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Numerical Instability in Delay Differential Equations

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ADAPTIVE AND OPTIMAL CONTROL ASSIGNMENT - STAGE 1 , JANUARY-2014 1 Numerical Stability Analysis of Simple Delay Differential Equations Surajkumar Harikumar (EE11B075) Anjali Ramesh (EE11B083) Gaurav Raina  Abstract—We analy se the intr insi c stabi lity of a line ar dela y differential equati on wi th a si ngle de lay . We nd the ti me step values that for ce the normally stable system to nume rica l instability, in the Runge-Kutta 2 and Runge-Kutta 4 numerical solvi ng techniqu es. We also arri ve at the optimal time step to avoid 4th decimal place errors in the numerical solution for RK2 and RK4. This ideal time step enables us to compute numerical solutions more quickly while avoiding error upto the 4th decimal place. We perform the necessary analysis of a linear delay differential equ ati on wit h a sin gle dis crete del ay . Follo win g thi s we sho w how dis cr eti zat ion of time int er val s can lea d to ins tabil iti es inde pende nt of the intrinsi c parameter s of the system. We nd the time ste p values tha t force the normall y stable syste m to nume rica l insta bili ty , in the Runge -Kut ta 2 nume rica l sche me. This is followed by an L A T E X reproduction of the Appendix section of  Loc al Bifurc ation Analysis of Some Dual Conges tion Control  Algorithms  by Prof. Gaurav Raina, IIT Madras.  Keywords Numerical Instability, Linear Differential  Equations,Runge-Kutta Eigenvalues I. I NTRODUCTION Most cont rol syst ems around us ca n be mode ll ed at a primiti ve lev el using simp le rst orde r delayed dif ferential equations. Various forms of stability of such systems have been well-studied. It is necessary for a given control system to be asymptoti call y stab le, as inst abil ity lead s to oscil lati ons and divergence. We star t this paper by determin ing the stab ilit y crit erio n for some common differential control systems - studying the eigenvalue characteristics to calculate necessary and sufcient conditions for asymptotic stability. In the systems we analyze, various intrinsic parameters can induce instability . Howeve r, we choose not to focus on a pa ra me te r  within  an y of the sys tems, but to ind uce ins tab il ity ext ern all y . We vary the discretization  step size in our mathematical tool (XPP), leading to numeri cal ins tab il ities dep end ing on the computationa l method. Our aim is not to intentionally destabilize the system, but to see how instabilities can be caused due to big step sizes and how analysis is always a trade off between computational power and processing time. In the el d of numeri cal analysis, most numeri cal alg o- rith ms would arrive at the correct solut ion if ther e were no approximation errors. But computational methods that are used for such anal ysis alwa ys discr etiz e var iabl es, con ver ting the dif fer ent ial sys tem int o a set of dif fer enc e equ ati ons. Thi s leads to truncation and round-off errors. Most algorithms are optimized to damp en these errors iter ati vel y , but some time s these errors magnify with each iteration, and can even grow exponentially. This leads to a different solution or sometimes, the complete lack of one. This paper is organized as follows. In Section I, we analyze the stab ilit y cond itio ns for a delay ed dif fere ntia l equat ion. In Section II, we add an instantaneous feedback component and perform stabili ty analy sis. In Secti on III, we intr oduce numerical instabilities in the system and Section IV concludes the discussion. For ease of expos ition, all the calculations associated with the bifurcation analysis are contained in the Appendix. II. DELAYED F EEDBACK Our dif ferential equation conta ins one disc rete delay with no instantaneous feedback. This is modelled as, d dt x(t) = κx(t τ )  (1) where κ, τ > 0. The corresponding characteristic equation for this system (substituting  x(t) =  e λt ) is λ = κe λτ (2) Here,  λ  is called the eigenvalue of the system. All eigenvalues with  Re(λ)  <  0  denote that the syst em is asymptot ical ly stable for all values of  κ, τ >  0. In our system, it is possible that  Re(λ)  >  0, which would cause our syst em to div erg e. Thus, we come up with a necessary condition for stability. Calculations:  For  τ  = 0,  λ  = −κ. Since  Re(λ) = −κ < 0, the system is asymptot ically stable. Now, as we incr ease τ , the system moves from stability to instability at a critical τ  = τ c . At this point,  Re (λ) = 0, i.e. the eigenvalue is purely imaginary. Any further increase in  τ c  will make  Re(λ)  > 0. Substituting λ  = j ω c  in (1)  jω c  + ae jωcτ c = 0 Equating the real and imaginary parts of the above equation to  0 , ω a sin(ω c τ c ) = 0  (3) a cos(ω c τ c ) = 0  (4) Squaring (3) and (4) and adding, we get ω 2 c  = a 2 (5)
Transcript
Page 1: Numerical Instability in Delay Differential Equations

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ADAPTIVE AND OPTIMAL CONTROL ASSIGNMENT - STAGE 1 , JANUARY-2014 1

Numerical Stability Analysis of Simple Delay

Differential EquationsSurajkumar Harikumar (EE11B075)

Anjali Ramesh (EE11B083)

Gaurav Raina

 Abstract—We analyse the intrinsic stability of a linear delaydifferential equation with a single delay. We find the timestep values that force the normally stable system to numericalinstability, in the Runge-Kutta 2 and Runge-Kutta 4 numericalsolving techniques. We also arrive at the optimal time step toavoid 4th decimal place errors in the numerical solution for RK2and RK4. This ideal time step enables us to compute numericalsolutions more quickly while avoiding error upto the 4th decimalplace.

We perform the necessary analysis of a linear delay differentialequation with a single discrete delay. Following this we showhow discretization of time intervals can lead to instabilities

independent of the intrinsic parameters of the system. We findthe time step values that force the normally stable system tonumerical instability, in the Runge-Kutta 2 numerical scheme.This is followed by an LATEX reproduction of the Appendix sectionof   Local Bifurcation Analysis of Some Dual Congestion Control 

 Algorithms  by Prof. Gaurav Raina, IIT Madras.

 Keywords— Numerical Instability, Linear Differential  Equations,Runge-Kutta Eigenvalues

I. INTRODUCTION

Most control systems around us can be modelled at aprimitive level using simple first order delayed differentialequations. Various forms of stability of such systems have been

well-studied. It is necessary for a given control system to beasymptotically stable, as instability leads to oscillations anddivergence.

We start this paper by determining the stability criterionfor some common differential control systems - studying theeigenvalue characteristics to calculate necessary and sufficientconditions for asymptotic stability. In the systems we analyze,various intrinsic parameters can induce instability. However,we choose not to focus on a parameter   within   any of thesystems, but to induce instability externally. We vary thediscretization  step size in our mathematical tool (XPP), leadingto numerical instabilities depending on the computationalmethod. Our aim is not to intentionally destabilize the system,but to see how instabilities can be caused due to big step sizes

and how analysis is always a trade off between computationalpower and processing time.

In the field of numerical analysis, most numerical algo-rithms would arrive at the correct solution if there were noapproximation errors. But computational methods that are usedfor such analysis always discretize variables, converting thedifferential system into a set of difference equations. Thisleads to truncation and round-off errors. Most algorithms are

optimized to dampen these errors iteratively, but sometimesthese errors magnify with each iteration, and can even growexponentially. This leads to a different solution or sometimes,the complete lack of one.

This paper is organized as follows. In Section I, we analyzethe stability conditions for a delayed differential equation.In Section II, we add an instantaneous feedback componentand perform stability analysis. In Section III, we introducenumerical instabilities in the system and Section IV concludesthe discussion. For ease of exposition, all the calculationsassociated with the bifurcation analysis are contained in the

Appendix.

II. DELAYED  F EEDBACK

Our differential equation contains one discrete delay withno instantaneous feedback. This is modelled as,

d

dtx(t) = −κx(t − τ )   (1)

where κ, τ >  0. The corresponding characteristic equation forthis system (substituting  x(t) =  eλt) is

λ = −κe−λτ  (2)

Here, λ  is called the eigenvalue of the system. All eigenvalues

with   Re(λ)   <   0   denote that the system is asymptoticallystable for all values of  κ, τ >  0. In our system, it is possiblethat   Re(λ)   >   0, which would cause our system to diverge.Thus, we come up with a necessary condition for stability.

Calculations:   For   τ   = 0,  λ  = −κ. Since  Re(λ) = −κ <0, the system is asymptotically stable. Now, as we increaseτ , the system moves from stability to instability at a criticalτ  = τ c. At this point,  Re(λ) = 0, i.e. the eigenvalue is purelyimaginary. Any further increase in   τ c   will make  Re(λ)  >  0.Substituting λ  =  jωc   in (1)

 jωc + ae−jωcτ c = 0

Equating the real and imaginary parts of the above equationto  0,

ω − a sin(ωcτ c) = 0   (3)

a cos(ωcτ c) = 0   (4)

Squaring (3) and (4) and adding, we get

ω2c  = a2 (5)

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ADAPTIVE AND OPTIMAL CONTROL ASSIGNMENT - STAGE 1 , JANUARY-2014 2

Also, from (4),

ωcτ c  = π

2  (6)

Therefore, from (5) and (6),

τ ca = π

2

As  τ < τ c  for stability, the required condition for (1) is,

τ c  <   π2a   (7)

III. DELAYED AND I NSTANTANEOUS  F EEDBACK

Adding to Section I, many control systems also have aninstantaneous component of feedback. Considering a moregeneric delay differential equation,

d

dtx(t) = −ax(t) − bx(t − τ )   (8)

where a, b, τ > 0. The characteristic equation of the system is(substituting λ  =  eλt)

λ + a + be−λτ 

= 0   (9)

Where λ   is again the eigenvalue of the system. Following thesame line of analysis as in (3) (Re(λ)   <  0), we proceed tofind the necessary and/or sufficient conditions for stability.

Calculations:   For   τ   = 0,   λ   = −(a +  b). Since   Re(λ) =−(a + b) <  0, the system is asymptotically stable. Now, as weincrease τ , the system moves from stability to instability at acritical  τ   =  τ c. At this point,  Re(λ) = 0, i.e. the eigenvalueis purely imaginary. Any further increase in   τ c   will makeRe(λ) >  0. Substituting  λ  =  jωc   in (9)

 jωc + a + be−jωcτ c = 0   (10)

Equating the real and imaginary parts of the above equationto  0,ω − b sin(ωcτ c) = 0   (11)

a + b cos(ωcτ c) = 0   (12)

On squaring and adding (11) and (12), we get

ω2c  +  a2 = b2

ω2c  = b2 − a2 (13)

Also, from (12),

ωcτ c  = arccos−a

b

  (14)

Substituting ωc   from (13),

τ c 

b2 − a2 = arccos−a

b

As   τ < τ c   for stability, the required condition for (8) to beasymptotically stable is,

τ c  <  1√ b2 − a2

 arccos−a

b

  (15)

Fig. 1. Convergence, Oscillatory behaviour and divergence for various valuesof  dt

IV. NUMERICAL  I NSTABILITY

We use the mathematical tool  XPP   to simulate and analysesuch differential equations (in Sections II and III). Even thoughwe have necessary and sufficient conditions, the computationalmethod by itself can cause numerical instabilities. This isprimarily due to the difference in the step sizes used inthese methods. In induce instability, we focus only on delayedfeedback systems (particularly (1)). A preliminary simulationof the system for the conditions  x(0) = 100, τ   = 1, κ  = 1.5was carried out for varying values of  dt in  RK-2 computationalscheme.

We find, by brute force, that  dtc  = 0.989  is the critical timestep interval for numerical instability. We then simulate thesystem for dt > dtc, dt =  dtc  and dt > dtc. We see in the Fig.1 that just varying the time step alone is enough to change therate of convergence considerably. For small values of  dt, thesolver takes a lot of processing time and gives us an accurate,exponentially decaying system response. As we increase the

value of  dt, we reach oscillatory stability and finally completedivergence of the system response. An ideal compromise isgenerally 0.2   times the  dt  causing numerical instability.

V. CONCLUSION

We studied the necessary and sufficient conditions for sta-bility of a delayed system with and without instantaneousfeedback. On the context of numerical instabilities, we showedhow analysis of a system can lead to erroneous results eventhough the system itself is stable.

APPENDIX  AHOP F B IFURCATION  A NALYSIS

The structure of this Appendix is as follows: In an au-

tonomous nonlinear equation with a single discrete delay wefirst give conditions for the loss of local stability to occur viaa   Hopf bifurcation   [3]. Then following the style of analysisoutlined in [3], we perform the necessary calculations todetermine the type of the Hopf bifurcation and the asymptoticform of the bifurcating solutions as local instability   just   setsin. For now, we will only be concerned with the   first   Hopf condition.

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ADAPTIVE AND OPTIMAL CONTROL ASSIGNMENT - STAGE 1 , JANUARY-2014 3

 A. Nonlinear Equation With a Single Discrete Delay

Consider the following nonlinear delay differential equation

d

dtx(t) =  κf (x(t), x(t − τ ))   (16)

where   f   has a unique equilibrium denoted by   (x∗, y∗)   andκ,τ > 0. Define u(t) =  x(t)

−x∗, and take a Taylor expansion

of (16) including the linear, quadratic, and cubic terms toobtain

d

dtu(t) =  κξ xu(t) + κξ yu(t − τ ) + κξ xxu2(t)

+ κξ xyu(t)(t − τ ) + κξ yyu2(t − τ ) + κξ xxxu3(t)

+ κξ xxxu2(t)u(t − τ ) + κξ xyyu(t)u2(t − τ )

+ κξ yyyu3(t − τ ) + O(u4)   (17)

where letting  f ∗ denote evaluation of  f   at  (x∗, y∗)

ξ x =  f ∗x , ξ y  =  f ∗y , ξ xx =  12

f ∗xx

ξ xy = f ∗xy, ξ yy  = 1

2f ∗yy, ξ xxx =

 1

6f ∗xxx

ξ xxy = 1

2f ∗xxy, ξ xyy  =

 1

2f ∗xyy, ξ yyy  =

 1

6f ∗yyy .

Consider the linearized form (17) [and (16)], namely

d

dtx(t) =  κξ xu(t) + κξ yu(t − τ )   (18)

It is well known [2] that the linearized stability of the fixedpoint of (16) is given by the stability of the trivial fixed point

(u = 0)  of (18). The stability of (18) is given by the roots of the assosciated scharacteristic equation (obtained by lookingfor exponential solutions). Characteristic equations arisingfrom first order scalar delay equations with a   single  discretedelay appear to be well understood (see [2], [3], and thereferences therein). We state here a result for convenience [2,Th. (A.5)].

 Result I.1:   Consider a linear autonomous delay equationwhose corresponding characteristic equation is given by

λ + a + be−λt = 0   (19)

where   a ≥   0, b >   0, b > a, τ >   0. Then the correspondingsystem is stable  if and only if 

τ <  arccos(−a/b)√ 

b2 − a2

We now outline some essential calculations for two character-istic equations.

 Result I.2:   Consider a linear autonomous delay equationwhose corresponding characteristic equation is given by

λ + κbe−λτ  = 0   (20)

where κ, b, τ >  0, then the trivial solution of the correspondingsystem is stable for all  κ ∈ (0, κc)   undergoes a Hopf bifurca-tion at  κ  =  κc, where

κτ b =  π/2Calculations:   Consider the following characteristic equa-

tion:λ + κbe−λτ  = 0   (21)

where  kappa,b >  0. With   τ   = 0, the characteristic equationhas a negative real root. For  τ > 0. let  λ  = ±iω, ω > 0  whichgives (after separating the real and imaginary parts)

κbcos(ωt) = 0

ω − κbsin(ωt) = 0

From the aforementioned system, we get

ωτ  = (2n + 1)π

2 , n = 0, 1, 2, . . .

which gives

(2n + 1) π

2τ  − κb(−1)n = 0

yielding the critical values of  κ  as

κn = (2n + 1)π

2bτ   , n = 0, 2, 4, . . .

We only treat the case   n  = 0. This gives   ω0   = (π/2τ )   anddenote the critical value of the gain as  κc  = (π/2bτ ). We nowsatisfy the transversality condition of the Hopf spectrum, i.e.,

Redλ

dκκ=κc

= 0

Therefore, evaluating

κ=κc

=  −be−λτ 

1 − κbτe−λτ 

we obtain

Re

κ=κc

=  bπ/2

1 + π2/4 > 0

 Result I.3:   Consider a linear autonomous delay equationwhose corresponding characteristic equation is given by

λ + κa + κbe−λτ  = 0   (22)

where   κ ,a,b,τ >   0   and   b > a, then the trivial solution of the corresponding system is stable for all   κ ∈   (0, κc)   andundergoes a Hopf bifurcation at  κ  =  κc  where

κcτ  =  arccos(−a/b)√ 

b2 − a2

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ADAPTIVE AND OPTIMAL CONTROL ASSIGNMENT - STAGE 1 , JANUARY-2014 4

Calculations:   Consider the following characteristic equa-tion:

λ + κa + κbe−λτ  = 0   (23)

where  κ,a,b >  0   and   b > a. With   τ   = 0, the characteristicequation has a negative real root. For  τ > 0  let  λ  = ±iω, ω >0  which gives

κa + κbcos(ωt) = 0   (24)

ω − κbsin(ωt) = 0   (25)

For ω > 0  we get  cos(ωτ ) <  0  and sin(ωτ ) >  0, giving

2nπ + π

2 < ωτ < 2nπ + π, n = 0, 1, 2, . . .

We only treat the case  n  = 0. From (24)-(25), this gives

ω0  =  κc(b2 − a2)1/2

κc(b2 − a2)1/2τ  = arccos(−a/b)

where   κc   denotes the critical value of   κ   at   ω   =   ω0. Wealso need to satisfy the transversality condition of the Hopf spectrum, i.e.,

Re

κ=κc

=  bπ/2

1 + π2/4 > 0

Therefore, evaluating

κ=κc

= −(a + be−λτ )

1 − κbτe−λτ 

we obtain

Redλ

dκκ=κc

=  κcτ (b2 − a2)

1 + 2κcaτ  +  κ2cb2τ 2  > 0.

The calculations the follow enable us to address equationsabout the form of the bifurcating solutions of (17) as it transitsfrom stability  to  instability  via a Hopf bifurcation. For this wehave to take higher order terms, i.e., the quadratic and cubicterms of (17) into consideration. Following the work of [3],we now perform the requisite calculations.

Consider the following autonomous delay-differential sys-tem:

d

dtu(t) = Lµ + F (ut, u)   (26)

t >  0, µ ∈ R, where for  τ > 0

ut(θ) =  u(t + θ)   u : [−τ, 0] → R, θ ∈ [−τ, 0].

Lµ  is a one-parameter family of continuous (bounded) linearoperators defined as Lµ   :   C [−τ, 0]   →   R. The operatorF (ut, u) :   C [−τ, 0]   →   R   contains the nonlinear terms.Further, assume that F (ut, u)   is analytic and that F   and Lµ

depend analytically on the  bifurcation parameter  µ   for small

mod  µ. Note that (17) is a type of the form (26). The objectiveis now to cast (26) into the form

d

dtut  = A(µ)ut + Rut   (27)

which has  ut   rather than both  u   and  ut. First, transform thelinear problem (d/dt)u(t) = Lut. By the  Riesz representationtheorem, there exists an  n  x  n  matrix valued function η(·, µ) :

[−τ, 0] −→ Rn

2

, such that each component of  η   has boundedvariation and for all  φ ∈ C [−τ, 0]

Lφ =

   0−τ 

dη(θ, µ)φ(θ)

In particular

Lµut  =

   0−τ 

dη(θ, µ)u(t + θ)   (28)

Observe that

dη(θ, µ) = (κξ xδ (θ) + κξ yδ (θ + τ ))dθ

where δ (θ)  is the Dirac delta function, would satisy (28).For φ ∈ C 1[−τ, 0], define

A(µ)φ(θ) =

  dφ(θ)

dθ   , θ ∈ [−τ, 0) 0−τ 

 dη(s, µ)φ(s) ≡ Lµφ, θ = 0(29)

and

Rφ(θ) =

  0, θ ∈ [−τ, 0)F (φ, µ), θ = 0

Then, as   dut/dθ  ≡   dut/dt, system (26) becomes (27) asdesired.

The bifurcating periodic solutions u(t, µ()) of (26) ( where ≥ 0   is a small parameter) have amplitude O(), period P ()and a non-zero Floquet parameter   β (), where O, P , β   havethe following (convergent) expansions:

µ =  µ22 + µ44 + . . .

P  =  2π

ω0(1 + T 22 + T 44)

β  =  β 22 + β 44

The sign of   µ2   determines the   direction of bifurcation: If µ2  > 0, the bifurcation is   supercritical  and  µ2  < 0   implies a

subcritical bifurcation. The sign of  β 2  determines the  stabilityof   u(t, µ()):   asymptotic orbital stability   if   β 2   <   0   andinstability if  β 2  >  0  These coefficients will now be determined.We only need to compute the expressions at  µ  = 0, hence, weset  µ = 0   in the following. Let  q (θ)  be the eigenfunction forA(0)  corresponding to  λ(0)  namely

A(0)q (θ) =  iω0q (θ)

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ADAPTIVE AND OPTIMAL CONTROL ASSIGNMENT - STAGE 1 , JANUARY-2014 5

and define the  adjoint  operator A∗(0)  as

A∗(0)α(s) =

 −dα(s)

ds  , s ∈ (0, τ ] 0

−τ  dηT (t, 0)α(−t), s = 0

where ηT  denotes the transpose of  η.Note that the domains of  A   and A∗ are   C 1[−τ, 0]   and

C 1[0, τ ]. As

Aq (θ) =  λ(0)q (θ)

λ(0)  is an eigenvalue for A∗, and

A∗q ∗ = −iω0q ∗

for some nonzero vector   q ∗. For   φ  ∈   C [−τ, 0]   and   ψ   ∈C [0,tau], define an inner product

ψ, φ = ψ(0)·φ(0)−   0θ=−τ 

   θζ =0

ψ(ζ −θ)dη(θ)φ(ζ )dζ   (30)

where  a · b  meansn

i=1 aibi. Then, ψ, Aφ  = A∗ψ, φ   for

φ ∈ Dom(A), ψ ∈ Dom(A∗). Let  q (θ) = eiω0θ and  q ∗(s) =Deiω0s be the eigenvectors for

Aand

A∗ corresponding to the

eigenvalues  +iω0   and −iω0. With

D =  1

1 + τ κξ yeiω0τ 

we get q ∗, q  = 1  and q ∗, q  = 0. Using (30), we first confirmq ∗, q  = 1

q ∗, q  = D − Dκ

   0θ=−τ 

θiω0θ(ξ xδ (θ) + ξ yδ (θ + τ ))dθ

= D + Dκτξ ye−iω0τ 

= 1.

Similarly we can show that

 q ∗, q 

= 0. Again using (30), we

get

q ∗, q  = D +  Dκ

2iω0

   0θ=−τ 

(e−iω0θ − eiω0θ)(ξ xδ (θ) + ξ yδ (θ + τ ))dθ

= D +  Dκ

2iω0ξ y(eiω0τ  − e−iω0τ 

= 0.

For µt, a solution of (27) at  µ  = 0,define

z(t) = q ∗, ut,and

W (t, θ) =  ut(θ) − 2Re{z(t)q (θ)}.

Then, on the manifold,  C 0, W (t, θ) =  W (z(t), z(t), θ)  where

w(z , z , θ) =  w20(θ)z2

2  + w11(θ)zz  + w02(θ)

z2

2  + . . .   (31)

In effect,   z   and   z   are local coordinates for   C 0   in   C   in thedirection of  q ∗ and  q ∗, respectively. Note that  w   is real if  µt

is real and we deal only with real solutions. The existence of the center manifold   C 0   enables the reduction of (27) to an

ordinary differential equation for a single complex variable onC 0.At  µ  = 0, this is

z

(t) = q ∗, Aut + Rut= iω0z(t) + q ∗(0).F (w(z , z , θ) + 2Re{z(t)q (θ)})

= iωz (t) + q ∗(0).F 0(z, z).   (32)

which is written in abbreviated form as

z

(t) =  iωz(t) + g(z, z).   (33)

The next objective is to expand   g   in powers of   z   and   z   .However, we also have to determine the coefcients  W ij(θ)   in(31) . Once the   W ij   have been determined, the differentialequation (32) for  z  would be explicit [as abbreviated in (33)]where expanding the function  g(z, z) in powers of  z  and z  wehave

g(z, z) =  q ∗(0).F (z, z)

= g20z2

2  + g11zz  + g02

z2

2  + g21

z2z

2  + . . .

Following [4], we write

w

= u

t − zq − zq.

and using (27) and (33) we obtain

w

=

  Aw − 2Re{q ∗(0).F q (θ)}, θ ∈ [−τ, 0)Aw − 2Re{q ∗(0).F q (θ)} + (F )0, θ = 0

which is written as

w

= Aw + H (z , z , θ)   (34)

using (31), where

H (z , z , θ) =  H 20(θ)z2

2  + H 11(θ)zz  + H 02(θ)

z2

2  + . . .   (35)

Now on  C 0, near the origin

w

= wzz

+ wzz

.

Use (31) and (33) to replace  wz, z

(and their conjugates bytheir power series expansion) and equating this with (34),weget [4]

(2iω0 − A)w20(θ) =  H 20(θ)

−Aw11(θ) =  H 11(θ)

−(2iω0 + A)w02(θ) =  H 02(θ)

We start by observing

ut(θ) =  w(z , z , θ) + zq (θ) + zq (θ)

= w20(θ)z2

2  + w11(θ)zz  + w02(θ)

z2

2+ zeiω0θ + ze−iω0θ + . . .

from which we obtain  ut(0)   and  ut(−τ )   . We have actuallylooked ahead and as we will only be requiring the coefficients

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of  z2, z z , z2 and z2z, hence we only keep these relevant termsin the following expansions:

u2t (0) = (w(z,z, 0) + z + z)2

= z2 + z2 + 2zz  + z2z(2w11(0) + w20(0)) + . . .

ut(0)ut(−τ ) = (w(z,z, 0) + z + z)

× (w(z,z, −τ ) + ze−iω0τ  + zeiω0τ )

= z2e−

iω0τ  + zz(eiω0τ  + e−

iω0τ ) + z2eiω0τ 

+ z2z

w11(0)e−iω0τ  +

 w20

2  eiω0τ 

+ w11(−τ ) + w20(−τ )

2

+ . . .

u2t (−τ ) = (w(z,z, −τ ) + ze−iω0τ  + zeiω0τ )2

= z2e−2iω0τ  + z2e2iω0τ  + 2zz

+ z2z(2e−iω0τ w11(−τ ) + eiω0τ w20(−τ )) + . . .

u3t (0) = (w(z,z, 0) + z + z)3

= 3z2z + . . .

u2t (−τ )ut(0) = (w(z,z, −τ ) + ze−iω0τ  + zeiω0τ )2

× (w(z,z, 0) + z + z)u2t (0)ut(−τ ) = (w(z,z, 0) + z + z)2

× (w(z,z, −τ ) + ze−iωτ  + zeiωτ )

= z2z(eiωτ  + 2e−iωτ ) + . . .

u3t (−τ ) = (w(z,z, −τ ) + ze−iωτ  + zeiωτ )3

= 3z2ze−iωτ  + . . .

Recall that

g(z, z) =  q ∗(0).F (z, z) and

g(z, z) =  g20z2

2  + g11zz  + g02

z2

2  + g21

z2z

2  + . . .

After collecting the coefficients of  z2, z z , z2,  and z2z  we arein a position to calculate the coefficients of  g20, g11, g02, g21,which we do as

g20  =  q ∗κ[2ξ xx + 2ξ xye−iω0τ  + 2ξ yye−2iω0τ ]   (36)

g11  =  q ∗κ[2ξ xx + 2ξ xy(eiω0τ  + e−iω0τ ) + 2ξ yy ]   (37)

g02  =  q ∗κ[2ξ xx + 2ξ xyeiω0τ  + 2ξ yye2iω0τ ]   (38)

g02  =  q ∗κ[2ξ xx(2w11(0) + w20(0))

+ ξ xy(2w11(0)e−iω0τ  + w20(0)eiω0τ )

+ 2w11(−τ ) + w20(−τ ))+ ξ yy(4w11(−τ )e−iω0τ  + 2w20(−τ )eiω0τ )

+ 6ξ xxx + ξ xyy(2e−2iω0τ  + 4)

+ ξ xxy(2eiω0τ  + 4e−iω0τ ) + 6ξ yyye−iω0τ ]   (39)

Observe that in the expression for   g21   we havew11(0), w−τ , w20(0),   and   w20(−τ )   which we still need

to evaluate.Now for  θ ∈ [−τ, 0)

H (z , z , θ) = −2Re{q ∗(0).(F )0q (θ)}= −2Reg(z, z)q (θ)

= −g(z, z)q (θ) − g(z, z)q (θ)

= −

g20z2

2  + g11zz  + g02

z2

2  + . . .

q (θ)

− g20

z2

2   + g11zz  + g02z2

2   + . . .

q (θ)

which, compared with (35), yields

H 20(θ) = −g20q (θ) − g02q (θ)

H 11(θ) = −g11q (θ) − g11q (θ)

We already noted that

(2iω0 − A)w20(θ) =  H 20(θ)   (40)

−Aw11(θ) =  H 11(θ)   (41)

−(2iω0 + A)w02(θ) =  H 02(θ).   (42)

From (29),(40), and (41), we get the following equations:

w

20(θ) = 2iω0w20(θ) + g20q (θ) + g02q (θ)   (43)w

11(θ) =  g11q (θ) + g11q (θ)   (44)

Solving the differential equations (43) and (44) gives us

w20(θ) = −g20

iω0q (0)eiωθ −   g02

3iω0q (0)e−iωθ + Ee2iω0θ(45)

w11(θ) =  g11iω0

q (0)eiωθ −  g11iω0

q (0)e−iω0θ + F    (46)

for some   E, F    which will soon be determined. Now,forH (z,z, 0) = −2Re(q ∗.F 0q (0)) + F 0

H 20(0) = −g20q (0) − g02q (0)

+ κ(2ξ xx + 2ξ xye−iω0τ  + 2ξ yye−2iω0τ )   (47)

H 11(0) = −g11q (0) − g11q (0)+ κ(2ξ xx + ξ xy(eiω0τ  + eiω0τ ) + 2ξ yy)   (48)

From (29),(40), and (41), we getκξ xw20(0) + κξ yw20(−τ ) − 2iω0w20(0)

= g20q (0) + g20q (0)

− κ(2ξ xx + 2ξ xye−iω0τ  + 2ξ yye−2iω0τ ))   (49)

κξ xw11(0) + κξ yw11(−τ )

= g11q (0) + g11q (0)

− κ(2ξ xx + ξ xy(eiω0τ  + e−iω0τ ) + 2ξ yy)   (50)

We have the solution for   w20(θ)   and   w11(θ)

from (45) and (46), respectively. Hence,evaluatew20(0), w20(−τ ), w11(0), w11(−τ ),   substitue into(49),respectively, and calculate  E , F   as and (50)

E  =  Φ1

κξ x + κξ ye−2iω0τ  − 2iω0

F  =  Φ2

κξ x + κξ y

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where

Φ1  = (κξ x − 2iω0)

g20iω0

+  g023iω0

+ κξ y

g20iω0

e−iω0τ  +  g023iω0

eiω0τ 

+ RHS of  (49)

Φ2  = −κξ x g11

iω0 − g11

iω0− κξ y

g11

iω0 e

−iω0τ 

+

 g11

iω0 e

iω0τ + RHS of  (50)

All the quantities required for the computations associated forthe stability analysis of the Hopf bifurcation are completed.The analysis can be performed using [4]

c1(0) =  i

2ω0

g20g11 − 2| g11 |2 −  1

3| g02 |2

+

 g21

2  (51)

µ2  = −Rec1(0)

α

(0)  (52)

P  = 2π

ω0(1 + 2T 2 + O(4))   (53)

T 2  = −

Imc1(0) + µ2ω

(0)ω0

  (54)

β  =  2β 2 + (O)(4) β 2  = 2Rec1(0)  =

  µ

µ2(55)

where   c1(0)   is the lyapunov coefficient and  g20, g11, g02, g21are defined by (36) - (37) , respectively.Note that to use κ   to induce instability let κ  =  κc + µ,where aHopf bifurcation takes place at  µ  = 0, where κc  may evaluated

from Result (I.2) or Result (I.3). Then  α

(0) and  ω

(0) are thereal and imaginary components of  (dλdκ)  evaluated at  κ  =  κc.We now state the conditions that will enable us to verify thestability of the Hopf bifurcation.

i) The sign of  µ2  determines the direction of bifurcation.If µ2   >   0   then the Hopf bifurcation is   supercritical   if µ2 <  0   it is  subcritical.

ii) The sign of (the Floquet exponent)   β 2   determinesthe stability of the bifurcating periodic solutions. Theperiodic solutions are  asymptotically orbitally stable   if β 2  <  0  and  unstable   if  β 2  >  0.

 Note: If   α

(0)   >   0   then from (52) and (55), super-criticality of the bifurcating solution also establishesasymptotic orbital stability.

iii) For small |   µ |   the period of the bifurcating periodic

solutions is given by (53) which reduces to2πω0

  as

tends to zero.

iv) Further, the bifurcating periodic solutions have theasymptotic formµ(t, µ()) = 2Re[q (0)eiω0t]+2Re[Ee2iω0t+F]+O(3)for 0 ≤ t ≤ P ().

As we have performed all the necessary calculations, we candetermine the stability of the bifurcating solutions. This can atleast be done numerically for a particular choice of parameter

0.0 0.5 1.0 1.5 2.0

        0  .

        0

        0  .        5

        1  .        0

        1  .

        5

        2  .

        0

        2  .

        5

        3  .

        0

0.0 0.5 1.0 1.5 2.0

        0  .

        0

        0  .        5

        1  .        0

        1  .

        5

        2  .

        0

        2  .

        5

        3  .

        0

a

        b

Hopf Surface (T=1)Hopf Surface (T=2)

Fig. 2. Hopf surface of   λ +  a  +  be−λτ  = 0  where   a  ≥   0, b >   0, b > a.The region below the lines is stable.

values. See Fig. 1, which plots the Hopf surface for character-istic equations considered in this Appendix.

 Example I.4:   Consider the following delay equation:

d

dtu(t) =  η(−au(t) − bu(t − τ ) + ξ xxu2(t) + ξ xyu(t)(t − τ )

+ ξ yyu2(t − τ ) + ξ xxxu3(t) + ξ xxxu2(t)u(t − τ )

+ ξ xyyu(t)u2(t − τ ) + ξ yyyu3(t − τ ))   (56)

where  η , τ , a, b >   0   and   b > a. Here,   η   is motivated to bea   nondimensional   parameter that drives the aforementioned

equation just  beyond the locally stable regime. The linearizsedequation associated with the previous equation is

d

dtu(t) = −ηau(t) − ηbu(t − τ )

with the following characteristic equation

λ + ηa + ηbe−λτ  = 0.

A Hopf bifurcation takes place at

ηcτ  

b2 − a2 = arccos(−a/b)

where ηc  denotes the critical value of  η  which induces a Hopf bifurcation.

Let   η   =   ηc  + µ   with a Hopf bifurcation taking place at

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µ = 0. We have

α(0) ≡ Re

η=ηc

=  ηcτ (b2 − a2)

1 + 2ηcaτ  +  η2cb2τ 2

  > 0

ω(0) ≡ Im

η=ηc

=

√ b2 − a2(1 + aτ ηc

1 + 2ηcaτ  +  η2cb2τ 2

  > 0

ω0  =  π

2τ   D =

  1 − ηcbτ e−iω0τ 

1 + 2ηcaτ  +  η2cb2τ 2

D̄ =  1 − ηcbτ eiω0τ 

1 + 2ηcaτ  +  η2cb2τ 2

= 1 − ηcbτ (−a/b + i

 1 − a2/b2)

1 + 2ηcaτ  +  η2cb2τ 2

We are now able to perform all the calculations. Due to theform of the Hopf bifurcation, unfortunately this yields anextremely lengthy expression and would, in general, have tobe evaluated numerically.

However, if we let  a  = 0, we get the following linearizedequation:

ddt u(t) = −ηbu(t − τ )

with the characteristic equation

λ + ηbe−λt = 0

Case (1):   With   a   = 0   in (56), treat   η   as the bifurcationparameter.

The Hopf condition is  ηcbτ   =  π/2, where  ηc  denoted thecritical value of  η  which induces a Hopf bifurcation. Let  η =ηc + u  with a Hopf bifurcation taking place at  µ  = 0. We get

α(0)

≡Redλ

dηη=ηc

=  bπ/2

1 + π2

/4

 > 0

ω(0) ≡ Im

η=ηc

=  b

1 + π2/4 > 0

ω0  =  π

2τ   D =

  1 + iπ/2

1 + π2/4D̄ =

  1 − iπ/2

1 + π2/4

The Hopf bifurcation is supercritical if  µ2  >  0  and subcriticalif  µ2  <  0, where

µ2  =  1

πb ×

ξ 2xx4(π − 9)

5b  + ξ 2xy

(3π − 2)

5b  = ξ 2yy

2(11π − 4)

5b

+ ξ xxξ xy(7π − 18)

5b

  + ξ xxξ yy2(7π − 18)

5b+ ξ yxξ yy

(7π − 18)

5b

− 6ξ xxx + πξ xxy − 2ξ xyy + 3πξ yyy

.

Further, the bifurcating periodic solutions have the asymptoticform

u(t) =

 4(η − ηc)

µ2cos

πt

2τ 

+

η − ηc

µ2

×

2ξ xx5b

2sin

πt

τ 

− cos

πt

τ 

+ 5

− 2ξ xy

5bsinπt

τ + 2cosπt

τ 

+ 2ξ yy

5b

cos

πt

τ 

− 2sin

πt

τ 

+ 5

+ O(3)

for 0 ≤ t ≤ P (), where   = 

(η − ηc)/µ2, ηcbτ  = π/2.The expression for the period is given by

P () = 4τ 

1 + ((η − ηc)/µ2)

  1

5πb

36

b ξ 2xx +

 2

bξ 2xy +

 8

bξ 2yy

+ 18

b ξ xxξ xy +

 36

b ξ xxξ yy  +

 18

b ξ xyξ yy  + 30ξ xxx

+ 10ξ yyy

+ O(4)

.

We now examine each of the nonlinear terms in  isolation.For example, if we only have the term   ξ xxu2(t),then thedirection of the bifurcation is determined by the sign of   µ2,which is given by

µ2  = −ξ 2xx4(9 − π)

5πb2  < 0

which yields a subcritical Hopf irrespective of the magnitudeof  ξ xx. If we consider the terms ξ xyu(t)u(t−τ ) or ξ yyu2(t−τ )in isolation, we get

µ2  =  ξ 2xy3π − 2

5πb2  > 0   and   µ2  =  ξ 2yy

2(11π − 4

5πb2  > 0

respectively, which yields a supercritical Hopf in both casesirrespective of the magnitude of  ξ xy   or xiyy .

If we only had cubic nonlinearities and considered eachof them in isolation, then the sign (not the magnitude) of the coefficients plays an important role in determining thedirection of the bifurcation.

In the previous example, we used a nondimensional pa-rameter, namely   η   to induce a Hopf bifurcation and theninvestigated the stability of the bifurcation. Using the discretedelay, i.e.,τ  as the bifurcation parameter, would yield the sameresults as long as the coefficients of the nonlinear equation (42)were not functions of the delay.

In order to allow us to compare different algorithms, theexposition of the analysis throughout this paper has beenmotivated using the exogeneous parameter  η.

 Numerics:   We use a numerical example to exemplify theanalysis. Consider the following equation:

d

dtu(t) =  η(bu(t − τ ) + ηxyu(t)u(t − τ ))   (57)

The analysis suggests that as local stability is just violated, theprevious equation will undergo a supercritical Hopf bifurcation

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Fig. 3. Numerical integration of (57) where   ξxy   =  b  gave the larger limitcycle and  ξxy   = 2b   produced the smaller one.

where the   stable   bifurcating solutions will take the followingform:

u(t) =  b

ξ xy

 20π(η − ηc)

3π − 2  cos

πt

2τ 

+ higher order terms.

We need to choose some numerical values. Let   τ   = 1   andb   =   π/2, so we get a Hopf bifurcation at   η   =   ηc   whereηc  = 1. We let  η  = 1.05   which renders the previous equationin a locally unstable state.

We consider 2 cases:  ηxy = b  and ηxy = 2b  and we shouldexpect with the latter choice to see oscillations of half theamplitude. As we can see in Fig. 2 this is indeed the case.1

We now proceed to examine another nonlinear delay equa-tion.

Case(2):   With  (a = 0, b = 1, τ   = 1)   in (56), treat  η  as thebifurcation parameter.

The Hopf condition is   ηc   =   π/2, where   ηc   denotes thecritical value of  η  which induces a Hopf bifurcation. Lettingη =  ηc + µ  with a Hopf bifurcation taking place at  µ  = 0, wehave

α(0) ≡ Re

η=ηc

=  1π/2

1 + π2/4 > 0

ω(0) ≡ Im

η=ηc

=  1

1 + π2/4 > 0

ω0  =  π2

  D =   1 + iπ/21 + π2/4D̄ =   1 − iπ/21 + π2/4

The Hopf bifurcation is supercritical if  µ2  >  0  and subcriticalif  µ2  <  0, where

1Our numerical computations used a fourth-order RungeKutta scheme witha step size of 0.005.

Fig. 4. Effect of   ξxx, ξxy   and   ξyy   on the type of the Hopf bifurcation:supercritical if  µ2  >  0  and subcritical if  µ2  <  0.(a)ξxx  and  ξxy . (b) ξxx  andξyy   (c)ξxy   and  ξyy

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µ2  =  1

πb ×

ξ 2xx2(π − 9)

5  + ξ 2xy

(3π − 2)

10  = ξ 2yy

(11π − 4)

5

+ ξ xxξ xy(7π − 18)

10  + ξ xxξ yy

(7π − 18)

5

+ ξ yxξ yy(7π − 18)

10

− 3ξ xxx + πξ xxy − ξ xyy + 3π/2ξ yyy

.

Further, the bifurcating periodic solutions have the asymptoticform

u(t) =

 4(η − ηc)

µ2cos

πt

2

+

η − ηc

µ2

×

2ξ xx5

  (2sin (πt) − cos (πt) + 5)

−  2ξ xy5

  (sin (πt) + 2cos (πt))

+ 2ξ yy

5   (cos (πt) − 2sin (πt) + 5)

+ O(3)

for 0 ≤ t ≤ P (), where    = 

(η − ηc)/µ2, ηc  =  π/2.The expression for the period is given by

P () = 4

1 + ((η − ηc)/µ2)

  1

5π(36ξ 2xx + 2ξ 2xy + 8ξ 2yy

+ 18ξ xxξ xy + 36ξ xxξ yy  + 18ξ xyξ yy  + 30ξ xxx

+ 10ξ yyy ) + O(4)

.

The quadratic terms, if considered in isolation have thefollowing types of bifurcations:   ξ xxu2(t)   induces a subcrit-ical Hopf whereas   ξ xyu(t)u(t

 − τ )   and   ξ yyu2(t

 − τ )   yield

a supercritical Hopf. It is interesting to observe the effectcombinations of quadratic terms have on the type of the Hopf bifurcation.We  µ2   for three pairs, i.e.,(ξ xx   and   ξ xy),(ξ xx   andξ yy) and (ξ xy   and ξ yy) in Fig.3. Note that the pair  ξ xyandξ yyappears to be an attractive combination to possess. plot

REFERENCES

[1] Gaurav Raina,   Local Bifurcation Analysis of Some Dual CongestionControl Algorithms, IEEE Transactions on Automatic Control, 2005.

[2] J. K. Hale and S. M. V. Lunel,   Introduction to Functional Differential Equations, New York: Springer-Verlag, 1993.

[3] B. D. Hassard, N. D. Kazarinoff, and Y.-H. Wan, Theory and Applicationsof Hopf Bifurcation. Cambridge, U.K.: Cambridge Univ. Press,1981.

[4] B. D. Hassard, N. D. Kazarinoff, and Y.-H. Wan, Theory and Applica-tions of Hopf Bifurcation . Cambridge, U.K.: Cambridge Univ. Press,1981.

[5] C. V. Hollot, V. Misra, D. Towsley, and W. B. Gong, On designingimproved controllers for AQM routers supporting TCP flows,in Proc.IEEE Infocom, 2001.


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