Transition from stable orbit to chaotic dynamics in hybrid systems of Filippov type with digital sampling

0.29 MB·English

by  Paul Glendinning

APS/123-QED
Transition from stable orbit to chaotic dynamics in hybrid systems of Filippov type
with digital sampling
∗
Paul Glendinning and Piotr Kowalczyk
Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA) and School of Mathematics,
University of Manchester, Oxford Road, Manchester M13 9PL, U.K.
(Dated: January 20, 2009)
We demonstrate on a representative example of a planar hybrid system with digital sampling a
sudden transition from a stable limit cycle to the onset of chaotic dynamics. We show that the
scaling law in thesize of theattractor isproportional tothedigital sampling time τ for sufficiently
smallvaluesofτ.Numericalandanalyticalresultsaregiven. Thescalinglawchangestoanonlinear
law for large values of the sampling time τ. This phenomenon is explained by the change in the
9 boundednessof theattractor.
0
0 PACSnumbers: 05.45.-a,05.45.Gg
2

n The control, design and analysis of many systems rel- and H(x(t),µ) is some smooth scalar function depend-
a evant to real world applications involves understanding ingonthesystemstatesx Rn,andparameterµ Rm;
J the interaction between continuous and discrete dynam- t R is the time variable∈. The boundary Σ on∈which
0 ∈
ics. For example, the automated controlof a car moving H(x,µ)=0isassumedtobeahyperplanewhichdivides
2
onaroadisimplementedbydigitalcomputerbutthemo- the region D into two subspaces, G1 (for H(x, µ) > 0)
] tion of a car is continuous in time [1]. Hence the design and G2 (for H(x, µ) < 0), in which the dynamics is
D ofthecontrolofsuchasystemneedstotakeintoaccount smooth and continuous. There may be extra specifi-
C the effects ofthe interactionbetweenthe continuousand cations which determine the motion across or in Σ, for
. discretedynamics. Otherexamplesincludethecontrolof example if there is sliding motion as is the case in our
n
the motion of digitally controlled machines, for instance example, but these are standard to include.
i
l inrobotics[2,3,4]. Thesetypesofcontrolsystemsareof- In (1) the control of the switching between the two
n
ten referredto in the controlliterature as hybrid control systems across Σ is instantaneous. The modified hybrid
[
systems [5]. In [2, 3] it has been shown that the digiti- Filippovsystemswestudyareobtainedbyassumingthat

1 zation of the spatial structure by the controller induces the control function H is evaluated at discrete times kτ,
v
micro-chaotic transient dynamics. Effects of digitization k = 0,1,2,..., for some constant τ > 0, and so the
9
on the stability of the solutions have been considered in decision to change evolution equation can only occur at
6
[6, 7], and in [8] the existence of different types of at- these discrete times. Thus for k =0,1,2,3,... we define
0
3 tractors in a simple model of a delta-modulated control a discrete variable ik by
. system has been shown. Here we consider another as-
1
pect of digitization in hybrid systems. We assume that 1 if H(x(kτ),µ)>0
0
9 the input to the controller is delivered at discrete times, ik+1 =2 if H(x(kτ),µ)<0 (2)
0 separated by a constant τ > 0, and show that for arbi- ik if H(x(kτ),µ)=0,
: trarily small τ the system can exhibit chaotic dynamics.
v
i Moreover, there are scaling laws relating the maximum with i0 = 1 (arbitrarily chosen) so that i1 is always well
X distance of the chaotic attractor due to digital sampling defined, and replace the evolution (1) by
r from the simple periodic attractor of the continuously
a
sampled system. It is linear for sufficiently small values x˙(t)=Fik(x(t),µ) if (k−1)τ ≤t<kτ. (3)
of τ but at largerτ there is a change in the propertiesof
Notethatthissystemexcludesthepossibilityofsliding
the boundedness of the chaotic attractorand the scaling
motion, that is a motion within the discontinuity set Σ.
becomes nonlinear. We will illustrate this effect by con-
The example which we will consider in the remainder
sidering simple Filippov systems for which the evolution
of a variable x in some region D Rn is determined by of this paper is planar. Set
⊆
the equations
F1 = −ωαxx11−αωxx22++xx2(1x(x2121++xx22)22,) F2 = ba, (4)
F1(x(t),µ) if H(x(t),µ)>0 (cid:26) − (cid:26)
x˙(t)= (1)
(F2(x(t),µ) if H(x(t),µ)<0, H(x) = x2 µ, and h = kτ (k = 0, 1, 2, ) where
− ···
α, ω, a, b, µ and τ are some chosen constants (system
where F1, F2 are sufficiently smooth vector functions parameters). The vector field F1 is the normal form for
asimple subcriticalHopfbifurcation,witha stablefocus
at the origin and an unstable periodic orbit with radius
√α if α > 0. For appropriate choices of µ, a and b
∗Electronicaddress: [email protected] thevectorfieldF2 canbeusedtocreateastableperiodic
2
x2 x2 (a) µ−xˆτΣ (b) µ−τΣ
1 1 x˜ x˜ x′
xˆ
x¯ x¯
0 0
xg xg
−1 −1 VR UR VR UR
−1 0 1 x1 −1 0 1 x1 (u, −R) (u, −R)
(a) (b)
FIG. 2: Schematic representation of the bounding regions
FIG. 1: Asymptotictrajectories in Filippov system (4) with-
along Σ and theboundaries UR and VR in thecase (a) when
out digital sampling (a), and with digital sampling τ = 0.01
xˆ>x˜, and (b) xˆ<x˜.
(b).
orbitfortheFilippovsystem(1)asshowninFigure1(a). let φτ1(u,v) = {(p,q) | (p,q) = φ1(u,v,t) for some 0 ≤
The stability is derived from the fact that part of the t ≤ τ}. Define V to be the set of points (x1,x2) which
cycle lies on Σ, and this segment of the orbit is called a canbereachedfromapoint(u,v)∈G1∪Σ,withv ≥−R
sliding segment [9]. This stable cycle may coexist with withintimeτ,andwhosetrajectoryintersectsG2 intime
the unstable cycle of the vector field F1. τ, i.e. the set of points (x1, x2) ∈ G2 that are reached
For all the following numerical computations we set fromG1∪Σwithintimeτ byfollowingφ1.Finally,letVR
be the right hand boundary of V, i.e. (u,v) V such
ω =15,α=1.Inthis casethe vectorfieldunstable limit ′ ′ ∈ R
thatif (u,v) V then u u. Along most(andin some
cycle of F1 is centered at the origin and has radius 1. ∈ ≥
examplespossibly all) ofits lengthV will be the time τ
The vector field F2 is assumed constant; we set a = −1 image of points on Σ, but close to xRg this might not be
and b= 1. Finally let us set µ= 1.1 so that Σ does not
the case. V therefore represents a boundary which no
intersecttheunstablelimitcycleofF1. Fortheseparam- R
eters(1)withvectorfieldsgivenby(4)admitsthe stable orbitwhichstartsinG1 above−Rcancrosswithin time
limit cycle with sliding already referred to as well as the τ under the flow φ1. The choice of R is determined by
later considerations,it needs to be large enoughto allow
unstable cycle of F1 (the main restriction in the choice
the argumentto close up below – numericalexperiments
of the parameters is that given µ, the angular velocity ω
show that R=2 is sufficiently large here.
in F1 is large enough for the sliding cycle to exist).
Let us now increase the sampling time τ from 0 using Now consider the effect of the flow generated by F2
(4). As Fig. 1(b) shows, there is an apparent thickening to points in G2 to the left of VR. The trajectories are
of the attractor: the stable limit cycle no longer exists straightlineswithslope−1,andasx˙2 =−1the furthest
and we observe an onset of more complex asymptotic to the left that an orbitfrom G2 canreachin time τ has
dynamics. It turns out that this complex dynamics is x1 = µ−τ. Let U be the union of VR and the set of
pointsonstraightlinesofslope 1from(u,v) V with
chaoticandsothere is atransitionfroma stableorbitto − ∈ R
u > µ τ to µ τ. Finally let U be the right hand
achaoticattractorduetoanintroductionofthesampling − − R
boundary of U. Note that U must be connected.
process. R
To summarise: by construction, in time τ, no solution
We will prove this in two parts. First we show that
if τ > 0 is sufficiently small then there is a compact set ofF1 abovethelinex2 = RcanmovetotherightofVR
−
(which is on the left of U or equal to it at places), and
which solutions cannot leave, and hence which contains R
at least one attractor. This part of the proof is based under F2 all such orbits remain to the left of UR until
on showing monotonic crossing of the local transversal they return to G1.
by a trajectory, similarly as in the Poincar´e- Bendixson Let x¯ be the highest point in UR with x¯1 = µ and
Theorem [10]. We present this argument in fairly gen- x˜ the highest point in UR with x˜1 = µ τ. Let xˆ be
−
eraltermsbelowsothattheextensiontosimilarsystems thefirstintersectionofthe solutionofF1 throughx¯ with
is clear. Second we show that any solution in this com- xˆ1 =µ τ.
−
pact set (and hence the attractor itself) has a positive If xˆ2 > x˜2 then the outer boundary of the bounding
Lyapunov exponent. region is the trajectory through xˆ under F1 until it hits
The compact invariantregionis annular,and its inner VR for the first time (see Figure 2(a)), a horizontal line
boundary is the unstable cycle of F1 with radius one. segment from VR to UR, and then UR back to x¯. Note
To begin the construction of the outer boundary, note that this requires R to be large enough so that the tra-
that the sliding segment of the (true) Filippov system jectory does hit VR. If not then a larger R needs to be
terminates at the point xg = (µ, xg) which is where the chosen.
2
′
solution of F1 is tangential to the surface Σ as shown If xˆ2 < x˜2 let x be the first preimage of x˜ on Σ un-
in Figures 2 and 3 (xg2 is close to 0 for α close to µ2). der F1, and note that this will lie above x¯. Then the
Now, let φ1(u,v,t) denote the flow generated by F1, i.e. outer boundary of the bounding region is the trajectory
′
solutions of F1 at time t with initial condition (u,v) and through x under F1 until it hits VR for the first time,
3
Σ
for non-zero τ is characterized by a positive Lyapunov
PSfrag exponent we consider the determinant of the linearized
mapthatmapstheneighborhoodofxg ontoitself. Hence,
we consider the determinant of the matrix composition
x of the solutions of the variationalequations for the flows
g
xp φ1 and φ2. Since the vector field F2 that generates the
XB xc ΣτF1 flow φ2 is a constant vector field then
τ xb ∂φ2
det =1,
∂x
(cid:18) (cid:19)t
FIG.3: Schematicrepresentation oftheboundingregion XB where t is the time corresponding to the evolution fol-
for small values of thesampling time τ.
lowing φ2, and ∂∂φx2 ≡Φ2(t) is the fundamental solution
matrix corresponding to the flow φ2.
a horizontal line segment from VR to UR, and then UR Usingtheexplicitexpressionfortheflowfunctionφ1 in
back to x˜ (see Figure 2(b)). As before R needs to be the polar co-ordinates we can compute the fundamental
large enough for the connections to work. solution matrix Φ1, corresponding to the flow φ1 :
In either case we will have created a compact region
wrehgiicohnncoonttraaijnesctaotrylecaasnt oexnietafrtotrma,ctaonrd. hencetheannular Φ1(tj)≡ ∂(∂ρφ,1θ) = fj(tj,ρj)0exp(2tj) 10 ,
Consider now a sufficiently small τ. Define Στ as the (cid:18) (cid:19)tj (cid:18) (cid:19)
image of Σ under the actionof φ1 for time τ. LeFt1xp Σ with j = 0, 1, 2 that correspond to the times of evo-
btheetnheeigphrbe-oimrhaogoedoofftxhge apsoisnhtoawtnwinhicFhigΣ.τF13.crDoessfienseΣ∈XiBn ltuotφio1n.Afotlljow=in0gwφ·e·1·ianfitteiarlijz−etthheswevitoclhuitniognfrforommththeeflnoewigφh2-
tobethesetofinitialpointsxintheneighborhoodofxg borhoodofxg,andρ denotes the radiusfromthe origin
j
esuvochlvetshatthrfoourgahnxygx.F∈orXτBsuaffitrcaiejnetcltyorsymgaellnΣerτFa1teids nbeyarFly1 fajttahreejm−otnhotsowniticcahlilnyginincsrteaanscineg. Ffuinnactlliyonfsj(o0f,tρjj.)=1and
tangenttoΣandthereexistsapointonΣτ ,sayxb,such Therefore,
F1
thatthetrajectorystartingatxb crossesΣatsomepoint
xc below xp, and the time of evolution from xb Στ to ∂φ1
X is τ (see Fig. 3). Therefore a trajectory∈staFrt1ing detΦ1(tj)=det ∂(ρ, θ) >1,
B (cid:18) (cid:19)tj
at xb must switch to φ1 to the left of set XB or on XB.
Moreover since xp lies above xc no trajectory generated and jtj > ωπ (note that the flow follows φ1 from xg
solely by F1 can lie to the right of XB – penetrate G2 until the first intersectionwith Σ for the amountof time
P
and return to G1 without switching to φ2. We further greater than π/ω). The resulting determinant of the
note that a trajectory rooted at any point within the composition of the fundamental solution matrices cor-
region bounded by Σ, ΣτF1, and the line segment joining responding to the flows φ1 and φ2 is Πjn=−01detΦ1(tj)×1
xb with xc switches to the vector field F1 in a region to wherenisthenumberofswitchingsthatarerequiredfor
the left of X – the time of evolution from any point the system trajectory to reach the neighborhood of xg.
B
in this region to reach some point in G1 to the left of We should note here that formally we should introduce
XB is less than τ. It then follows that for sufficiently a co-ordinate transformation to the flow φ2 and use the
small τ, X is a bounding set for the attractor. To find fundamentalsolutionmatrixcorrespondingtothevector
B
the bounding set to the left of Σ we note that the set field F2 in polar co-ordinates. However, since the flow
of points furthest to the left of Σ, which can be reached φ2 is a constant flow the determinant of Φ2 is always 1
by a trajectory generated by F2, has co-ordinates (µ regardless on the co-ordinate set.
τ, x2).This impliesthatalongthex1 co-ordinate,inth−e Therefore, we are only interested in Πjn=−01detΦ1(tj).
neighborhood of xg, the difference between the largest Since this productis greaterthan1 andthe determinant
and the smallest values of x1 on the attractor is τ. of a matrix is the product of its eigenvalues we conclude
Forlargervaluesofthesamplingtimeτ pointxp might thatthere isatleastoneeigenvalueofΠjn=−01Φ1(tj)which
no longer lie above xc, andthere existtrajectoriesin the ischaracterizedbythemagnitudegreaterthanone. This
neighborhood of xg solely generated by F1 that lie to eigenvalue is the exponential of the Lyapunov exponent.
the right of X . In this case the difference between the Let us determine the size of the attractor, measured
B
largest and the smallest values of x1 on the attractor is as the distance from the largestto smallest values which
different from τ. Therefore, we expect to see a change in the attractor attains on a Poincar´e section defined on a
the size of the attractor as a function of the sampling set x2 = 0, 1.3 < x1 < 1 . For sufficiently small
{ − − }
time, that changes from a linear law for small τ, to a values of the sampling time τ the size of the attractor
different scaling not linearly proportionalto τ. along the x1 co-ordinate is proportional to τ around the
To see that the attractor born out of the stable cycle point xg. However, the size of the attractor is measured
4
reticalpredictioncoincideswiththenumericalresultsfor
100
small values of τ. For τ sufficiently small the attractoris
width ∆ x bounded by the trajectory leaving Σ at xg.
1
We notice that the increase in the value of the sam-
10−1
pling time τ above τ = 0.01 results in the growing dis-
crepancy between the numerical and theoretical values.
This comes from the fact that the attractor is no longer
10−2
bounded by the trajectory leavingΣ at xg. Other effects
such as resonances between the sampling time τ and the
10−3 rotation ω of the flow φ1 produce other local variations
in the scaling law visible in Fig. 4.
Inconclusion,wehaveshownonaplanarexamplethat
10−4 the digital sampling applied to the decision function in
10−4 10−3 time τ 10−2 10−1 Filippov type systems leads to the onset of chaotic dy-
namics. Further on we have shown that for sufficiently
smallvaluesofthesamplingtimeτ thesizeofthechaotic
FIG.4: Sizeoftheattractorversusthesamplingtimeτ.The
attractor scales linearly with the sampling time τ. The
dashed line refers to theoretical predictions of the size of the
attractor. mechanism that leads to the onset of chaos is triggered
by the expansion of the volume of phase space produced
by the flow φ1 combined with the re-injection triggered
on section x2 = 0, 1.3 < x1 < 1 . Therefore, we bythesubsequentapplicationoftheswitchingsalongthe
{ − − }
have to determine the expansion of the attractor along manifoldΣ.Thescenarioobservedinourmodelexample
the flow after time π/ω (which is the time required to will be present in a larger class of systems with swith-
map the points from the neighborhood of xg using flow ings. The essential ingredients of these systems will be
φ1 onto our chosen Poincar´e section). This expansion thepresenceofastablecycle(whennodigitizationisap-
is captured by the non-trivial Floquet multiplier of the plied) and the presence of an expansion of a volume of
fundamental solution matrix. To find this multiplier we phase space in the presence of digitization.
usetheexplicitsolutionsofthedifferentialequationsthat
The onset ofchaotic dynamics triggeredby this mech-
define F1 in polar co-ordinates. We find that anism is similar to an abrupt transition from a stable
periodic orbit with sliding to a small scale chaotic dy-
α
ρ(t)= (1 αρ−2)exp(2αt) 1, θ =θ0+ωt. namics that might occur in Filippov type systems under
r − 0 − an introduction of an arbitrarily small time delay in the
switching function [11]. On the practical side, in spite
Differentiating ρ(t) with respect to ρ0, and after substi- of the fact that these oscillations are micro-chaotic,they
tuting for α = 1, t = π/15, and for ρ0 = 1.1 we get cancanbehighlyharmfultocontrolelements,induceex-
dρ
= 1.8084. In Fig. 4 using the logarithmic scales we cessive wear to machine tools [3], and therefore it is im-
dρ0
are depicting how the size of the attractor scales against portanttounderstandthemechanismsthatmighttrigger
the sampling time τ. The dashed diagonal line refers to this type of complex dynamics.
the linear scaling proportional to τ obtained using the Research partially funded by EPSRC grant
above theoretical prediction. We can see that the theo- EP/E050441/1and the University of Manchester.
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