Haddad W.M. Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 125

Moreover, show that x

∗

is an equilibrium point of (2.136).

Problem 2.104 (Gronwall-Bellman Lemma). Assume there exist

continuous functions α, x : R → R and β : R → R

such that

x(t) ≤ α(t) +

β(s)x(s)ds, t ≥ t

. (2.266)

Show that

x(t) ≤ α(t) +

α(s)β(s)e

β(τ )dτ

ds, t ≥ t

. (2.267)

Furth ermore, if α(t) ≡ α ∈ R, then show th at

x(t) ≤ αe

β(s)ds

, t ≥ t

. (2.268)

Finally, if, in addition, β(t) ≡ β ≥ 0 show that x(t), t ≥ t

, satisﬁes (2.153).

Problem 2.105. Consider the nonlinear dynamical system

˙x(t) = f(x(t)), x(0) = x

, t ≥ 0, (2.269)

˙x(t) = g(x(t)), x(0) = x

, t ≥ 0, (2.270)

where f, g : D → R

are continuously diﬀerentiable functions on D. T he

dynamical systems (2.269) and (2.270) are topologically equivalent on D if

there exists a homeomorphism H : D → D which maps trajectories of (2.269)

onto trajectories of (2.270) and p reserves th eir orientation in time. Show

that the nonlinear dynamical sys tem

˙x(t) = f (x(t))[1 + kf(x(t))k]

−1

, x(0) = x

, t ≥ 0, (2.271)

where f : D → R

is continuously diﬀerentiable on D, has a unique solution

deﬁned for all t ≥ 0. Moreover, show that (2.271) is topologically equivalent

to (2.269) on R

Problem 2.106. Consider the nonlinear time-varying dynamical sys-

tem (2.211) where f(t, ·) : D → R

is Lipschitz continuous on D f or all

t ∈ [t

, t

] and f(·, x) : [t

, t

] → R

is piecewise continuous on [t

, t

] for all

x ∈ D. Furtherm ore, assume there exist constants α, β ∈ R such that

kf(t, x)k ≤ α + βkxk, (t, x) ∈ [0, ∞) × D . (2.272)

Show that th e solution x(t), t ∈ I

, to (2.211) satisﬁes

kx(t)k ≤ kx

β(t−t

)

β(t−t

)

−1], t ∈ I

, (2.273)

and h en ce, cannot exhibit ﬁnite escape time.

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126 CHAPTER 2

Problem 2.107 (Comparison Principle). Cons ider the scalar nonlin-

ear d y namical system

˙x(t) = f (t, x(t)), x(t

) = x

, t ≥ t

, (2.274)

where f(t, ·) : D → R is Lipschitz continuous on D for all t ∈ [t

, t

] and

f(·, x) : [t

, t

] → R is continuous on [t

, t

] f or all x ∈ D. Let v : [t

, t

] → D

be such that

˙v(t) ≤ f (t, v(t)), v(t

) ≤ x

, t ≥ t

. (2.275)

Show that v(t) ≤ x(t) for all t ∈ [t

, t

Problem 2.108. Prove Theorem 2.25 using Picard’s method of su c-

cessive approximations (see Problem 2.101).

Problem 2.109. Prove Theorem 2.37.

Problem 2.110. Prove Theorem 2.38.

Problem 2.111. Prove Theorem 2.39.

Problem 2.112. Consider the nonlinear dynamical s ys tem (2.136).

Let f : D → R

be continuously diﬀerentiable on D and let s : S → D

be the ﬂow generated by (2.136) and S

△

= {(t, x

) ∈ R ×D : t ∈ I

}. Show

that s : S → D is continuously d iﬀerentiable on S.

Problem 2.113. Show that if th e solution x(t) to (2.136) correspond-

ing to an initial condition x(0) = x

is bounded for all t ≤ 0, then the

negative limit set α(x

) of x(t), t ≤ 0, is a nonempty, compact, invariant,

and connected set. Fur thermore, show that x(t) → α(x

) as t → −∞.

Problem 2. 114. Consider the nonlinear dynamical system (2.136) and

let s(t, x), t ≥ 0, denote the solution to (2.136) with the initial condition

x(0) = x. Show that the positive limit set of (2.136) is given by ω(x

) =

∩

t≥0

, where O

△

= {s(t, x) : t ∈ [0, ∞)} denotes the positive orbit of

(2.136).

Problem 2.115. Consider the nonlinear dynamical s ys tem (2.136).

Let R ⊂ D and let M be the largest invariant set in R. Show that:

i) M is the union of all motions deﬁ ned on (−∞, ∞) that remain in R

for all t ∈ R.

ii) x ∈ M if and only if I

= (−∞, ∞) and s(t, x) ∈ R for all t ∈ R.

iii) If R is compact, then M is compact.

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 127

Problem 2. 116. Consider the nonlinear dynamical system G given by

(2.215). Show that M ⊂ D is invariant with respect to G if and only if M

is positively and negatively invariant with respect to G.

Problem 2. 117. Consider the nonlinear dynamical system G given by

(2.215) with m aximum interval of existence I

= R for all x

∈ R

. Show

that the sets D and Ø are positively and negatively invariant with respect

to G.

Problem 2. 118. Consider the nonlinear dynamical system G given by

(2.215). Show that for every x ∈ D, O

, O

, and O

−

are, respectively,

invariant, positively invariant, and negatively invariant w ith respect to G.

Problem 2. 119. Consider the nonlinear dynamical system G given by

(2.215), let M ⊂ D, and let O

(M) = ∪{O

: x ∈ M}. Show th at M is

invariant, positively invariant, or negatively invariant with respect to G if

and only if, respectively, O

(M) ⊆ M, O

(M) ⊆ M, or O

−

(M) ⊆ M.

Problem 2. 120. Consider the nonlinear dynamical system G given by

(2.215) and let M ⊂ D. Show that M is invariant, positively invariant, or

negatively invariant with respect to G if and only if, respectively, each of the

connected components of M is invariant, positively invariant, or negatively

invariant with respect to G.

Problem 2. 121. Consider the nonlinear dynamical system G given by

(2.215). Show that if x ∈ D is such that x = s(t, x) for some t ∈ R, then

x = s(nt, x) for all n ∈ Z.

Problem 2. 122. Consider the nonlinear dynamical system G given by

(2.215). Let x ∈ D. Show that the following statements are equivalent:

i) x is an equilibrium point.

ii) {x} = O

iii) {x} = O

iv) {x} = O

−

v) {x} = {y ∈ D : y = s(t, x), t ∈ [a, b]} for a < b.

vi) There exists a sequ en ce {t

}

∞

n=0

, with t

> 0 and t

→ 0 as n → ∞,

such that x = s(t

, x) f or each n.

(Hint: Use Problem 2.121 to establish the equivalence of i) with vi).)

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128 CHAPTER 2

Problem 2. 123. Consider the nonlinear dynamical system G given by

(2.215). Show that if x 6= s(t, x) for some x ∈ D and t ∈ R, then there

exists open neighborhoods U of x and V of s(t, x) such that V = {s(t, y) :

y ∈ U and t ∈ {t}} and U ∩ V = Ø.

Problem 2. 124. Consider the nonlinear dynamical system G given by

(2.215) with t ≥ 0. Show that x ∈ D is an equilibrium point of G if and only

if every neighborhood of x contains a positive orbit.

Problem 2. 125. Consider the nonlinear dynamical system G given by

(2.215). Show that the set of all equilibrium points x ∈ D of G is closed.

Problem 2. 126. Consider the nonlinear dynamical system G given by

(2.215). Show that for every x ∈ D:

i) ω(x) =

: y ∈ O

} =

: n ∈ Z}.

ii) ω(x) = ω(s(t, x)) for t ∈ R.

Problem 2. 127. Consider the nonlinear dynamical system G given by

(2.215). Sh ow that x ∈ D is a periodic point of G if and only if ω(x) =

α(x) = O

Problem 2.128. Consider the nonlinear dynamical system

˙x

(t) = x

(t) + x

(t)x

(t), x

(0) = x

, t ≥ 0, (2.276)

˙x

(t) = −x

(t) + x

(t)x

(t), x

(0) = x

. (2.277)

Show that the linearization of (2.276) and (2.277) at (x

, x

) = (0, 0)

possesses a continuum of periodic solutions whereas (2.276) and (2.277) has

no periodic solutions.

Problem 2.129. Consider the nonlinear dynamical system given by

(2.220) and (2.221) in Example 2.38. Show th at the d ivergence ∇f(x) < 0

in the annular region D = {x ∈ R

: 2/3 < x

+ x

< 2} and yet as shown

in Example 2.38, (2.220) and (2.221) possesses a limit cycle in D. Why does

this contradict Bendixson’s theorem?

Problem 2.130. Consider the second-order nonlinear mechanical sys-

tem consisting of a unit mass with a nonlinear spring and a nonlinear damper

given by

¨x(t) + f ( ˙x(t)) + g(x(t)) = 0, x(0) = x

, ˙x(0) = ˙x

, t ≥ 0, (2.278)

where f(·) is the damping force exerted by the damper and g(·) is

the restoring force of the nonlinear spring. Suppose f(·) and g(·) are

continuously diﬀerentiable.

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 129

i) Show that this system has no periodic solutions if f

′

( ˙x) 6= 0 for all

˙x ∈ R. What does this mean physically?

ii) Use the Poincar´e-Bendixson theorem to show that the undamped

nonlinear mechanical system (f( ˙x) ≡ 0) always has a continuum of

periodic solutions if xg(x) > 0, x 6= 0.

Problem 2.131. Consider the nonlinear dynamical system

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (2.279)

˙x

(t) = x

(t) −x

(t) −αx

(t) + x

(t)x

(t), x

(0) = x

, (2.280)

where α > 0. Does (2.279) and (2.280) possess a periodic solution?

Problem 2. 132. Show that if ∇f (x) = 0 for all x ∈ D, w here f : R

→

, then Bendixson’s theorem implies that there may be a trivial periodic

orbit in D but there is no limit cycle in D.

Problem 2.133. Consider the nonlinear dynamical system

˙x

(t) = x

(t) cos x

(t), x

(0) = x

, t ≥ 0, (2.281)

˙x

(t) = sin x

(t), x

(0) = x

. (2.282)

Show that (2.281) and (2.282) does n ot possess a limit cycle.

Problem 2.134. Consider the nonlinear dynamical system

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (2.283)

˙x

(t) = −x

(t) −x

(t) + x

(t), x

(0) = x

. (2.284)

Use Dulac’s theorem to show that (2.283) and (2.284) does not possess a

periodic solution. (Hint: Use the Dulac function g(x

, x

) = e

−2x

Problem 2.135. Consider the nonlinear dynamical system

˙x

(t) = x

(t) −x

(t)x

(t), x

(0) = x

, t ≥ 0, (2.285)

˙x

(t) = x

(t) −x

(t)x

(t), x

(0) = x

. (2.286)

Use Dulac’s theorem to show that (2.285) and (2.286) does not possess a

periodic solution in the nonnegative orthant R

. (Hint: Use the Dulac

function g(x

, x

) = 1/x

, x

6= 0 and x

6= 0.)

Problem 2.136. Consider the second-order nonlinear dy namical sys-

tem (2.230) and (2.231) with vector ﬁeld f : D → R

, where D is not a

simply conn ected region in R

. Show that in this case Dulac’s theorem is

no longer valid.

Problem 2.137. Consider Dulac’s theorem with D being topologically

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130 CHAPTER 2

equivalent to an annulus, th at is, D has exactly one hole in it. Show that in

this case (2.230) and (2.231) possesses at most one closed orbit in D. (Hint:

Use Green’s theorem.)

Problem 2.138. Prove Dulac’s theorem.

Problem 2.139. Consider the nonlinear dynamical system

˙x

(t) = x

(t) −x

(t), x

(0) = x

, t ≥ 0, (2.287)

˙x

(t) = x

(t) + x

(t) −x

(t), x

(0) = x

. (2.288)

Use the Poincar´e-Bendixson theorem to show that (2.287) and (2.288) has

a periodic orbit.

Problem 2.140. Consider the nonlinear dynamical system

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (2.289)

˙x

(t) = −x

(t) + x

(t)[1 − x

(t) −x

(t)], x

(0) = x

. (2.290)

Use the Poincar´e-Bendixson theorem to show that (2.289) and (2.290) has a

unique limit cycle. Furthermore, show that every trajectory of (2.289) and

(2.290), except for the equilibrium point, approaches this limit cycle.

Problem 2.141. Consider the nonlinear dynamical system

˙x

(t) = −x

(t) + x

(t)[1 −x

(t) −x

(t)], x

(0) = x

, t ≥ 0, (2.291)

˙x

(t) = x

(t) + x

(t)[1 − x

(t) −x

(t)], x

(0) = x

(2.292)

˙x

(t) = αx

(t), x

(0) = x

, (2.293)

where α ∈ R. Use the Poincar´e-Bendixson theorem to show that (2.291)–

(2.293) has a periodic orbit.

Problem 2.142. Consider the nonlinear dynamical system

˙x

(t) = x

(t) + αx

(t)[β

− x

(t) −x

(t)]

, x

(0) = x

, t ≥ 0, (2.294)

˙x

(t) = −x

(t) + αx

(t)[β

− x

(t) − x

(t)]

, x

(0) = x

, (2.295)

where α, β ∈ R and γ ≥ 1. For γ = 1, ﬁnd the solution of (2.294) and

(2.295). Does (2.294) and (2.295) exhib it periodic solutions for γ = 1?

Repeat the problem for γ = 2.

Problem 2.143. Consider the nonlinear dynamical system

˙x

(t) = x

(t) + x

(t) −x

(t)[|x

(t)| + |x

(t)|], x

(0) = x

, t ≥ 0, (2.296)

˙x

(t) = −2x

(t) + x

(t) − x

(t)[|x

(t)| + |x

(t)|], x

(0) = x

. (2.297)

Use the Poincar´e-Bendixson theorem to show that (2.296) and (2.297) has

a periodic orbit.

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 131

Problem 2.144. Consider the nonlinear dynamical system

˙r(t) = r(t)[1 − r

(t)] + µr(t) cos θ(t), r(0) = r

, t ≥ 0, (2.298)

θ(t) = 1, θ(0) = θ

. (2.299)

As shown in Example 2.38, when µ = 0 (2.298) and (2.299) possesses a

stable limit cycle. Show that a closed orbit still exists for suﬃciently small

µ > 0.

Problem 2.145. Consider the nonlinear dynamical system

˙x

(t) = x

(t) − x

(t)[x

(t) + 5x

(t)], x

(0) = x

, t ≥ 0,

(2.300)

˙x

(t) = x

(t) + x

(t) − x

(t)[x

(t) + 5x

(t)], x

(0) = x

. (2.301)

Use the Poincar´e-Bendixson theorem to show that (2.300) and (2.301) h as

a periodic orbit. (Hint: Rewrite the system in polar coordinates using

r ˙r = x

˙x

+ x

˙x

and

θ = (x

˙x

− x

˙x

)/r

Problem 2.146. Consider the nonlinear dynamical system

˙x

(t) = x

(t) −x

(t), x

(0) = x

, t ≥ 0, (2.302)

˙x

(t) = x

(t) + x

(t) −x

(t), x

(0) = x

. (2.303)

Use the Poincar´e-Bendixson theorem to show that (2.302) and (2.303) h as

a periodic orbit.

Problem 2.147. Consider the nonlinear dynamical system

˙x

(t) = x

(t)[1 − 4x

(t) −x

(t)] −

(t)[1 + x

(t)], x

(0) = x

, t ≥ 0,

(2.304)

˙x

(t) = x

(t)[1 − 4x

(t) −x

(t)] + 2x

(t)[1 + x

(t)], x

(0) = x

. (2.305)

Use the Poincar´e-Bendixson theorem to show that all trajectories of (2.304)

and (2.305) approach the ellipse E

△

= {(x

, x

) ∈ R

: 4x

+ x

= 1} as

t → ∞.

Problem 2.148. Consider the nonlinear dynamical system

˙x

(t) = −x

(t) −x

(t) + x

(t)[x

(t) + 2x

(t)], x

(0) = x

, t ≥ 0,

(2.306)

˙x

(t) = x

(t) − x

(t) + x

(t)[x

(t) + 2x

(t)], x

(0) = x

. (2.307)

Use the Poincar´e-Bendixson theorem to show that (2.306) and (2.307) h as

a periodic orbit.

Problem 2.149. Consider the second-order dynamical system (2.230)

and (2.231). Let V : D ⊆ R

→ R be conserved along the ﬂow of (2.230)

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132 CHAPTER 2

and (2.231), that is,

V (x

, x

) =

∂V

∂x

, x

) +

∂V

∂x

, x

) = 0. (2.308)

Show that if V is not constant on any open set U ⊂ D, then (2.230) and

(2.231) does not possess any limit cycle.

Problem 2.150 (Isocline Method). The method of isoclines is a

technique for generating solution cur ves in the two-dimensional state space

(that is, the phase plane) f or various initial conditions. In particular,

given the second-order nonlinear dynamical system (2.230) and (2.231), and

assuming f

, x

) 6= 0, it follows that th e slope of the system trajectories

of (2.230) and (2.231) in the phase plane passing through a point (x

, x

) is

given by

, x

)

, x

)

= s(x

, x

). (2.309)

Isoclines correspond to curves x

= g(x

) in the phase plane on which the

slope s(x

, x

) = c, where c is a constant. These curves can be obtained by

solving the equation

, x

) = cf

, x

). (2.310)

A family of phase p lane trajectories corresponding to various initial condi-

tions is called a phase portrait of (2.230) and (2.231). If the slope at points

, x

) and (x

, −x

) is such that s(x

, x

) = −s(x

, −x

), then the phase

portrait is symmetric about the x

-axis. Similarly, if s(x

, x

) = −s(−x

), then the phase portrait is symmetric about the x

-axis. Finally, if

s(x

, x

) = s(−x

, −x

), then the phase portrait is symmetric about the

origin. Using the method of iso clines, sketch the phase portraits of the

following dynamical systems.

¨x(t) + x

(t) − x(t) = 0, x(0) = x

, ˙x(0) = ˙x

, t ≥ 0, (2.311)

¨x(t) + sin x(t) = 0, x(0) = x

, ˙x(0) = ˙x

, t ≥ 0, (2.312)

¨x(t) − 0.2[1 − x

(t)] ˙x(t) + x(t) = 0, x(0) = x

, ˙x(0) = ˙x

, t ≥ 0,

(2.313)

¨x(t) +

˙x(t) + 2x(t) + x

(t) = 0, x(0) = x

, ˙x(0) = ˙x

, t ≥ 0.

(2.314)

Verify your sketches by simulations.

Problem 2.151. Consider the nonlinear Lienard system given by

¨x(t) + f (x(t)) ˙x(t) + g(x(t)) = 0, x(0) = x

, ˙x(0) = ˙x

, t ≥ 0. (2.315)

Suppose that f and g satisfy the following conditions:

i) f (x) and g(x) are continuously diﬀerentiable f or all x ∈ R.

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 133

ii) g(−x) = −g(x) for all x ∈ R.

iii) xg(x) > 0, x ∈ R, x 6= 0.

iv) f(−x) = f(x) for all x ∈ R.

v) F (x) =

f(σ)dσ is such that F (0) = 0, F

′

(0) < 0, F (·) h as a single

positive zero at x = a, and F (x) → ∞ for x ≥ a as x → ∞.

Show that (2.315) has a unique, stable limit cycle.

2.15 Notes and References

The qualitative analysis of diﬀerential equations was ﬁrs t developed by Henri

Poincar´e [358–360] with furth er developments given by Birkhoﬀ [60,61]. The

material on existence, uniqueness, and continuity of solutions with respect to

system in itial conditions of nonlinear diﬀerential equations is standard and

can be found in most textbooks on diﬀerential equations. Notable textbooks

include those by Hartman [185], Coddington and Levinson [96], Hirsch and

Smale [196], Agarwal and Lakshmikantham [4], Lefschetz [264], Nemytskii

and Stepanov [333], Hale [179], and Miller and Michel [315]. The topics on

matrix analysis are also standard and can be found in Bellman [37], Horn

and Joh nson [201,202], Lancaster and Tismenetsky [256], Gantmacher [132],

Stewart and Su n [417], and Bernstein [45]. For a thorough presentation

on advanced calculus and mathematical analysis the reader is r eferred to

Rudin [373], Royden [371], Apostol [12], Graves [139], Edwards [114], Naylor

and S ell [332], Fleming [120], Munkres [323], Hoﬀman [198], and Bartle

[30]. See also Halmos [182] and Luenberger [288]. Finally, the study of the

existence and absence of perio dic orbits in nonlinear dynamical systems was

fathered by Poincar´e [358–360] and further developed by Bendixson [39].

For a modern treatment of these results, see Hirsch and Smale [196], Hale

and Kocak [180], Wiggins [454], and Perko [349].

NonlinearBook10pt November 20, 2007