48 Int. J. Dynamical Systems and Differential Equations, Vol. 3, Nos. 1/2, 2011

Stability results for higher dimensional equations

on time scales

Joan Hoffacker

Department of Mathematical Sciences,Clemson University,Clemson, SC 29634, USAE-mail: johoff@clemson.eduWebsite: http://www.math.clemson.edu/facstaff/johoff.htm

Billy Jackson*

School of Mathematical Sciences,University of Northern Colorado,Greeley, CO 80639, USAE-mail: billy.jackson@unco.edu*Corresponding author

Abstract: The stability of the equilibrium solution of x∆ = f(x) wheref : D ⊂ R

n → Rn and x : T → R

n, T a time scale, with regards to theHilger Circle is examined.

Keywords: dynamic equations on time scales; Jacobian; stability;instability.

Reference to this paper should be made as follows: Hoffacker, J. andJackson, B. (2011) ‘Stability results for higher dimensional equationson time scales’, Int. J. Dynamical Systems and Differential Equations,Vol. 3, Nos. 1/2, pp.48–58.

Biographical notes: Joan Hoffacker got her PhD Degree underthe direction of Allan Peterson in 2001 at the University ofNebraska-Lincoln. Currently, she is an Associate Professor ofMathematics at Clemson University. Her research interests includedynamic equations on time scales and problems that occur in biology.

Billy Jackson got his PhD Degree under the direction of JohnDavis at Baylor University. He currently is an Assistant Professor ofMathematics at Saint Xavier University in Chicago, Illinois. His researchinterests include control theory, dynamic equations on time scales, andmathematical biology.

Herein we examine the stability of the equilibrium solution of x∆ = f(x) wheref : D ⊂ R

n → Rn and x : T → R

n, T a time scale, with regards to the Hilger Circle.These results extend the results of Gard and Hoffacker (2003) to higher dimensional

Copyright © 2011 Inderscience Enterprises Ltd.

Stability results for higher dimensional equations on time scales 49

systems. While we include introductory material in the appendix, more informationcan be found in Bohner and Peterson (2001, 2003). In the following we assumethat T is a time scale which is unbounded above, and has µ∗ = lim sup

t→∞µ(t) being

finite. The difficulty arises in unbounded graininess as seen in Gard and Hoffacker(2003) since the equation x∆ = px has no stable solutions other than the trivial oneif lim sup

t→∞µ(t) is unbounded, no matter the regressive value of p.

1 Stability via the Jacobian

Definition 1.1: We call x an equilibrium solution of x∆ = f(x) if f(x) = 0.

Definition 1.2: We say the equilibrium solution x of x∆ = f(x) is exponentiallystable if there exist δ, λ, β > 0 with −λ ∈ R+ such that ||x(0) − x|| < δ implies||x(t) − x|| < βe−λ(t, 0)||x(0) − x||.

The matrix equation

AT V + [I + µAT ]V A + [I + µAT ]V ∆ + µ[I + µAT ]V ∆A = −I (1)

will be useful in proving results regarding the Jacobian. In Davis et al. (not dated),this particular equation is studied in some depth and is referred to as the TimeScale Dynamic Lyapunov Equation (TSDLE).

One must be careful in saying that all eigenvalues “lie inside the Hilger Circle”as the Hilger Circle changes with respect to the graininess function µ(t). Thus wetake

I = lim supt→∞

Iµ(t)

and

I = lim inft→∞

Iµ(t).

Recall that if A has all of its eigenvalues lying in I, then the matrix equation (1)has solution

V (t) =∫ ∞

t

eAT (s, t)eA(s, t)∆s.

Further, we have

V ∆ = −[I + µAT ]−1AT V [I + µA]−1 − V A[I + µA]−1

−[I + µAT ]−1[I + µA]−1.

Note that if A has eigenvalues in I, then since V is positive definite, the followingbounds hold for some M, N ∈ R

+:

N ||x||2 ≤ xT V x ≤ M ||x||2.

50 J. Hoffacker and B. Jackson

Let

D = supt∈T

||(I + µA)−1|| and D = supt∈T

||(1 + µAT )−1||

so that if T has µ∗ < ∞, we can bound V ∆ in norm by

||V ∆|| ≤ DD||A||M + M ||A||D + DD.

Theorem 1.1: Let T be a time scale with µ∗ < ∞. Given the state equation x∆ =f(x), let f : D ⊂ R

n → Rn be such that f ∈ C1(D), and let x be an equilibrium

solution of this equation. Then for A = ∂f∂x (x) regressive and having eigenvalues

all residing in I, it follows that the equilibrium solution x is exponentially stable.

Proof: Since f ∈ C1(D), we can use Lagrange’s formula for every y such thaty + x ∈ D to obtain

f(x + y) − f(x) =∫ 1

0

∂f

∂x(x + sy) y ds

=∫ 1

0

[∂f

∂x(x + sy) − ∂f

∂x(x)

]y ds +

∂f

∂x(x)y

= Ay + g(y),

where

g(y) =∫ 1

0

[∂f

∂x(x + sy) − ∂f

∂x(x)

]y ds.

Note that

||g(y)|| ≤∫ 1

0

∥∥∥∥∂f

∂x(x + sy) − ∂f

∂x(x)

∥∥∥∥ ds||y||.

Since ∂f∂x is continuous, it is uniformly continuous in the unit ball ||y|| ≤ 1. Let ω(r)

be the corresponding modulus of continuity. Thus, we have

||g(y)|| ≤∫ 1

0ω(s||y||) ds ||y|| ≤ ω(||y||) ||y||.

Since limr→0

ω(r) = 0, it follows that for any γ > 0, there exists δ(γ) > 0 such that

if ||y|| ≤ δ(γ), then ||g(y)|| ≤ γ||y||. Let x be an arbitrary solution of the stateequation and set y(t) = x(t) − x. Then

y∆(t) = x∆(t) = f(x(t)) = f(x + y(t)) − f(x) = Ay(t) + g(y(t)).

Set

V (t) =∫ ∞

t

eAT (s, t)eA(s, t)∆s,

Stability results for higher dimensional equations on time scales 51

and let V ∗ be an upper bound for the norm of V ∆(t), which is guaranteed by ourdiscussion above.

Now define

V (t) = yT (t)V (t)y(t).

V is delta differentiable with

V ∆ = y∆T

V y + yT σ

V y∆ + yT σ

V ∆yσ

= [yT AT + gT (y)]V y + [yT + µ(yT AT + gT (y))]V [Ay + g(y)]

+[yT + µ(yT AT + gT (y))]V ∆[y + µ(Ay + g(y))]

= yT AT V y + gT (y)V y + yT V Ay + yT V g(y)

+µyT AT V Ay + µgT (y)V Ay + µyT AT V g(y) + µgT (y)V g(y)

+yT V ∆y + µyT V ∆Ay + µyT V ∆g(y) + µyT AT V ∆y + µ2yT AT V ∆Ay

+µ2yT AT V ∆g(y) + µgT (y)V ∆y + µ2gT (y)V ∆Ay + µ2gT (y)V ∆g(y)

= yT [AT V + V A + µAT V A + V ∆ + µAT V ∆ + µV ∆A + µ2AT V ∆A]y

+gT (y)V y + yT V g(y) + µX + µ2Y

= −||y||2 + gT (y)V y + yT V g(y) + µX + µ2Y,

where

X = gT (y)V Ay + yT AT V g(y) + gT (y)V g(y) + yT V ∆g(y) + gT (y)V ∆y,

and

Y = yT AT V ∆g(y) + gT (y)V ∆Ay + gT (y)V ∆g(y).

Thus, we have

V ∆(t) ≤ −||y||2 + 2M ||g(y)|| ||y|| + µ∗X∗ + µ∗2Y ∗,

where

X∗ = 2||A|| ||M || ||y|| ||g(y)|| + M ||g(y)||2 + 2V ∗||y|| ||g(y)||,Y ∗ = 2||A|| M ||y|| ||g(y)|| + ||g(y)||2V ∗.

Let 0 < λ < 1M be such that −λ ∈ R+. Choose γ > 0 such that

(µ∗M + µ∗2V ∗)γ2 + (2M + 2||A||Mµ∗ + 2V ∗µ∗ + 2µ∗2 ||A||M)γ − 1 ≤ −λM.

Note that λ < 1M guarantees the existence of γ > 0 that solves this inequality.

Then for ||y|| ≤ δ(γ), it follows that V ∆(t) ≤ −Mλ||y||2. But we have theinequality

||y||2 ≥ 1M

V ⇒ −Mλ||y||2 ≤ −λV ,

52 J. Hoffacker and B. Jackson

and so

V ∆(t) ≤ −λV (t)

for ||y|| ≤ δ(γ).Since −λ ∈ R+, the following inequalities are equivalent:

V ∆(t) ≤ −λV (t) (2)1

1 − µλe�(−λ)(t, 0)V ∆(t) +

λ

1 − µλe�(−λ)(t, 0)V (t) ≤ 0 (3)

e�(−λ)(σ(t), 0)V ∆(t) +λ

1 − µλe�(−λ)(t, 0)V (t) ≤ 0 (4)

[e�(−λ)(t, 0)V (t)]∆ ≤ 0. (5)

Thus, the function e�(−λ)(t, 0)V (t) is decreasing and therefore

e�(−λ)(t, 0)V (t) ≤ V (0) ≤ M ||y(0)||2 ≤ M ||x(0) − x||2

implies

N ||y||2 ≤ V (t) ≤ Me−λ(t, 0)||x(0) − x||2.

Hence, for ||x(0) − x|| < δ(γ), we have

||x(t) − x|| ≤√

M

Ne−λ(t, 0)||x(0) − x|| ≤

√M

Ne−λ(t, 0)||x(0) − x||,

where λ = 1−√

1−µ∗λµ∗ . (Note that λ > 0 and −λ ∈ R+.) Thus, the exponential

stability of x is established. �

The regressivity condition is necessary to prevent solutions from going to zero infinite time. In Doan et al. (not dated) Example 12, the authors show that thereexists a time scale T which has bounded graininess such that the system

x∆ = −x

is exponentially stable and the system

x∆ =(

−1 10 −1

)x

is not exponentially stable. However in their example, λ = −1 is not in I for theirparticular choice of T, so this is not a contradiction to the above theorem.

Stability results for higher dimensional equations on time scales 53

2 Instability

On R and Z, when one looks at stability, the regions of stability can be foundwith sharp boundaries: outside of this region, the system becomes unstable. So onthese two particular time scales, biological models can be analysed in such a waythat sharp bounds on quantities that one is trying to stabilise can be found. Withthe above theorem, bounds may not be sharp, because instability is not guaranteedoutside I. However, we can guarantee instability in some instances as the followingtheorems show.

Theorem 2.1: Let T be a time scale with µ∗ < ∞. Given the state equation x∆ =f(x), let f : D ⊂ R

n → Rn be such that f ∈ C1(D), and let x be an equilibrium

solution of this equation. Then for A = ∂f∂x (x) regressive and having either

1 an eigenvalue λ with Re(λ) > 0, or

2 a negative real eigenvalue λ with lim inft→∞

µ(t) > − 2λ

it follows that the equilibrium solution x is unstable.

Proof: Suppose x were stable. Then the Jordan form can be used to rewrite

x∆ = Ax + f(t, x)

in the form

y∆ = By + g(t, y)

where

B =(

B1 00 B2

)

with B1 having k columns of eigenvalues which satisfy conditions (1) or (2), andB2 containing all other eigenvalues. Apply the standard basis vector ej to obtainyj for j ≤ k. Then by Gard and Hoffacker (2003), if λ ∈ R and λ > 0, or if λ ∈ R

with lim inft→∞

µ(t) > − 2λ , we have that |yi(t)| → ∞, and so ||y(t)|| → ∞ and thus

||x(t)|| → ∞ as well. On the other hand, if λ is complex valued with Re(λ) > 0then (Pötzsche et al., 2003) gives the result. �

3 Examples

Example 3.1: We first consider the system

x∆ =(

− 12 0

0 − 52

)x.

It is natural to ask on what kinds of time scales is the trivial solution to thisdynamic equation exponentially stable or unstable. Clearly the eigenvalues λ1 = − 1

2

54 J. Hoffacker and B. Jackson

and λ2 = − 52 are both real. Thus if T = R, this equation is exponentially stable.

However if the time scale has

lim inft→∞

µ(t) > min{

− 2λ1

, − 2λ1

}=

45

then in fact the trivial solution is unstable. It is well known that this equationis unstable if T = Z, and in fact in this case lim inf

t→∞µ(t) = 1 > 4

5 . Note that this

“change in stablility” is actually the result of the Hilger Circle being dependent onµ(t).

Example 3.2: Now, we examine the stability of the system

x∆ = (x2 + y2 − 1)x − y

y∆ = x + (x2 + y2 − 1)y

for the time scale T = P[1, 12 ] = [0, 1] ∪ [ 32 , 5

2 ] ∪ . . . Note that on this time scale, µ∗ =12 . The only equilibrium for this system is (0, 0). The Jacobian matrix evaluated atthis equilibrium has eigenvalues of λ1,2 = −1 ± i. Since λ1,2 ∈ I, the origin is stable.In fact, the origin is stable for this system for any time scale T with µ∗ < 1.

Example 3.3: We conclude by looking at the system

x∆ = x2 + y2 − 5y∆ = x2 − y2 − 3

defined on the time scale T = C3 + Z (that is, the integer translates of theCantor tertiary set). On this time scale, we have µ∗ = 1

3 . The equilibriaare (2, 1), (2, −1), (−2, 1) and (−2, −1). The corresponding Jacobian eigenvaluesassociated with each are λ1,2 = 1 ±

√17, λ1,2 = 3 ± i

√7, λ1,2 = −3 ± i

√7,

and λ1,2 = −1 ±√

17, respectively. Hence, (−2, 1) is stable since −3 ± i√

7 ∈ I

for this choice of T, while the other three equilibria are unstable for alltime scales.

4 Appendix on time scales

A time scale is simply any nonempty closed set of real numbers, and the time-scalecalculus is the unification and extension of discrete calculus, quantum calculus,continuous calculus, and indeed arbitrary real-number calculus to a new, moregeneral and overarching theory (Hilger, 1990). The definitions that follow here willserve as a short summary of the time-scale calculus; they can be found in Bohnerand Peterson (2001, 2003) and the references therein.

Definition 4.1: Define the forward (backward) jump operator σ(t) at t fort < sup T (respectively ρ(t) at t for t > inf T) by

σ(t) = inf{τ > t : τ ∈ T}, (ρ(t) = sup{τ < t : τ ∈ T}, ) for all t ∈ T.

Stability results for higher dimensional equations on time scales 55

Also define σ(sup T) = sup T, if sup T < ∞, and ρ(inf T) = inf T, if inf T > −∞.Define the graininess function µ : T → R by µ(t) = σ(t) − t.

Throughout this work the assumption is made that T is unbounded aboveand has the topology that it inherits from the standard topology on the realnumbers R. Also assume throughout that a < b are points in T and define the timescale interval [a, b]T = {t ∈ T : a ≤ t ≤ b}. The jump operators σ and ρ allow theclassification of points in a time scale in the following way: If σ(t) > t, then thepoint t is right-scattered, while if ρ(t) < t, then t is left-scattered. If σ(t) = t, thent is right-dense; if t > inf T and ρ(t) = t, then t is left-dense. The following definesthe so-called delta derivative.

Definition 4.2: Fix t ∈ T and let y : T → R. Define y∆(t) to be the number (if itexists) with the property that given ε > 0 there is a neighbourhood U of t such that,for all s ∈ U ,

|[y(σ(t)) − y(s)] − y∆(t)[σ(t) − s]| ≤ ε|σ(t) − s|.

Call y∆(t) the (delta) derivative of y(t) at t.

Definition 4.3: If F∆(t) = f(t), then define the (Cauchy) delta integral by

∫ t

a

f(s)∆s = F (t) − F (a).

Similar definitions hold for the nabla derivative and integral. The following theoremis due to Hilger (1990).

Theorem 4.1: Assume that f : T → R and let t ∈ T.

1 If f is differentiable at t, then f is continuous at t.

2 If f is continuous at t and t is right-scattered, then f is differentiable at twith

f∆(t) =f(σ(t)) − f(t)

σ(t) − t.

3 If f is differentiable and t is right-dense, then

f∆(t) = lims→t

f(t) − f(s)t − s

.

4 If f is differentiable at t, then f(σ(t)) = f(t) + µ(t)f∆(t).

Next we define the important concept of right-dense continuity. An important factconcerning right-dense continuity is that every right-dense continuous function hasa delta antiderivative (Bohner and Peterson, 2001, Theorem 1.74). This implies thatthe delta definite integral of any right-dense continuous function exists.

Definition 4.4: The function f : T → R is right-dense continuous, denotedf ∈ Crd(T; R), provided f is continuous at every right-dense point t ∈ T, andlims→t− f(s) exists and is finite at every left-dense point t ∈ T.

56 J. Hoffacker and B. Jackson

A function p : T → R is regressive provided 1 + µ(t)p(t) = 0, ∀t ∈ T. Let

R := {p ∈ Crd(T; R) : 1 + µ(t)p(t) = 0, t ∈ T}.

Also, p ∈ R+ if and only if p ∈ R and 1 + µ(t)p(t) > 0, ∀t ∈ T. Then if p ∈ R,t0 ∈ T, one can define the generalised exponential function ep(t, t0) to be the uniquesolution of the initial value problem

x∆ = p(t)x, x(t0) = 1.

Many of the properties of this generalised exponential function ep(t, t0) listed inTheorem 4.2 below are employed throughout this work.

Theorem 4.2 (Bohner and Peterson, 2001, Theorem 2.36): If p, q ∈ R and s, t ∈ T,then

1 e0(t, s) ≡ 1 and ep(t, t) ≡ 1;

2 ep(σ(t), s) = (1 + µ(t)p(t))ep(t, s);

3 1ep(t,s) = e�p(t, s), where p := −p

1+µp ;

4 ep(t, s) = 1ep(s,t) = e�p(s, t);

5 ep(t, s)ep(s, r) = ep(t, r);

6 ep(t, s)eq(t, s) = ep⊕q(t, s), where p ⊕ q := p + q + µpq;

7 ep(t,s)eq(t,s) = ep�q(t, s).

Again a similar list of properties for the nabla exponential function e exists; seeBohner and Peterson (2003, Chapter 3).

Definition 4.5: For h > 0, we define the Hilger complex numbers, the Hilger realaxis, the Hilger alternating axis, and the Hilger imaginary circle as:

Ch ={

z ∈ C : z = − 1h

}

Rh ={

z ∈ Ch : z ∈ R and z > − 1h

}

Ah ={

z ∈ Ch : z ∈ R and z < − 1h

}

Ih ={

z ∈ Ch :∣∣∣∣z +

1h

∣∣∣∣ =1h

}.

For h = 0, let C0 := C, R0 = R, I0 = iR, and A0 = ∅.

In Pötzsche et al. (2003) the set of exponential stability S(T) is defined. We areinterested in a subset of the set S(T), defined as

SC(T) ={

ξ ∈ C : α = lim supt→∞

1t − t0

∫ t

t0

ln |1 + µ(τ)ξ(τ)|µ(τ)

∆τ < 0}

. (6)

Stability results for higher dimensional equations on time scales 57

The region defined by |1 + ξµ| < 1 in (6) is termed the Hilger Circle, H, a circlein the left-hand complex plane that passes through the origin, with centre at − 1

µ .Note that, for systems of non-constant graininess, the Hilger Circle changes radiusdynamically.

We now consider the linear system of dynamic equations

y∆ = Ay (7)

where A is an n × n matrix valued function on T.

Definition 4.6: An n × n matrix valued function A on a time scale T is calledregressive provided

I + µ(t)A is invertible for all t ∈ T.

Definition 4.7: Let t0 ∈ T and assume that A is a regressive n × n matrix valuedfunction. The unique matrix solution of the initial value problem

Y ∆ = AY, Y (t0) = I,

where I denotes the usual identity matrix, is called the matrix exponential functionand is denoted by eA(t, t0).

The following theorem contains useful properties of the matrix exponentialfunction.

Theorem 4.3: If A, B are regressive matrix valued functions on T, then

1 e0(t, s) = I and eA(t, t) = I;

2 eA(σ(t), s) = (I + µ(t)A(t))eA(t, s);

3 e−1A (t, s) = e∗

�A∗(t, s);

4 eA(t, s) = e−1A (s, t) = e∗

�A∗(s, t);

5 eA(t, s)eA(s, r) = eA(t, r);

6 eA(t, s)eB(t, s) = eA⊕B(t, s) if eA(t, s) and B(t) commute.

References

Bohner, M. and Peterson, A.C. (2001) Dynamic Equations on Time Scales, An Introductionwith Applications, Birkhäuser, Boston.

Bohner, M. and Peterson, A.C. (Eds.) (2003) Advances in Dynamic Equations on TimeScales, Birkhäuser, Boston.

Davis, J.M., Gravagne, I.A., Marks, R.J. and Ramos, A.A. (not dated) ‘Algebraic anddynamic Lyapunov equations on time scales’, submitted.

Doan, T.S., Kalauch, A. and Siegmund, S. (not dated) ‘Exponential stability of lineartime-invariant systems on time scales’, Nonlinear Dynamics and Systems Theory, toappear.

58 J. Hoffacker and B. Jackson

Gard, T. and Hoffacker, J. (2003) ‘Asymptotic behavior of natural growth on time scales.Special issue: dynamic equations on time scales’, Dynam. Systems Appl., Vol. 12,Nos. 1–2, pp.131–147.

Hilger, S. (1990) ‘Analysis on measure chains – a unified approach to continuous anddiscrete calculus’, Results Math., Vol. 18, pp.18–56.

Pötzsche, C., Siegmund, S. and Wirth, F. (2003) ‘A spectral characterization of exponentialstability for linear time-invariant systems on time scales’, Discrete Contin. Dyn. Syst.,Vol. 9, pp.1223–1241.