The Dynamics of Hybrid Systems
by John C. Eckalbar
Department of Economics
CSU, Chico
February 2009
This paper is concerned with the dynamic properties of what have come to be
called “hybrid” or “switching” systems. Without getting overly specific yet, a hybrid
system often has the following general properties: The dynamic variables are given by
the vector z  R n , and z (t) is the vector giving dz/dt. Define a switching surface S = {z |
s(z) = 0}, where s(z): Rn → R is a continuous function which divides Rn into regions Sa =
{z | s(z) < 0}, Sb = {z | s(z) > 0}, and S. In a hybrid system, z (t) is governed by one set
of equations when z (t)  Sa and by a different set when z (t)  Sb , and there is often a
discontinuity in the vector field at S. We will leave aside for the moment the value
of z (t) on S.
Figure 1 illustrates a sample hybrid. If the initial point z(0) is in Sa, the positive
semi-path from z(0) will be governed by fa, where z (t)  f a (z (t)), f a : Sa  R n .
Suppose that fa drives z(t) onto S at t = t1, then fb (defined similarly) may take over and
drive the system back onto S at t = t2 > t1. And so on.
Exploring a system of this sort raises a number of interesting questions. Is
stability of the fa and fb subsystems sufficient to ensure stability of the hybrid? Is
subsystem stability necessary? Could two stable systems form an unstable hybrid? Can
the hybrid exhibit more exotic behavior (like limit cycles or chaos) than either subsystem
individually?
1
Among economists, the original motivation for study of switching systems came
from disequilibrium and monetary theorists. Imagine, for example, that we build a model
that allows trading outside equilibrium, and that output, Q, is a function, F, of the actual
quantity of labor hired, L. Suppose that labor demand, Ld, is given by F-1(e), where e is
the firm’s expected sales, and labor supply, Ls, is a function of the real wage, G(w). If
the actual quantity of labor hired is the minimum of labor supply and labor demand, we
S
Sb
z(t1)
z(t2)
Sa
S
z(0)
Figure 1
have Q = F(min{ F-1(e), G(w)}) = min{e, F(G(w))}. Thus, changes in Q would be
governed either by changes in e or by changes in w, depending upon whether Ls is
greater or less than Ld. In a simple model such as this, the switching line would be {(e,
w) | F-1(e) = G(w)}. (See Varian 1977, Eckalbar 1981, and Ito 1980.) Similarly, if we
modeled a liquidity constrained, cash on demand, economy, we would have different
2
dynamics according to whether or not cash constraints are binding at our current position
in the phase space. (See Caballe, Jarque, and Michetti, 2006.)
Other disciplines are also actively involved in the study of hybrid systems. In
physics, Murry Milgrom has offered an alternative to the dark matter hypothesis used to
explain the observed anomaly in the rotation of galaxies. Under Milgrom’s theory,
Newtonian dynamics (F = m a) is replaced by F = m μ(a/a0) a, where |a| = a and μ(a/a0) =
1 if a > a0 and μ(a/a0) = a/a0 otherwise. Hence, the classical equation is revised at low
accelerations. (Milgrom, 1983.) And in mathematics, engineering and control theory,
the study of systems with friction encounter similar problems. (Stewart and Anitescu,
2005 or di Bernardo, et al., 2003.)
In this paper we initially study the stability properties of a two-dimensional
switching system composed of two linear subsystems with a common equilibrium and a
linear switching surface that passes through the equilibrium. We explore a number of
issues, among them the following: if A and B are stable matrices, i.e., have eigenvalues
with negative real parts, is that sufficient for stability of the patched together hybrid
system, or is there something in the interplay as the governing systems switch that can
lead to instability?
We are able to show that stability of the A and B subsystems separately is
sufficient, but not necessary, for stability of the hybrid system. And further, we are for
the first time able to identify the necessary and sufficient conditions for stability. Our
method of proof, following Poincare, involves transformation of the patched together
differential equation switching system in R2 into a companion difference equation system
on S.
3
We later explore the effect of alternative assumptions regarding S. The most
interesting and novel feature of this analysis is that we model a process where switching
takes place gradually throughout a two-dimensional switching region, so that the effect
from one subsystem gradually diminishes while the effect of the other grows as the path
transitions from the domain of one subsystem, through the switching region, and into the
domain of the next subsystem. This is no doubt quite plausible in certain applications,
and it has the advantage of producing a continuous vector field. It can also give rise to
fascinating dynamics, including multiple locally stable equilibria and limit cycles.
The layout of the paper is as follows. In section I, we organize notation, establish
some definitions, give preliminary assumptions, and review some basics regarding
solutions for two-dimensional linear autonomous systems. In section II, we review some
pertinent facts regarding simple non-hybrid two dimensional linear dynamical systems,
and we investigate the manner in which each subsystem can be transformed from a
differential equation to a companion difference equation system, which provides a
“strobe-like” view of the path of the differential system over a succession of fixed time
intervals. In section III we show that by carefully selecting the unit time interval, we can
create a single continuous difference system from the hybrid switching system. Once this
is accomplished, our main theorem is easily established. In section IV, we explore
alternative assumptions on the nature of the switching line. And in section V we consider
a system which has a two dimensional switching surface with gradual transition from one
regime to another.
4
I. Preliminaries
We will model a two dimensional linear system, where our variables are x and y.
For ease of notation, we will sometimes use z = (x, y) and z  (x , y ) . We begin with the
specification of the switching surface, S,
Definition S: Let S = {(x, y}| s(x, y) = - k ∙ x + y = 0, (x, y) ≠ (0, 0)}, Sb = {(x, y)| s(x,y)
= - k ∙ x + y > 0}, and Sa = {(x, y)| s(x, y) = - k ∙ x + y < 0}, k  R.
We
will
shortly
have
occasion
to
use
s  (s/x, s/y)  (k, 1), which is the gradient of s. The gradient of the function s(x,
y) is a vector orthogonal to S and pointing into the half-space Sb .
Assumption M: We assume that z (t)  A z (t)  z (t)  Sa, and z (t)  Bz(t)  z (t)  Sb. A
and B are real, distinct 2 × 2 matrices with constant coefficients.
We now discuss the behaviour on S, beginning with a few definitions:
Definition AP: We will say that two subsystems, A and B, “agree perfectly” on S if
x 
x 
A i   B i  for all (x i , y i )  S or equivalent ly Az i  Bz i for all z i  S.
 yi 
 yi 
Under AP, at every point z in S the vectors Az and Bz have the same direction
and magnitude. Assumption AP may seem quite strong, but it is sensible in a wide class
of economic models. (For more elaborate models where AP holds, see Eckalbar 1981
and 1985.) An obvious advantage of AP is that it results in a continuous vector field
everywhere in R2.
A less stringent assumption might be that two systems “weakly agree
directionally,” which we define as follows:
5
Definition WAD:
Two subsystems, A and B, will be said to “weakly agree
directionally” at point z i  S if sgn s  Az i   sgn s  Bz i , where the term in square
brackets is the scalar product of a vector normal to S, i.e. the gradient of S or (-k, 1), with
the vector giving the motion of the A or B subsystem at zi. When two subsystems satisfy
WAD, then they both point across S to the same side of S, i.e., either both flow into Sa or
both into Sb. Under WAD, the two vectors do not have the same exact direction and
magnitude, and as a result the hybrid may have a vector field which lacks continuity at S.
Note that if the two vectors Azi and Bzi point into Sb, then under WAD
sgn s  Az i   sgn s  Bz i   1,
and if they both point into Sa,
sgn s  Az i   sgn s  Bz i    1.
Let us pause to consider those facts. Figure 2 gives the sense of Definition WAD.
The figure shows a subset of S near the origin.
s is a vector normal to S, with
coordinates (-k, 1). The vector (1, k) is shown along S, since S is defined by y = k x.
Note that these two vectors are orthogonal and that their scalar product is zero. The
vector s is also translated to point zi = (xi, yi). The vectors Azi and Bzi give the flow of
the A and B subsystems (respectively) from point zi. Both systems flow into the domain
of the B subsystem, Sb, i.e., they agree directionally. The angle between vector Azi and
s is δ, and the angle between Bzi and s is γ. Note that if the angles are acute, both
systems must point toward Sb. Also, both the angles are acute if and only if both Cos(δ)
and Cos(γ) are positive. The scalar product of the vectors s and Azi is given by | s |
|Azi | Cos (δ), so the sgn[ s ∙ Azi] = sgn[Cos(δ)] = 1. Hence, if both scalar products
s ∙Azi and s ∙Bzi are positive, they both point into Sb, i.e., they agree directionally.
6
Similarly, if both had negative sign, they would agree directionally and point into Sa. If
the signs of the two scalar products are opposite, then the two systems “disagree” on the
direction of flow on S.
s
Sb
s
Bz
S


1
Az
zi = (xi , yi)
(1, k)
s)
S
-k-
1
Sa
Figure 2
The next result shows that if two subsystems have AP or WAD at any point on S,
then they have those properties at every point on S.
Theorem A: If A and B are matrices with fixed real coefficients and S is as
defined by Definition S, then if there is any point on S at which the two subsystems
weekly agree directionally, then they weakly agree directionally at all points on S.
Proof: We assume that we have a single point (xi, yi) in S with


 x 
 x 
sgn ( k ,1)  A i   sgn ( k ,1)  B i  . But since (xi, yi) is in S, we can wright
 yi 
 yi 




1 
1 
sgn (k ,1)  A  xi   sgn (k ,1)  B  xi  , which could be written
k  
k  


7


 1 
 1 
sgn (k ,1)  A  sgn[ xi ]  sgn (k ,1)  B  sgn[ xi ].
 k 
 k 


But then it is obvious from
inspection that if the above holds for some xi, it will also hold for any xj, j ≠ i. Hence, if
the subsystems weakly agree directionally anywhere on S, they weakly agree everywhere
on S. q.e.d.
Corollary A: If WAD is violated at (xi, yi)  S, then WAD is violated at all other points
in S.
Proof: Suppose not. Then there is a (xi, yi)  S not satisfying WAD and another (xj, yj)
 S, i ≠ j, which satisfies WAD. But by Theorem A, if (xj, yj) satisfies WAD, so do all
other points in S. A contradiction. q.e.d.
Since systems satisfying AP are a subset of all systems satisfying WAD, the
Theorem A and Corollary A also apply when WAD is replaced by AP.
There are two types of directional disagreement:
Type I. Near S, Az points toward Sa, while Bz points toward Sb. In the
case of Type I disagreement, all points off S but near the point of disagreement on
S flow away from S. Thus, we would never observe a point on S with Type I
disagreement except as an initial condition.
Systems are sometimes said to
“branch” in this case.
Type II. Az points toward Sb, while Bz points toward Sa. In this case
points on S will have the neighboring vector field pointing toward S from both
sides. This is likely to lead to “sliding” along S. In fact if there is Type I
8
disagreement on the side of S where x > 0 (x < 0), there will be Type II
S
Sb
0
Sa
S
Figure 3.
disagreement on S when x < 0 (x > 0). See Figure 3.
The upshot of the above is that if WAD holds, trajectories will cross S, while if
WAD does not hold, paths will not cross S. If WAD does not hold, a path beginning in
Sa (Sb) will stay in Sa (Sb), and there will be no switching from one regime to another.
Hence, stability will be completely determined by the initial condition and the stability
properties of the host sub-system. Since we are interested here in switching, we make the
following:
Assumption WAD: The two subsystems weakly agree directionally.
9
1
y
S
0.75
θ
0.5
0.25
-1
-0.5
θ
θ
0.5
1
1.5
x
-0.25
-0.5
-0.75
Figure 4
The above results follow critically from the linearity of the A and B subsystems
together with the linearity of S and the fact that S passes through the origin. Consider
Figure 4. Notice trajectories passing through the right side of S, where x > 0. It is easy
to verify that all paths leave this segment of S at the same angle and that the magnitude of
the vectors diminishes toward zero as x drops toward zero. At the other end of S, where
x < 0, the vectors are directed 180 degrees from the right-hand side vectors, and their
magnitudes increase as x → -∞.
We are now ready to specify our hybrid dynamic system:
10
 x (t) 
 x(t) 

  A
 if (x, y)  Sa  {(x, y) | y  k  x} ,
 y (t) 
 y(t) 
 x (t) 
 x(t) 

  B
 if (x, y)  Sb  {(x, y) | y  k  x}, and
 y (t) 
 y(t) 
(W)
 x (t) 
 x(t) 

  B
 if (x, y)  S and sgn s  Az i   sgn s  Bz i  1
 y (t) 
 y(t) 
 x (t) 
 x(t) 

  A
 if (x, y)  S and sgn s  Az i   sgn s  Bz i    1
 y (t) 
 y(t) 
System (W) has the property that when a trajectory reaches S, its flow is instantly
handed over to the subsystem toward which it is heading. In other words, the “receiving”
system determines the flow at z on S.
We turn now to a brief review of results from the theory of isolated linear systems
which are essential to establishing our first main finding.
II. Simple Linear Non-hybrid Systems
We begin by defining some notation and exploring the means by which a
conventional (non-switching) linear autonomous differential equation system in R2 can be
transformed into a companion difference equation system. For the moment, assume that
the domain of A is all of R2. Thus we have system (1):
(1)
a 
a
 x (t ) 
 x(t ) 

  A
 , where A   11 12  .
 y (t ) 
 y(t ) 
 a21 a22 
The aij are assumed to be real with at least one non-zero entry.
Assume for the moment that the eigenvalues of A are real, and consider a line, L,
which contains the origin and an eigenvector of A. In this case, all trajectories beginning
from points on L will remain on L, and no points originating off L will ever cross L. In
11
contrast, if the eigenvalues of A are complex or complex conjugates, then trajectories on
the non-hybrid, stand alone A subsystem spiral around the origin and cross L infinitely
many times.
Though we are not yet adding a B subsystem and exploring regime
switching, it should be clear that the opportunities for switching are quite limited when
the eigenvalues of A are real. Thus, switching is much more problematic and interesting
when the eigenvalues contain imaginary parts. In the interest of studying the more
interesting case, we will assume:
Assumption R: The eigenvalues of A (and, with suitable changes in notation, later
B) are complex conjugates, λA1 = αA + βA·i and λA2 = αA - βA·i (with βA > 0 and i =
 1 ).
(Where clarity is not compromised, we will simplify our notation in this section
by dropping the A subscript as long as it is clear that we are referring to matrix A and its
associated eigenvalues, eigenvectors, and other related terms and parameters.)
Define the real matrix V such that
1  v
V    11
 i   v12
v21 1
 
v22  i 
is an eigenvector of A corresponding to λ1 = α + β i. Be sure to note that all vij are real
numbers. Now define the real and imaginary parts of the eigenvector by
v 
v 
V1  Re( V)   11  and V2  Im( V)   21 .
 v12 
 v22 
Then the general solution for the system is given by:
(2)
 x(t ) 

  [(c1  V1  c 2  V 2)Cos(  t )  (c 2  V1  c1  V 2) Sin (  t )]et ,
 y (t ) 
12
where the scalars c1 and c2 are determined by the initial values of x and y, denoted x0
and y0.
In some applications, it is more convenient to write equation (2) as:
(2' )
 x(t )   c11 
 c12 

  [
Cos(  t )  
 Sin (  t )]et ,
 y (t )   c 21
 c 22 
where
c11  c1 v11  c 2 v 21
c 21  c1 v12  c 2v 22
c12  c 2 v11  c1 v 21
c 22  c 2 v12  c1 v 22.
The system is asymptotically stable if α < 0, unstable if α > 0, and has closed
elliptical orbits when α = 0.
While still limiting ourselves to the behavior of a single system (A) without
switching, let us establish a useful fact about the time required to complete a 180 degree
(π radians) revolution about the origin. We will look at the time required to move from
one point on the line L = {(x, y)| g x – y = 0} at time t = 0, to the next point where the
path hits the line.
Whenever (x(t), y(t)) is on L, we have y(t) = g·x(t), which is equivalent to
(4)
[(c1  v12  c 2  v 22 )Cos(  t )  (c 2  v12  c1  v 22 ) Sin (  t )]et 
g[(c1  v11  c 2  v 21 )Cos(  t )  (c 2  v11  c1  v 21 ) Sin (  t )]et ,
or
(5)
[(c1  v12  c 2  v 22 )  g (c1  v11  c 2  v 21 )]Cos(  t ) 
[(c 2  v12  c1  v 22 )  g (c 2  v11  c1  v 21 )]Sin (  t )  0.
Without loss of generality, suppose that (x0, y0) is on L. Now since, x0 =
c1 v11  c2  v21 , and y0 = c1 v12  c2  v22 , it follows that the first term in equation (5),
[(c1  v12  c2  v22 )  g(c1  v11  c2  v21)] , equals zero.
Thus we can simplify (4) and
conclude that a trajectory emanating from (x0, y0) crosses L whenever
13
(5)
[(c2  v12  c1 v22 )  g(c2  v11  c1 v21)]Sin( β t)  0.
This implies that a path crosses L when Sin(βt) = 0.
Since Sin(h π) = 0 for all integer values of h, we see that the trajectory coming
from (x0, y0) crosses L when t = 0, π/β, 2π/β, 3π/β,… It follows that the path (x(t), y(t))
always takes the same amount of time as it spirals from one side of L to another. If the
path “ticked” every time it crossed L, it would sound like a metronome.
Consider the path of x(t):
x(t )  [(c1 v11  c2  v21 )Cos(  t )  (c2  v11  c1 v21 )Sin(  t )]e t .
As t advances from 0 to π/β to 2π/β to 3π/β, Cos(βt) alternates from 1 to -1 to 1, so the
value of x(t) at successive crossings of L is given by
x(t )  [(c1  v11  c2  v21)( 1) t/ ]e t  x0(1) t/ e t
t  0,  /  , 2 /  , 3 /  , ...
If we change our time unit from t to T = tβ/π, then (x(t), y(t)) re-crosses L
whenever T = 0, 1, 2, 3,…
Following Poincare’s idea of a “section,” this fact can be utilized to derive a
difference equation on L for x which takes the form
(6)
απ
β
x(T)  e x(T  1)  (1) e
T
απT
β
x0
T  1, 2, 3...
This difference equation yields successive values of x(.) as the trajectory crosses and recrosses L. Clearly, with β > 0, α < 0 means that x(T) → 0 as T → ∞, which is the same
as the stability condition for the continuous differential equation system.
With obvious changes, it is easy to show that
(7 )
απ
β
y(T)  e y(T  1)  (1)  e
T
απT
β
y0
for y(T) on L. Thus, the companion discrete system to (1) is
14
(8)
 
 x(T )    e 

  
 y(T )   0

T


T

x
(
T

1
)



(1)  e 
0 


  
 
  y (T  1) 

e 
0


 x0 
  T 
 y 0 .
 
(1)T  e  
0
Note that the origin is an asymptotically stable equilibrium to the difference
system (8) if and only if α < 0, and that the differential equation system defined by A in
(1) also has a stable equilibrium at the origin if and only if α < 0. This means that the
differential equation system has exactly the same stability requirement as its companion
difference equation system.
One other point of interest: With β > 0, a system that begins at (x0, y0) with t = T
= 0 will complete a 180 degree revolution around the origin at t = π/β (or T = 1) with both
x0 and y0 then equal to eαπ/β times their original values. Thus, if α < 0 (α > 0), each 180
degree revolution ends closer to (further from) the origin than its starting point. And the
“strength” of the stability (or instability) will be governed by the magnitude of the ratio
of α to β. This will be useful later.
As an aid to intuition, Figure 5 provides a sketch of the graph of the differential
equation solution (x(t), y(t)) together with dots for each point visited by the companion
  .4 4 
 and the initial
difference equation with initial. The assumption is that A  
  2  .3 
value is (1, 1).
As we will see in the next section, these facts are very useful in the analysis of
switching systems like (W).
The above discussion dealt with repeated crossings of a line L by a single linear
autonomous system. In the following section, we return to a discussion of switching on
the line S, but now we know how long each subsystem takes between the time it initiates
15
governance of z(t) and the time it surrenders control to the other subsystem. And, more
importantly, we know how much closer (further) we are to (from) the equilibrium at each
switching time.
L
t = 0; T = 0
t = 2/b; T = 2
t = /b; T = 1
Figure 5.
III. A Necessary and Sufficient Condition for Hybrid Stability
The main result now comes easily.
Theorem W: Given the WAD and R assumptions, system (W) has a unique
asymptotically stable equilibrium at (0, 0) if and only if αB/βB + αA/βA < 0.
Proof: Consider the path (x(t), y(t)) from the initial point (x0, y0) on S, and
without loss of generality, assume that (x(t), y(t)) initially moves from (x0, y0) into Sa.
We know that (x(t), y(t)) remains in Sa and is governed by A until t = π/βA. At this time
a A
x(.)  e
A
a A
x0 and y(.)  e  A y0. Given our WAD assumption, system B then takes
over and drives (x(t), y(t)) through Sb and back onto S, hitting S at time t = π/βA + π/βB,
16
 B
at which time x(
y(


 )  (e  )( e
B  A
B
A 
A
) x0  e M x0, where M 
 B  A

.
B
A
And



)  e M y 0.
B  A
Let T = t /( π /βA + π /βB), which is the period for a complete 360 degree rotation
from (x0, y0) on S, then through Sa, across S, through Sb, then back to S. We now have
the difference equations x(T )  e M T x0 and y(T )  e MT y0 , T = 0, 1, 2, 3…, which give
the x and y co-ordinates of every other point of intersection between the S and the (x, y)
trajectory as determined by the switching system (W). Thus, (x(T), y(T)) → (0, 0) as T→
∞ if and only if M < 0. q.e.d.
Thus, since βA and βB are positive, stability of the two subsystems (i.e., αA, αB <
0) is a sufficient condition for the stability of (W). But interestingly, it is not necessary
that both subsystems be stable—all that is necessary for stability of W is that M be
negative. If, for example, A is “very stable,” meaning αA < 0 with |αA|/βA being “large,”
then B can be “weakly” unstable, meaning αB > 0 with αB/βB being “small” in the sense
that αB/βB < |αA|/βA.
Note that αi/βi plays a role in the stability of the hybrid system, whereas in a nonhybrid system only αi matters for stability. In non-hybrid systems, the imaginary parts of
the eigenvalues (the βi) only determine whether or not the solution path rotates around the
equilibrium. With the hybrid, the situation is more complex. Recall that the β i term
determines the average angular velocity of the solution path, it governs the length of time
it takes a subsystem to rotate 180 degrees from a starting point on S at, say, t1 to the next
arrival on S, at t2. Specifically, the time required is π/βi, and the system arrives on S at t2
at a distance from (0, 0) which is equal to e  i /  i times its distance at t1. If αi > 0, the
17
system will be further from the origin when it re-intersects S at time t2. But the smaller βi
happens to be (with a given αi), i.e., the more time the system takes between t1 and t2 in
the unstable region, the further the path will be from the origin at t2. So a small βi
coupled with αi > 0 can lead to hybrid instability, whereas a large βi and resulting quick
transition time through the i region may still allow hybrid stability with one αi being
positive.
(The above implies that W has a unique, continuous solution z(t, z(0)), relying on
the fact that linear autonomous differential equations have unique, continuous solutions.
See Guckenheimer and Holmes, p. 8. Note that we would run into trouble, however, if
Az disagreed directionally with Bz on S. But even then, other plausible assumptions can
yield unique continuous solution paths. See Ploycarpou and Ioannou’s discussion of
Filippov solutions, 1993.)
The above result depends critically upon the assumption that the eigenvalues of A
and B are complex conjugates. Suppose, for example, that the eigenvalues of A and B
are real and distinct, with the eigenvalues of A given by λA1 and λA2 and for B by λB1 and
λB2. Suppose also that λA1 < λA2 < λB1 < 0 < λB2. Then there will be an eigenvector for B
associated with λB2 giving a line through the origin with flow away from the origin in two
directions which are 180 degrees apart. There is then no possible position for S which
prevents the hybrid from being unstable. So with real distinct eigenvalues, stability
requires λA1, λA2, λB1, λB2 < 0.
IV. Alternative Assumptions on the Nature of the Switching Surface
The main result at this point depends critically upon the assumptions regarding S,
i.e., that S is linear, that it passes through a unique equilibrium point that is common to
18
the A and B subsystems, and that the systems “agree” on how W should move on S.
These assumptions are certainly defensible in many cases (see Eckalbar 1985), but it is
interesting to see how the dynamics of the hybrid system are affected by other
assumptions. We begin with the alternative assumption that S is piecewise linear.
It is known that when our initial assumptions on S are relaxed, two asymptotically
stable subsystems can be patched together to form an unstable hybrid system and that two
unstable systems can patch to a stable hybrid. (Utkin, 1977.) Many other interesting
outcomes are possible. We begin with a pair of examples from Branicky (1998) which
illustrate the first two cases.
In the following we will assume that S is piecewise linear, with the switching line
bounding the first quadrant, running horizontally along the x axis when x > 0 and
vertically along the y axis from x = 0 upward. That is, S = {(x, y)| x ≥ 0 and y = 0} U
{(x, y)| y ≥ 0 and x = 0}. Furthermore, we will assume that B is the operative subsystem
in quadrant 1, while A is operative elseqhere.
Let our piecewise linear system be
governed by:
  1 10 
  1 100 
 and B  
.
( PL1) A  
  100  1
  10  1 
Note that on the vertical leg of S, the two subsystems agree that the flow should
be into quadrant 1, and on the horizontal leg of S, they agree that the flow should be into
quadrant 2. For instance, at (0, 1), the A subsystem has (x , y )  (10,  1) , while the B
subsystem has (x , y )  (100,  1) . Thus, both systems “push” from the vertical leg of S
into quadrant 1, but there is a discontinuity in the magnitude and exact direction of the
push at the border. The details of the process being modeled should dictate whether the
19
instantaneous flow on S should be governed by A, B, or some combination of the two.
As before we assume that the “receiving” system takes over on S.
The eigenvalues for both A and B are λ = -1 ± 101.5∙i, so both systems have
asymptotically stable spiral points at (0, 0). Also, both systems have ellipse-like orbits,
but the orbit on A has its “major axis” oriented vertically, while B’s is oriented
horizontally, as you see in Figure 6.
A Sub-system
B Sub-system
Figure 6.
The resulting hybrid system together with the switching boundary is shown in
Figure 7. The hybrid is unstable in spite of the fact that the two subsystems are stable.
Speaking loosely, this is due to: (i) the degree of eccentricity of the ellipse-like orbits, (ii)
the difference in the orientation of the “major axes” of the two systems, and (iii) the
locations and dimensions of the domains of the two subsystems. We will take these
points up in a moment, but first let us note that it is also possible to create an example in
20
which two unstable systems patch together to form an asymptotically stable hybrid
system. This would be the case, for example, if we reversed the flow used in the previous
example (by using –A in place of A and –B in place of B). This will yield trajectories
like those shown in Figure 7, except that the direction of flow is reversed. The reader can
verify that the –B subsystem operating in quadrant 1 now forces the hybrid ever closer to
the origin with each revolution.
Now let us consider the factors (i) through (iii) mentioned above.
S
-2
2
4
6
8
S
-5
-10
-15
-20
-25
Hybrid System
Figure 7
(i) When an orbit is strongly “eccentric,” there are intervals during which the
distance to the equilibrium is increasing substantially. That is true for subsystem A in
quadrants 2 and 4, for example.
If the subsystem is stable, then these periods of
increasing distance to the equilibrium are overcome by subsequent periods of decreasing
distance, so that the subsystem is always closer to the equilibrium after 180 degrees of
21
revolution. By contrast, a less eccentric, more circular, orbit does not have periods where
it moves strongly away from the equilibrium.
(ii) Part of the reason for the instability of the hybrid system in our example is
that the “major axes” of the two subsystems are orthogonal. If you combine this with the
eccentricities, you see that the source of the problem is that switching causes the
convergent motion of the A subsystem to shut off in quadrant 1, substituting the
divergent motion of the B subsystem. The result is that with each rotation through the
horizontal axis, the hybrid system is further from the origin. If the major axes were more
closely aligned, or if the eccentricities were less, this problem would be less serious.
(iii)
If we continue to assume for the moment that the switching border is
piecewise linear, then the smaller the angle between the two legs of the switching border
in quadrant 1, i.e., the smaller the domain of applicability of the troublesome B
subsystem, the lower the likelihood that it can produce instability in the hybrid when both
subsystems are stable. Consider the hybrid system in our current example, Figure 7. The
angle between the two legs of S is 90 degrees, and this gives the B subsystem maximum
opportunity to displace the trajectory away from the equilibrium. If we reduce the angle
between the two legs gradually by rotating the horizontal leg of S counterclockwise
through the origin, we will reach an angle where the phase space of the hybrid system
becomes dense with closed orbits, one of which is shown in Figure 8. This figure is
drawn in the following way: System B is given initial point (0, 1) and it is run forward in
time from t = 0 to t = (π/2βB), where the semi-path strikes the horizontal axis at x = 3.
Then the A system is run backward in time from the point (0, 1) to t = -2(π/βA), where the
path hits the y-axis again. Since A is stable, this occurs above (0, 1), where (x, y) ≈ (0,
22
1.22). Also, the path of the A subsystem (running backward) crosses the horizontal axis
at (x, y) ≈ (0, .367). Given the continuity of the paths, they must cross somewhere in
quadrant 1, as shown by point W in Figure 8. Thus, if the lower leg of the switching line,
labeled S1, runs through point W as depicted, if B operates in the narrow wedge between
S0 and S1, and if the A subsystem operates elsewhere, then there is a closed orbit running
under B from (0, 1) to W and thence under A back to (0, 1). In fact, for any μ > 0, there
will be a closed orbit running under B from (x, y) = (0, μ), switching to A on S1, and
returning to (0, μ).
If it were possible to control the location of S1, one could make the hybrid system
asymptotically stable by rotating S1 slightly counterclockwise from the position shown in
Figure 8.
S0
S1
W
B sub-system
A sub-system
Figure 8.
The conclusion is that with piecewise linear switching borders almost anything
can happen, even when the two subsystems are individually stable and share a common
equilibrium.
V. When Switching Happens Gradually Throughout a Region
Up to this point we have considered one-dimensional switching lines, where one
subsystem is fully operative on one side of the line, and the other subsystem is fully
23
operative on the other side. In some applications it may be more appropriate to consider
two-dimensional switching regions within which both subsystems simultaneously
influence the path. Consider an epidemic model of the spread of a newly mutated virus.
If policy makers and health officials fail to take action and the general population fails to
change its behavior, the dynamics of the system may drive it to an endemic equilibrium.
But if the spread of the virus causes a sufficient level of alarm, society may wage all-out
war on the virus, and this may give rise to a dynamic that yields an equilibrium rate of
infection which is quite low. And between these extremes of passivity and extreme
mobilization, intermediate levels of effort may be dedicated to fighting the spread of the
virus. Thus we might model this viral epidemic dynamical system using two subsystems
(one with no social mobilization to fight the spread of the virus and one with maximal
effort) and a switching region within which the quantity of resources devoted to fighting
the epidemic is a function of the present state of the system.
Or one can imagine an analogous economic model, wherein policymakers,
businesses and households behave according to one set or rules when the system is near a
full-employment equilibrium, follow another set of rules when there is a serious
recession, and behave in an intermediate fashion when conditions are neither dire nor
ideal. The range of possible models is, of course, immense, but we can learn a great deal
by careful examination of an example.
Consider the following Switching Region (SR) system:
24
( SR)
( A)
 x (t )    .4 4  x(t )  10 

  

 if ( x, y )  Sa  {( x, y ) | x  24},
 y (t )    2  .3  y (t )  20 
B 
 x (t )    .4  3  x(t )  40 

  

 if ( x, y )  Sb  {( x, y ) | x  28}, and
 y (t )   2  .5  y (t )  8 
 x (t ) 
  .4 4  x(t )  10 
  .4  3  x(t )  40 

  k (t )

  (1  k (t ))


 y (t ) 
  2  .3  y (t )  20 
 2  .5  y (t )  8 
if ( x, y )  S  {( x, y ) | 24  x  28}, where k (t )  7  .25 x(t ).
Note that k(t) varies from 1 when x(t) = 24 to 0 when x(t) = 28. This means that
inside the switching region, subsystem A dominates toward the left, B dominates toward
the right, and the relative influence of the A subsystem versus the B subsystem varies
continuously across the switching region according to the distance between x(t) and the
left and right switching region borders Sl and Sr, respectively.
Furthermore,
x (t) and y (t) are continuous as a path moves between Sl and Sr, which insures continuity
of the vector field. Continuity can be seen easily if we define the continuous function
k(t) such that:
1


k (t )  7  .25 x(t )

0

if x(t)  24
if 24  x(t)  28
28  x(t)
and then re-write (SR) as
 x (t ) 
  .4 4  x(t )  10 
  .4  3  x(t )  40 

  k (t )

  (1  k (t ))

,
 y (t ) 
  2  .3  y (t )  20 
 2  .5  y (t )  8 
( SR' )
since the right side of (SR') is seen to be merely the sum and product of continuous
functions.
Figure 8 is offered to help fix ideas. The switching region is a vertical column
25
bounded by x = 24 on the left and x = 28 on the right. These lines are marked Sl and Sr.
On the left side of the switching region, the A subsystem is operative. The A subsystem
has a unique, locally asymptotically stable, spiral point, with a clockwise flow toward the
equilibrium Ea = (x, y) = (10, 20). The B subsystem operates to the right of the switching
region, and it too has a unique locally asymptotically stable equilibrium at (40, 8), which
is a spiral point with surrounding counterclockwise flow. There is also a saddle point, Es,
in the “mixing region” S, i.e., between Sl and Sr, at x ≈ 26.44 and y ≈ -58.25, and an
unstable limit cycle shown as the closed path running from Sb to S and back to Sb. Paths
which originate inside the limit cycle spiral into the B subsystem equilibrium, Eb, with
the force on the trajectories sometimes coming from B alone and sometimes oscillating
from B to the mixed system in S.
Figure 9
26
The interior of the limit cycle constitutes the basin of attraction of the equilibrium
Eb. All paths other than those paths which originate on or inside the limit cycle, at the
saddle point, or on the stable manifold of the saddle point approach the A subsystem
equilibrium, Ea.
The existence of the unstable limit cycle can be verified by a careful examination
of the trajectories in the neighborhood of the cycle. Figure 10 shows a dashed Poincare
section line Σ, which drops vertically from the Eb equilibrium. The local flow at points
on Σ is everywhere left to right as time increases. Also shown are segments of two
trajectories that originate on Σ: one starting at b0, spiraling inward toward Eb, and making
its first return to Σ at b1, and the other starting at c0, spiraling through Sb, crossing into S,
re-entering Sb, and then returning to Σ at c1. We define set P to be the set enclosed by the
trajectory from b0 to b1 together with the line segment on Σ from b0 to b1. Set Q is a
“disjointed annulus” bounded on the inside by P and on the outside by the trajectory from
c0 to c1 and the line segment on Σ from c0 to c1. We let set Q include its borders. Let z0 =
(x0, y0) be any point in Q, and set t such that z(t) = z0 when t = 0. z(t) is the path through
z0, where t  (-∞, ∞).
Consider this path as t runs backward from 0 toward - ∞. First, notice that the
path z(t) cannot wander backward in time from set Q into set P, since this would require
either crossing the trajectory from b0 to b1 or crossing left to right on the line segment
from b0 to b1 as time decreases. But we have already seen that the flow is everywhere
left to right on Σ as time increases. Hence, paths cannot move backward in time from set
Q to set P. Similarly, no path beginning at a point in Q can move backward in time to
27
cross the outer boundary of Q, since that would imply either that trajectories cross or that
there is a violation of the known flow on the line segment from c0 to c1.
The upshot of this is that Q is shown to be a negatively invariant set, i.e., if z0 
Q, then z(t)  Q for all t  (-∞, 0]. Since set Q is closed, bounded, and negatively
invariant, we can apply Theorem 1 from Hirsch and Smale (p.251): A non-empty
compact set K that is positive or negative invariant contains either a limit cycle or an
equilibrium. And finally, since there is no equilibrium in A, there must be a limit cycle,
which in this case is an α-limit cycle.
Models of this type could have novel economic applications. For example, Axel
Leijonhufvud has proposed that economic systems might exhibit “corridor phenomena,”
y
Eb
P
b1
x
Q
b0
Q
c0
c1

Figure 10.
28
which might be interpreted as a set of conditions wherein the system has two equilibria,
one “better” than the other under some metric.
According to Leijonhufvud, small
displacements from the good equilibrium would be followed by convergence back to the
good equilibrium.
These displacements were said to leave the system within its
“corridor,” where the system is self-adjusting. Large shocks on the other hand would
send the system on a path to the inferior equilibrium. Under Leijonhufvud’s argument,
liquidity constraints played a role in the corridor phenomena. If traders hold liquid
balances in accordance with the anticipated strengths and durations of disruptions to their
ability to sell commodities, they may maintain effective demand in the face of normal
shocks to the demands for the things they sell. This will help the system self-correct back
to the good equilibrium. But if the shock is large or long enough to exhaust liquid buffer
stocks for traders who cannot effectuate their intended sales, then this will push the
system outside its corridor, and it will move off to an inferior equilibrium.
Leijonhufvud did not propose an analytical model, and the above summary of his
ideas does not constitute a model either, but the bi-stability displayed in our example has
a certain resonance with his idea. If Eb is the good equilibrium, it is easy to see that
small displacements from Eb give rise to paths back toward Eb, while larger
displacements (those outside the limit cycle or a stable manifold of the saddle) lead to Ea
instead.
VI. Conclusion
We have established necessary and sufficient conditions for the stability of linear
autonomous hybrid differential equation systems in R2.
Though several sufficient
conditions have been known for years (Ito 1980, Eckalbar 1981, Branicky 1997), our
29
finding here is novel. We have also created an example that explores some features of
hybrids with two-dimensional switching regions rather than one-dimensional switching
lines. This has the twin advantages of having a continuous vector field plus plausible
(but as yet unexplored) structure. To this author, the last point seems to be the most
interesting for further research.
References
di Bernardo, M., P. Kowalezyk, and A. Nordmark, “Sliding Bifurcations: A Novel
Mechanism for the sudden Onset of Chaos in Dry-Friction Oscillators,” International
Journal of Bifurcation and Chaos, 13(10), pp.2935-2948, October 2003.
Bignami, F., Luca, C., Weinrich, “Complex Business Cycles and Recurrent
Unemployment in a Non-Walrasian Macroeconomic Model,”
Journal of Economic
Behavior & Organization, Vol. 53, Issue 2, pp. 173-191, February, 2004.
Branicky, Michael S., “Stability of Hybrid Systems: State of the Art,” Proceedings of
the 36th Conference on Decision & Control, San Diego, 1997.
Branicky, Michael S., “Multiple Lyapunov Functions and Other Analysis Tools for
Switched and Hybrid Systems,” IEEE Transaction on Automatic Control, Vol. 42, No. 4,
pp. 475-482, April 1998.
30
Caballe, J., Jarque, X.,
and E. Michetti, “Chaotic Dynamics in Credit Constrained
Emerging Economies,” Journal of Economic Dynamics and Control, Vol. 30, Issue 8, pp.
1262-1275, August, 2006.
Eckalbar, J., "Stable Quantities in Fixed Price Disequilibrium," Journal of Economic
Theory, Vol. 25, No. 2, pp. 302-313. 1981.
Eckalbar, J., “Inventory Fluctuations in a Disequilibrium Macro Model,” The Economic
Journal, Vol. 95, pp. 976-991, December 1985.
Guckenheimet, J, and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and
Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
Ito, Takatoshi, "Disequilibrium Growth Theory," Journal of Economic Theory, vol. 23, pp. 380409, 1980.
Leijonhufvud, A., “Effective Demand Failures,” The Swedish Journal of Economics, Vol. 75,
No. 1, Stabilization Policy, pp. 27-48, March 1973.
Milgrom, M., “A Modification of the Newtonian Dynamics as a Possible Alternative to
the Hidden Mass Hypothesis,” The Astrophysical Journal, 270, pp. 365-370, July, 1983.
31
Polycarpou, M. and P. Ioannou, “On the Existence and Uniqueness of Solutions in
Adaptive Control Systems,” IEEE Transactions on Automatic Control, Vol. 38, No. 3,
pp. 474-479, 1993.
Stewart, D. and M. Anitescu, “Optimal Control of Systems with Discontinuous
Differential Equations,” Preprint ANL/MCS-P1258-0605, Math and Comp. Sci. Division,
Argonne Natl. Lab., 2005.
Utkin, V., “Variable Structure with Sliding Modes,” IEEE Transactions Automatic
Control, 22(2), pp. 212-222, 1977.
Varian, H., “The Stability of a Disequilibrium IS-LM Model,” Scandinavian Journal of
Economics, 1977.
32