is
[Aa, B3]. Let p o = ( x o , O) be any point on the x-axis.
Straightforward computation shows the following.
If x o > - I z / c (that is, P0 is to the right of B 0) then p~ = (cx o + I~, dxo), P2 = ([ac + d]x o + (a + 1)/z,0), and P3 = (a[ac + d]x o + [a(a + 1) + 1]. /z,0). Hence, both P2 and P3 are on the x-axis for X o > - I ~ / c . If 0 < x 0 < - t z / c (that is, P0 is on the x-axis between A 0 and B 0) then 191 = (¢X 0 d- /~, d x 0 ) , 19 2 = ([C 2 d- d]x o + (c + 1)~, cdx 0+d/x) , and since (c 2 + d ) x o + ( c + 1 ) ~ <
- c l z - ( d / c ) ~ + c/z + ~ = (1 - d /c ) l z < 0, we have P3 = ([ ac2 + ad + cd]x 0 + [ac + a + d + 1]/z,0), so the point P3 is on the x-axis. If Xo < 0
then P l = (aXo + lz, O), P2 = (aCXo + (c + 1)/~, adx 0 + dtz), and P3 = (a[ac + d]x 0 + [ac + a + d + 1]/~, 0), so the point 19 3 is on the x-axis. The conclusion is that for each point Po-- (Xo, 0) on the x-axis, the third iterate of Po is also on the x-axis, that is, GL3(Po) -- (x 3, 0). Hence, a return map of GL~ exists on the x-axis. We call this return map H, so H ( x ) = GL3(x,0). The above results imply
and
H ( x ) = (a2c + a d ) x + (a 2 + a + 1) ./~
1 f°r x° >-- - c "/~"
The graph of H is similar to fig. 13. The map H has three fixed points, namely
qs = (ac + a + d + 1 ) / z / [ 1 - a ( a c + d ) ] < 0 ,
Pu = - ( ac + a + d + 1 ) l~ / [ c ( ac + d) + a a - 1],
and
p~ = (a 2 + a + 1) /z / [1 - a ( a c + d ) ] > 0.
The fixed point Pu is unstable since the slope a c 2 + ad + cd of H in Pu is bigger than 1, and the two fixed points qs and Ps is stable since the slope a2c + ad of H at both qs and Ps is between 0 and 1. Furthermore, for all x with x <Pu we have l im, _.® H n ( x ) = qs, and for all x with x >pu we have l i m , _ ~ H ' ( x ) = p s. The properties (1) H has slope between 0 and 1 for x < 0, (2) H has slope bigger than 1 for 0 < x < - t z / c , (3) H has slope between 0 and 1 for x > - I ~ / c , and (4) H ( 0 ) < 0 and H ( - I ~ / c ) > - i x ~ c , imply that GLg has a period 3 attractor consisting of the points
S, = (( ac + a + d + 1)Ix/[1 - a( ac + d ) ] , 0 ) ,
S 2 = ( ( a 2 + a + 1) /z / [1 - a ( a c + d ) ] , O ) ,
and
S 3 = ( { c ( a 2 + a + 1 ) / [ 1 - a ( a c + d ) ] + 1} "ix,
d(a 2 + a + 1) /z / [1 - a(ac + d ) ] ) .
H ( x ) = (a2c + a d ) x + (ac + a + d + 1) "tz
for x < 0 ,
H ( x ) = (ac 2 + a d + c d ) x + (ac + a + d + 1 ) . g
for 0 < x < - I x ~ c ,
Notice that the norms of all three points converge to zero as p, goes to zero, that is, all three
IIS~ll--, 0, 1[S2[[ ~ 0, and IlSal[ ~ 0 as /x ~ 0. Hence, the period 3 attractor shrinks to a point as /z goes to zero; this point to which the period 3
H.E. Nusse, J.A, Yorke / Border-collision bifurcations for piecewise smooth systems 57
attractor converges is the fixed point of GL~, at
/ z = 0 . The point (pu,0) is a point of a period 3 orbit
which is a regular saddle of the map GL~,. We
conclude: at ~ = 0, there is a "period two to
period three" border-collision bifurcation. This completes the proof of the proposition.
5. Discussion and concluding remarks
Question 2. More generally, is it possible to give a classification of the border-collision bifur-
cations for the piecewise linear map P~,? Question 3. When the plane is subdivided in N
regions, where N is at least 3, do there exist
border-collision bifurcations that do not occur when there are only 2 regions, and in particular
bifurcations that persist despite small perturba-
tions?
We have presented bifurcation phenomena, which we call "border-collision bifurcations".
These bifurcations occur when the nature of a fixed point (or periodic point) of a piecewise
smooth system changes when it collides with the border of two regions. An interesting case occurs
when the fixed point changes from being a flip saddle to a repellor with complex eigenvalues at
the parameter value where it collides with the
border of two regions. We have presented a vari- ety of examples based on the piecewise linear
map P~ and the H6non map. In particular, we
have shown the occurrence of a "period two to period three" border-collision bifurcation for
maps in the class C.
We point out that the border-collision bifurca-
tion can be expected to occur in many piecewise smooth models. In particular, the "period two to
period three" bifurcation phenomenon can be expected to occur in many linear models with
constraints. Assume for the piecewise linear map P, that
the fixed point E~ is a flip saddle in the left half plane and a repellor with complex eigenvalues in the right half plane,
Question 1. Does there exist a classification of the border-collision bifurcations for P, in the
case where a period 2 attractor converges to the fixed point (0, 0) when /z goes to 0?
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