Prediction and measurement of transient responses of first difference based chaos control for 1-dimensional maps

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by  Edward H. Hellen

Prediction and Measurement of Transient Responses of First Difference Based Chaos
Control for 1-dimensional Maps
Edward H. Hellen∗ and J. Keith Thomas
University of North Carolina Greensboro, Department of Physics and Astronomy, Greensboro, NC 27402
(Dated: January 14, 2010)
Chaotic behavior can be produced from difference equations with unstable fixed points. Differ-
ence equations can be used for algorithms to control the chaotic behavior by perturbing a system
parameterusingfeedbackbased onthefirstdifferenceof thesystem value. Thisresultsinasystem
of nonlinear first order difference equations whose stable fixed point is the controlled chaotic be-
havior. Basing the feedback on the first difference produces distinctly different transient responses
thanwhenbasingfeedbackontheerrorfromthefixedpoint. Analogelectroniccircuitsprovidethe
0 experimental system for testing the chaos control algorithm. The circuits are low-cost, relatively
1 easy toconstruct,and therefore providea usefultransition towards morespecialized real-world ap-
0 plications. Herewepresentpredictionsandexperimentalresultsforthetransientresponsesofafirst
2 difference based feedback control method applied to a chaotic finite difference 1-dimensional map.
Theexperimentalresultsareingoodagreementwithpredictions,showingavarietyofbehaviorsfor
n
thetransient response, including erratic appearing non-steady convergence.
a
J

4 I. INTRODUCTION where the erroris ∆x=x x∗ and the fixed pointis x∗.
1 −
Thus DFC has the desired property of the perturbation

] May showed that the logistic equation, a recursionre- vanishing when the system value attains the fixed value.
D lation based on a quadratic return map, can generate Stabilization is attempted only when the magnitude of
C chaotic behavior.1 Proportional feedback methods can d(xn) is within a specified window for control.
. controlchaosinthesesystems.2–5 Typicallyinthesecon- Weperformalinearstabilityanalysistodeterminethe
n
trol methods a system parameter is perturbed by an conditionsforstabilizationofthechaoticbehaviorandto
i
nl amount proportional to the ”error”, the difference be- calculatethe formofthe transientresponse. The control
[ tween the current system value and the unstable fixed algorithmis implemented in ananalogelectronic circuit.
point. This control method is simple proportional feed- Measurements from the circuit compare well with pre-
3 back (SPF). Pyragas introduced an alternative in which dictions. A variety of transient responses are found de-
v
the perturbation is proportional to the first difference of pendingonthegainofthefeedback. Someshowasteady
7
3 the system value (difference between current and previ- convergenceof the system value to the fixed value, while
6 ous values), and therefore does not depend on the un- othercasesappeartoconvergesomewhaterratically. The
2 stablefixedpoint.6 This delayedfeedbackcontrol(DFC) behaviors are distinctly different from the transient re-
7. methodsuccessfullycontrolschaoticbehaviorinavariety sponses obtained when using SPF control based on the
0 of cases.7 Here we apply DFC to a chaotic finite differ- error ∆x. The electronic circuit provides a real-world
8 encereturnmapandderivethemathematicalformofthe system for testing implementation of the control algo-
0 transient response and verify it experimentally with an rithm.
v: analog electronic circuit.
i The system whose chaotic behavior is to be stabilized
X II. PREDICTIONS
is a 1-dimensional finite difference map for system value
ar x with system parameter a0
Here we do linear stability analysis in order to deter-
xn+1 =f(xn,a0). (1) minetheconditionsunderwhichchaoticbehaviorcanbe
stabilized and we find the form of the transient response
Stabilization is accomplished using DFC by perturbing
when stabilization is turned on. We use the first differ-
parameter a0 by an amount proportionalto the first dif- ence d(x ) as our variableof interest since the perturba-
n
ference, d(xn) = (xn xn−1). Thus when stabilization tion of system parameter a is ∆a = Kd(x ). Using
− n n n
isturnedonwehaveasystemoftwofirstordernonlinear
d(x ) can be useful in real-world applications since the
n
finite difference equations
system values are easily obtainable, whereas the fixed
value may be unknown or may drift.
xn+1 =f(xn,an) (2a) The fixed point of Eq. (2) satisfies x∗ =f(x∗,a0) and
an+1 =a0+K(f(xn,an)−xn). (2b) a∗ =a0. The Jacobian for linearization of Eq. (2) about
the fixed point is
K is the gain for the feedback that perturbs system pa-
rameter an from a0. The first difference can be written J = fx fa (4)
in terms of the error term K(f 1) Kf
x a
(cid:18) − (cid:19)
d(xn)=∆xn ∆xn−1 (3) where fx and fa are the partial derivatives evaluated at
−
2
the fixed point. We point out that knowledge of the where
fixedpointisrequiredtoevaluatethe partialderivatives.
However during the attempt to control chaotic behavior c= y0 +i y0Re(λ+)−y1 =c0eiθ. (12)
it is not necessary to monitor and update a potentially 2 2Im(λ+)
(cid:18) (cid:19)
drifting fixed point, as long as the partial derivatives do
Thus the solution for the first difference is
not change too much. In contrast, a feedback method
basedonthedifferencefromthesystemvaluetothefixed yn =c0eiθ reiφ n+c0e−iθ re−iφ n =2c0rncos(nφ+θ)
point requires monitoring and updating if the system’s
(13)
fixed point can drift. (cid:0) (cid:1) (cid:0) (cid:1)
where
The trace and determinant of the Jacobian are (f +
x
Kfa)andKfa,respectively. Thesegivetheconditionfor r = Kfa, (14)
a stable fixed point,
p
f +Kf <1+Kf <2. (5)
x a a 2
| | 4Kf (f +Kf )
a x a
φ=arctan − , (15)
The right-side inequality requires Kfa < 1. We are in- q fx+Kfa 
terestedinsituationswhenthefixedpointisunstable,so
f < 1. Therefore Kf >0, and the left-side condition  
x − a and
requires ( f Kf ) < (1+Kf ). Thus the condition
x a a
− −
on Kfa for a stable fixed point is y0Re(λ+) y1
θ =arctan − . (16)
fx 1 (cid:18) y0Im(λ+) (cid:19)
− − <Kf <1. (6)
a
2 Thecosinetermcancauseerraticappearingconvergence
This gives the result that DFC can not control chaos if distinctly different from the steady geometric conver-
the slope of the return map at the fixed point is steeper gence for the real multipliers in Eq. (8).
than 3. The result is that we predict convergence (control)
Now−we find the form of the transient response of the for feedback gainK between (1+fx)/( 2fa) and1/fa,
−
firstdifference. ItfollowsfromEq.(3)thatEq.(4)isthe with Eq. (10) giving the transition from geometric con-
Jacobianford(x ). Lety =d(x )andlookforsolutions vergence [Eq. (8)] to sinusoidal convergerence[Eq. (13)].
n n n
y =λn. Characteristic multipliers are As an example we consider the 1-dimensional H´enon
n
map:
2
f +Kf f +Kf
λ± = x 2 a ±s x 2 a −Kfa (7) f(x,a)=1−ax2 (17)
(cid:18) (cid:19)
with fixed point
giving solution
yn =c1λn++c2λn−. (8) x∗ = −1+√1+4a0 (18)
2a0
The coefficients c1 andc2 are determined by two consec- ∗
utiveobservedvaluesy0 =c1+c2 andy1 =c1λ++c2λ−. and partial derivatives evaluated at (x ,a0)
Solving for the coefficients gives fx = 2a0x∗ =1 √1+4a0 (19)
− −
λ−y0+y1
c1 = − (9a)
λ+−λ− fa = (x∗)2. (20)
λ+y0 y1 −
c2 = − (9b)
λ+ λ− For values of a0 between 1.4 and 2 Eq. (17) displays a
−
variety of unstable behavior including high period oscil-
Setting the discriminant in Eq. (7) to zero gives the lationsandchaos.10 Fora0 =1.9wefindfx = 1.93and
value of Kfa for transition from geometric to sinusoidal fa = 0.259,soconvergenceofd(xn)ispredict−edforval-
behavior, ues of−K between-1.8and-3.87with the transitionfrom
geometric to sinusoidal convergence at -1.96.
Kf =2 f 2 1 f . (10)
a x x
− − −
For values of Kf larger than inpEq. (10) and less than
a III. CIRCUIT AND MEASUREMENTS
onethecharacteristicmultipliersarecomplexconjugates
and the convergence is sinusoidal. Using the Euler rep-
resentation, λ± = re±iφ, and taking the form of Eq. (8) Figure 1 shows the circuitry used to apply the control
the solution is algorithm to a function block circuit f(x,a) that per-
forms analog computation of a chaotic return map. The
y =c reiφ n+c∗ re−iφ n (11) voltage V corresponds to the system value x, where the
n
(cid:0) (cid:1) (cid:0) (cid:1)
3
FIG. 1: The circuit for controlling chaotic behavior of thereturn map xn+1=f(xn,an). Op amps are LF412.
scaling factor of the AD633 multiplier integrated circuit tem value voltage V , to the f(x,a) circuit block which
n
mustbeusedsothatxn =Vn/(10volts). Hereweusethe produces the next system value voltage Vn+1. The sub-
function blockcircuit showninFig. 2that calculatesthe traction op amp creates the first difference ∆Vn+1 =
1-dimensionalH´enonmap Eq. (17).10 We have also used Vn+1 Vn that is used to create the perturbation for
−
functionblockcircuitsthatproducetheLogisticmapand the next iteration, ∆an+1. ∆Vn+1 is passed to an abso-
the tent map. lute value/comparator stage and to a gain stage which
At the upper left in Fig. 1 the unperturbed param- produces ∆an+1. The output of the comparison stage
eter value a0 is added to the perturbation ∆an to cre- (LM339) controls the gate of the FET in the gain stage
ate system parameter a . This is input, along with sys- sothatif ∆V islargerthanthecontrolwindowthenthe
n
| |
gate goes to 5 volts turning off the FET and thereby
−
setting feedback gain K = 0. A nonzero value for K
is determined by the inverting op amp adjacent to the
FET. For the values shown, 47kΩ and 13kΩ, K = 3.6.
−
ThesignofK iseasilyswitchedbychangingthe orderof
inputs Vn and Vn+1 to the subtraction amplifier. Prior
to the FET ∆V is divided by 10, the scaling factor of
| |
the AD633 multiplier used in the f(x,a) circuit block,
to convertfromvoltage ∆V to ∆x. The sample/holds
| | | |
(LF398) act as shift-registers for the iteration of n un-
der the control of the 555 timer circuit. With the 68kΩ
FIG. 2: Henon function block circuit for f(x,a). Relation and 0.001µf shown in the schematic the period is about
between voltage and system value is x = V/10. The ×10 100µs.
noninvertingamplifierat theoutput accountsfor both a and
the +1 not being multiplied by 10, the scaling factor of the DatawerecollectedwithaTektronixTDS3000oscillo-
AD633 multiplier. scope. The controlcircuit was periodically gated on and
off (circuitry not shown) so that it was possible to trig-
4
ger from the gating signal in order to capture the entire IV. RESULTS AND DISCUSSION
control of chaotic behavior. Figures 3 and 4 show data
andthe effect ofthe gating. Also apparentis the control
Figures 3 and 4 show the effectiveness of the control
window. Control was gated on at t = 0 in both cases,
circuit in Fig. 1 when applied to the H´enon circuit with
but in Fig. 3 ∆V was not within the control window
until about t=| 4.5|ms. parameter a0 =1.9, a value that gives chaotic behavior.
The figures show the measured voltages for the system
valuesV =10x andparametervaluesa . Figure3uses
n n n
feedbackgainK = 1.96correspondingtothetransition
−
10 betweengeometricandsinusoidalconvergence[Eq.(10)],
and Fig. 4 uses K = 3.53 corresponding to non-steady
−
sinusoidal convergence.
5 The DFC method uses feedback proportional to the
first difference of system values, d(xn) = xn xn−1.
−
ge When the system is successfully controlled d(xn) → 0.
a In Section II we showed that the nature of the conver-
t 0
ol genceofd(x )depends onthe feedback gainK. The ap-
V n
proach to zero for d(x ) may be steady [Eq. (8)] or may
n
appear somewhat erratic [Eq. (13)]. Figures 5, 6, and 7
(cid:0)5 show data and prediction for yn (actually ∆Vn = 10yn)
forthreevalues ofK showingthe varietyofconvergence.
Predictionsweremadebyusingtwosuccessivemeasured
(cid:0)100.005 0.000 0.005 0.010 0.015 0.020 system value differences for y0 and y1 in Eqs. (9) and
(cid:0) Time (sec) (12) to determine the coefficients for Eqs. (8) or (13).
The convergence in Fig. 5 is steady, in Fig. 6 it appears
FIG. 3: Data for control of chaotic H´enon system with feed- somewhaterratic,andin Fig.7 a pattern is apparental-
back gain K = −1.96, a value predicted to result in steady though the convergence is not steady. In all cases there
convergence of the system value. Control was gated on at is good agreement between the prediction and measure-
t=0andoffatt=0.012. Alsoshownistheparamatervalue ment.
a=a0+∆a with a0 =1.9.
0.6
0.4
10
)
s
t
ol 0.2
v
(
)
5 (x 0.0
d
0
1
ge V=(cid:2)0.2
a
t 0
ol (cid:3)
V (cid:2)0.4
0.6
5 (cid:2)
(cid:1)
0.002 0.003 0.004 0.005 0.006
Time (sec)
10 FIG.5: Data(dots)andprediction(opencircles)showingthe
(cid:1)0.005 0.000 0.005 0.010 0.015 0.020
(cid:1) Time (sec) firstdifferenceofsystemvaluesd(xi)=(xi−xi−1)forsteady
geometric convergence, K = −1.9. The connecting dashed
lines are for visual aid only.
FIG. 4: Data for control of chaotic H´enon system with feed-
back gain K = −3.53, a value predicted to result in non-
steady convergence of the system value. Control was gated
onatt=0andoffatt=0.012. Alsoshownistheparamater
valuea=a0+∆a with a0=1.9.
When SPF is used to controlchaos,system parameter
aisperturbedbyanamount∆aproportionaltotheerror
∆x from the fixed value. The equations for the system’s
5
Eq. (4) except for J21 = Kfx. This makes det(J) = 0
resulting in stability when 1 f < Kf < 1 f
x a x
2 and solution ∆x = A(f +−Kf−)n. Thus the beha−vior
n x a
of the transient response for SPF is quite different from
that for DFC. The SPF transient response has a single
s) 1 characteristic multiplier so it is a steady geometric con-
t
ol vergence,eitheralternatingsignormonotonicdepending
v
( onwhether(f +Kf )ispositiveornegative. Immediate
) x a
d(x 0 convergence occurs when K = −fx/fa since this causes
0 the characteristic multilplier to be zero. DFC has either
1
= a more complicated geometric convergence governed by
V
two characteristic multipliers, or the sinusoidal conver-
1
(cid:5) (cid:4)
gencethatisresponsibleforerraticappearingnon-steady
convergence. DFC can not have immediate convergence
becauseaccordingtoEqs.(10)and(14)rinEq.(13)can
2
(cid:4) 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 neverbe zero. The fastestconvergenceoccurswhenKfa
Time (sec)
has its minimal value given by Eq. (10).
FIG.6: Data(dots)andprediction(opencircles)showingthe For SPF applied to the H´enon map with a0 = 1.9
first difference of system values d(xi) = (xi−xi−1) for non- the value of K for immediate convergence is −fx/fa =
steady sinusoidal convergence,with feedback gain K =−2.3. 1.93/0.259= 7.45. TherangeofK thatgivestability
− −
Connecting dashed lines are for visual aid only. is ( 1 fx)/fa = 3.59 to (1 fx)/fa = 11.3. For
− − − − −
DFC, Eq. (6) predicts that the range of gain giving sta-
bilityis 1.8to 3.87. Thisisreasonablesincetheerror
(x x∗−) can be−expected to typically be about half of
n
−
the firstdifference (xn xn−1), so that the SPF method
0.6 −
needs a gain about twice that of the DFC gain in order
to get a similar perturbation ∆a.
0.4
We have shown that there are a variety of tran-
)
s sient responses when applying delayed feedback control
t
ol 0.2 to a chaotic system governed by a finite difference 1-
v
) ( dimensional map. In addition, the behaviors of the con-
x
( 0.0 vergencearedistinctlydifferentfromthesimplergeomet-
d
10 ric convergence obtained with simple proportional feed-
=
V(cid:6)0.2 back. Knowledge of this variety is important in situa-
tions in which the system’s response to application of
(cid:7)
a control algorithm is being closely monitored since er-
0.4
(cid:6)
ratic and non-steady convergence could be mistaken for
a faulty control mechanism. The important advantage
0.6
(cid:6)0.000 0.001 0.002 0.003 0.004 0.005 DFC has overSPF is that DFC uses anerrorsignalthat
Time (sec)
does not depend on the unstable fixed point. This sim-
plifies experimentalimplementationandalsomeansthat
FIG.7: Data(dots)andprediction(opencircles)showingthe
DFC will continue to stabilize chaos even if the unsta-
fistresatddyiffseinruensociedoafl csoynstveemrgevnaclue,eswdit(hxif)ee=db(axcik−gaxiin−1K) f=or−n3o.n4-. ble fixed point drifts by small amounts due to noise or
changing system parameters.
Connecting dashed lines are for visual aid only.
value and parameter are,
Acknowledgments
xn+1 =f(xn,an) (21a)
an+1 =a0+K(f(xn,an) x∗). (21b) This research was supported by an award from the
− Research Corporation. Corey Clift contributed to early
The Jacobian for linear stability analysis is the same as work on this project.
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