Stability of distributed pendulum-like systems. Geotropic circumnutations of helianthus annus^{1}

G.A.Leonov^{2} and V.B.Smirnova

leonov@math.spbu.ru root@AL2189.spb.ru

1 Introduction

In this paper the advantage of a certain mathematical model for the stability study of pendulum-like systems is demonstrated. Frequency-domain criteria of global stability for distributed pendulum-like systems are obtained. The results are applied to the sunflower equation.

2 Frequency-domain stability criteria

Let us consider the integro-differential equation

σ̇(t) = α(t) + ρΦ(σ(t − h)) − ∫_{0}^{t} g(t − τ)Φ(σ(τ)) dτ  (1)

where α, σ, g: R_{+} → R, and Φ: R → R is Δ-periodic. Equation (1) possesses the main property of a pendulum-like mathematical model, i.e. if σ(t) is a solution of (1) then σ(t) + Δk (k ∈ Z) also is a solution of (1).

The most important problem in connection with a pendulum-like system is the problem of its asymptotic behaviour. In the simplest case of mathematical pendulum it is the alternative of damped oscillations about the stable equilibrium or the so-called circular movements.

Qualitative investigation of pendulum-like equations was started in 1933 by the paper of F.Tricomi [1] devoted to the second-order equation of mathematical pendulum:

σ̈ + aσ̇ + b sin σ = γ (a, b, γ > 0).  (2)

Then a number of publications devoted to this equation and to the equation

σ̈ + aσ̇ + Φ(σ) = 0  (3)

succeeded. The results were summarized in [2]. The phase portrait of (3) was exhaustively investigated by the end of the seventies. It is known that there exists a bifurcational value a_{cr} such that for a > a_{cr} every solution of (3) tends to a certain equilibrium and for a < a_{cr} equation (3) has circular solutions.

The research of second-order equation (3) was followed by the research of second-order systems [3, 4], of third-order systems [5, 6] and at last by the research of multidimensional systems [7, 8, 9] and equations with time-delay [10–12]. Various methods were exploited here: qualitative-numerical methods, approximate method of harmonic balance, Lyapunov direct method.

1Based on the lecture “Attractors of Volterra integro-differential equations” hold at the DFG-Workshop “Dimensions of attractors”, St. Petersburg, Russia, October 11-12, 1999

2Partly supported by the Russian Research Fund 98-01-01005

Lyapunov’s direct method occurred to be rather fruitful. But it turned out that classical Lyapunov functions are of no use here. They often give an “empty” stability domain in the space of parameters of a pendulum-like system. So certain new methods and approaches were elaborated in the frame of Lyapunov direct method, especially for the investigation of pendulum-like systems [8].

Equation (1) can be used as mathematical description of pendulum-like systems with time-delay or with distributed components. Stability problems for Volterra equation (1) are often successfully solved by the method of a priori integral estimates which was introduced in stability theory of control systems by V.M.Popov [13]. But similarly to the Lyapunov method the traditional Popov method does not give good results for pendulum-like systems. It turned out that the new technique elaborated in the case of ordinary differential equations can be adapted to the case of integro-differential equations. A number of stability theorems proved by means of Popov method joined with certain new approaches are cited in books [14, 15].

In this article we consider the simplest case when the nonlinear function Φ(σ) has the zero average over the period [0, Δ], i.e.

∫_{0}^{Δ} Φ(σ) dσ = 0.  (4)

On the one hand the proof of stability theorems in this case is rather transparent and on the other hand there exist a number of engineering, mechanical and bio-mechanical problems which can be described by (1), (4).

We suppose also that Φ(σ) is a continuously differentiable function with two zeros σ_{1}, σ_{2} on the period [0, Δ). We suppose further that α(t) is uniformly continuous on [0, +∞), g(t) is measurable and there exists a positive number κ such that

α(t)e^{κt}, g(t)e^{κt} ∈ L_{2}[0, +∞).  (5)

We shall use denotations

μ_{1} := inf_{σ∈[0,Δ)} Φ′(σ), μ_{2} := sup_{σ∈[0,Δ)} Φ′(σ).  (6)

(Note that μ_{1}μ_{2} < 0.)

Since the method of a priori integral estimates is applied here all the results are formulated in terms of the transfer function of the linear part of (1)

K(p) = ∫_{0}^{∞}g(t)e^{−pt}dt − ρe^{−ph} (p ∈ C).

Theorem 2.1 Suppose that all the relations (4)–(6) are valid. Suppose also that there exist numbers ε ≥ 0, δ ≥ 0, τ ≥ 0 such that for all ω ∈ R the following inequality is true

Ω(ω) := Re {K(iω) − ε|K(iω)|^{2} − τ[K(iω) + μ_{1}^{−1}iω]^{∗}[K(iω) + μ_{2}^{−1}iω] ≥ δ.  (7)

Then for any solution of (1)

σ̇ ∈ L_{2}[0, +∞),  (8)
σ̇(t) → 0 as t → +∞,  (9)
Φ(σ(t)) → 0 as t → +∞,  (10)
σ(t) → σ̂ as t → +∞,  (11)

where σ̂ is a zero of Φ.

Proof. Let η(t) := Φ(σ(t)) and let us construct the functions

η_{T} (t) := μ(t) \{ \begin{matrix}0 & for t < 0, \\ η(t) & for t ∈ [0, T], (T > 0, c > κ), \\ η(T)e^{c(T −t)} & for t > T \end{matrix}

where

μ(t) = \{ \begin{matrix}0 & for t < 0, \\ t & for t ∈ [0, 1], \\ 1 & for t > 1, \end{matrix}

ξ_{T} (t) := ρη_{T} (t − h) − ∫_{0}^{t} g(t − τ)η_{T} (τ) dτ.

Note that η_{T} is piece-wise continuously differentiable and

η_{T} (0) = 0.  (12)

We also introduce the functions η^{μ}(t) := μ(t)η(t),

ζ^{μ}(t) := ρη^{μ}(t − h) − ∫_{0}^{t} g(t − τ)η^{μ}(τ) dτ

and

σ_{0}^{μ}(t) := α(t) − ρ[μ(t − h) − 1]η(t − h) − ∫_{0}^{t} g(t − τ)(1 − μ(τ))η(τ) dτ.

Thus

ζ_{T} (t) = ζ^{μ}(t) for t ∈ [0, T]

and

σ̇(t) = σ_{0}^{μ}(t) + ζ^{μ}(t).  (13)

It is obvious that

η_{T} (t), η̇_{T} ∈ L^{1}[0, +∞) ∩ L^{2}[0, +∞).

The same inclusion is true for ζ_{T} (t). Let us consider for instance

∫_{0}^{∞}[ ∫_{0}^{t} g(t − τ)η_{T} (τ) dτ]^{2}dt = ∫_{0}^{∞}e^{−2κt}[ ∫_{0}^{t} (e^{κ(t−τ)}g(t − τ))(e^{κτ}η_{T} (τ))]^{2} dt.

Note that

\begin{matrix}∫_{0}^{t} (e^{κ(t−τ)}g(t − τ))(e^{κτ}η_{T} (τ)) dτ = ∫_{0}^{T} (e^{κ(t−τ)}g(t − τ))(e^{κτ}η^{μ}(τ)) dτ+ \\ +η(T) ∫_{T}^{t} (e^{κ(t−τ)}g(t − τ))(e^{κτ}μ(τ)e^{c(T −τ)}) dτ. \end{matrix}  (14)

The first term in the right side of (14) is a constant which depends on T. For the second term we have

\begin{matrix}( ∫_{T}^{t} (e^{κ(t−τ)}g(t − τ))(e^{κτ}μ(τ)e^{c(T −τ)}) dτ)^{2} ≤ \\ ≤ ∫_{T}^{t} e^{2κ(t−τ)}g^{2}(t − τ) dτ ∫_{T}^{t} e^{2cT} e^{2(κ−c)τ} dτ \end{matrix}

which leads by virtue of (5) to the inequality

( ∫_{T}^{t} (e^{κ(t−τ)}g(t − τ))(e^{κτ}μ(τ)e^{c(T −τ)}) dτ)^{2} ≤ C(T) [e^{2(κ−c)t} − e^{2(κ−c)T}] ,

where constant C depends on T. Hence ζ_{T} ∈ L^{2}[0, +∞) for every T > 0. Just in the same way one can demonstrate that ζ_{T} ∈ L^{1}[0, +∞).

Let us consider a set of functionals

\begin{matrix}B_{T} := ∫_{0}^{∞}\{ζ_{T} (t)η_{T} (t) + δ(η_{T} (t))^{2} + ε(ζ_{T} (t))^{2}+ \\ +τ[ζ_{T} (t) − μ_{1}^{−1}η̇_{T} (t)][ζ_{T} (t) − μ_{2}^{−1}η̇_{T} (t)]\} dt. \end{matrix}

and apply the Parseval equality to each of them.

We shall use the designation

F[f](iω) := ∫_{−∞}^{+∞}f(t)e^{−iωt}dt (f ∈ L^{1}(R) ∩ L^{2}(R))

for the Fourier transform of function f.

Note that the Fourier transforms of ζ_{T}, η_{T} and η̇_{T} satisfy the equalities

\begin{matrix}F[ζ_{T}](iω) = −K(iω)F[η_{T}](iω), \\ F[η̇_{T}](iω) = iωF[η_{T}](iω). \end{matrix}

So we get

\begin{matrix}B_{T} = −\frac{1}{2π} ∫_{−∞}^{+∞}\{Re K(iω) − δ − ε|K(iω)|^{2} − τRe \{(K(iω) + μ_{1}^{−1}iω)^{∗}· \\ ·(K(iω) + μ_{2}^{−1}iω\} |F[η_{T}](iω)|^{2}\} dω. \end{matrix}

It follows from (7) that

B_{T} ≤ 0 for all T > 0.  (15)

Let us represent B_{T} in the form

B_{T} = R_{T} + R_{1} + ∫_{T}^{∞}(ε + τ)(ζ_{T} (t))^{2} dt

with

\begin{matrix}R_{T} := ∫_{0}^{T} \{ζ^{μ}η^{μ} + δ(η^{μ})^{2} + ε(ζ^{μ})^{2} + τ[ζ^{μ} − μ_{1}^{−1}η̇^{μ}][ζ^{μ} − μ_{2}^{−1}η̇^{μ}]\} dt, \\ R_{1} := ∫_{T}^{∞}\{ζ_{T} η_{T} + δ(η_{T})^{2} + τ(η̇_{T})^{2}μ_{1}^{−1}μ_{2}^{−1} − τ(μ_{1}^{−1} + μ_{2}^{−1})ζ_{T} η̇_{T} \} dt. \end{matrix}

Note that by virtue of the very choice of η_{T} we have that

|R_{1}| < q_{1}  (16)

where q_{1} does not depend on T and consequently

B_{T} ≥ −q_{1} + R_{T} .

With the help of (13) we obtain

R_{T} = I_{T} + J_{T} + R_{0} + R_{2} + (ε + τ) ∫_{0}^{T}(σ_{0}^{μ})^{2} dt,  (17)

where

I_{T} := ∫_{0}^{T} {σ̇(t)Φ(σ(t)) + εσ̇^{2}(t) + δΦ^{2}(σ(t))} dt,

J_{T} := τ ∫_{0}^{T}(σ̇ − μ_{1}^{−1}η̇)(σ̇ − μ_{2}^{−1}η̇) dt,

R_{0} := ∫_{0}^{T} { − σ_{0}^{μ}η^{μ} − 2σ_{0}^{μ}σ̇(ε + τ) + τ(μ_{1}^{−1} + μ_{2}^{−1})η̇^{μ}σ_{0}^{μ}} dt,

R_{2} := ∫_{0}^{T} {(μ − 1)σ̇η + δ(μ^{2} − 1)η^{2} − μ_{1}^{−1}μ_{2}^{−1}τ(η̇^{2} − (η̇^{μ})^{2}) + τ(μ_{1}^{−1} + μ_{2}^{−1})(η̇ − η̇^{μ})σ̇} dt.

The functional J_{T} is non-negative since μ_{1} ≤ Φ′(σ) ≤ μ_{2}. The sum of functionals R_{2} and R_{0} is modulo bounded by a constant q_{2} independent of T. Then it follows from (15),

(16) and (17) that for all T > 0

I_{T} < q_{2} + q_{1}.  (18)

Let us consider now the functional I_{T}. Note that

∫_{0}^{T} σ̇(t)Φ(σ(t)) dt = ∫_{σ(0)}^{σ(T)}Φ(σ) dσ.

From this and from (4) it follows that

∫_{0}^{T} σ̇(t)Φ(σ(t)) dt < q_{3},  (19)

where q_{3} is independent of T.

Then by virtue of (18) we have that

σ̇(t) ∈ L_{2}[0, +∞)  (20)

and

Φ(σ(t)) ∈ L_{2}[0, +∞).  (21)

It is obvious that η(t) = Φ(σ(t)) is uniformly continuous on [0, +∞). The function σ̇(t) is uniformly continuous on [0, +∞) as well. Indeed, it is sufficient to consider the function

ζ(t) = ∫_{0}^{t} g(t − τ)η(τ) dτ = ∫_{0}^{t} η(t − τ)g(τ) dτ.

Note that

|ζ(t + Δt) − ζ(t)| ≤ ∫_{t}^{t+Δt}|η(t + Δt − τ)g(τ)| dτ + ∫_{0}^{t} |η(t + Δt − τ) − η(t − τ)| |g(τ)| dτ ≤

≤ sup_{σ∈[0,Δ)} Φ(σ) ∫_{t}^{t+Δt}|g(τ)| dτ + (L ∫_{0}^{∞} |g(τ)| dτ) Δt,

where L is a Lipschitz constant for η(t). From this and from (5) it follows that ζ(t) is uniformly continuous on [0, +∞).

So according to the Barbalat lemma [16] the inclusion (20) and (21) imply (9) and (10). From Lemma 2.5.1 [15] it follows that (10) implies (11) with Φ(σ̂) = 0. Theorem 2.1 is

proved.

Theorem 2.2 Suppose K(0) ≠ 0 and all the hypotheses of Theorem 2.1 are valid. Let there exist numbers ε ≥ 0 and τ ≥ 0 such that for all ω ∈ R^{1}

Ω(ω) ≥ 0.

Then all the relations (8)–(11) are true.

The proof of Theorem 2.2 is based on the following property of solutions of the equation (1) which is demonstrated in [14].

Lemma 2.1 Suppose Φ is continuously differentiable and inclusions (5) hold. Suppose also K(0) ≠ 0. Then for an arbitrary solution of (1) the validity of the property (8) implies the validity of the property (10).

The proof of Theorem 2.1 gives the opportunity to establish certain additional properties of solutions of (1).

Let us consider once more the inequality (18). Note that we can make the value of q_{1} as small as we wish by the choice of the value of c in the formula for η_{T} (t). Note also that q_{1} = q_{1}(δ, τ) and q_{2} = q_{2}(ε, δ, τ).

Theorem 2.3 Suppose that all the hypotheses of Theorem 2.1 are valid for certain values of ε, δ, τ. Suppose also that for a certain integer k the following inequality is true

2\sqrt{εδ} kM > q_{2}(ε, δ, τ),  (22)

where

M := ∫_{0}^{Δ} |Φ(σ) |dσ.

Then for all t > 0 the following estimate is true:

|σ(t) − σ(0)| < kΔ.  (23)

Proof. Suppose there exists a moment t_{1} such that

σ(t_{1}) = σ(0) + kΔ.  (24)

Then it follows from (18) that

∫_{σ(0)}^{σ(t_{1})}Φ(σ) dσ + ∫_{0}^{t_{1}} {εσ̇^{2}(t) + δΦ^{2}(σ(t))} dt < q_{1}(δ, τ) + q_{2}(ε, δ, τ).  (25)

But by virtue of (4) we can affirm that

∫_{σ(0)}^{σ(t_{1})}Φ(σ) dσ = 0.  (26)

On the other hand

εσ̇^{2}(t) + δΦ^{2}(σ(t)) ≥ 2\sqrt{εδ} |Φ(σ(t)) |σ̇(t)

and

∫_{0}^{t_{1}} \{εσ̇^{2}(t) + δΦ^{2}(σ(t))\} dt ≥ 2\sqrt{εδ} ∫_{σ(0)}^{σ(0)+kΔ}|Φ(σ) |dσ = 2\sqrt{εδ} kM.  (27)

Let c from the proof of Theorem 2.1 be so large that

q_{1}(δ, τ) < 2\sqrt{εδ} kM − q_{2}(ε, δ, τ).

Then the relations (26) and (27) contradict inequality (25). So for all t > 0

σ(t) < σ(0) + kΔ.

Just in the same way one can prove that for all t > 0

σ(t) > σ(0) − kΔ.

Theorem 2.3 is proved.

Theorem 2.3 gives for equation (1) a solution of the problem of ”cycle slipping” [17]. Definition 2.1 We say that σ(t) is a solution of (1) with k slipped cycles if there exists a T > 0 such that

| σ(T) − σ(0) | = kΔ and | σ(t) − σ(0) | < (k + 1)Δ for all t ≥ 0.

So if the hypotheses of Theorem 2.3 are true we can say that σ(t) slips less than k cycles.

3 Sufficient conditions of global stability for the sunflower equation.

Theorem 2.2 can be applied to a mathematical model of helical movements of tips of growing plants which is borrowed from [18]:

σ̇(t) = −\frac{b}{h} ∫_{−∞}^{t−h}e^{−a \frac{t−h−τ}{h}} sin σ(τ) dτ.  (28)

The model supposes that the cause of the movements is nonuniform auxin concentration around the stem. Here σ(t) is the angle of deviation of the stem from the vertical line and a, h and b are constants (h is the geotropic reaction time and b is inversely proportional to the diameter of the stem).

Equation (28) corresponds to equation (1) with

α(t) = −\frac{b}{h} ∫_{−∞}^{0} e^{−a \frac{t−h−τ}{h}} sin σ(τ) dτ, ρ = 0,

g(t) = \frac{b}{h} \{ \begin{matrix}e^{−a \frac{t−h}{h}} , & t ≥ h \\ 0, & t < h \end{matrix} \} , Φ(σ) = sin σ.

The transfer function for (28) is

K(p) = \frac{\frac{b}{h} e^{−ph}}{p + \frac{a}{h}}

and by the change of variables p̂ = ph we get

K̂(p̂) = \frac{Be^{−p̂}}{p̂ + a}

with B = bh. The symbol ”ˆ ” will be omitted in the following text.

Let us apply Theorem 2.2. Let μ_{1} = −1, μ_{2} = 1. Then

Ω(iω) ≡ τω^{2}(ω^{2} + a^{2}) − (τ + ε)B^{2} + B(a cos ω − ω sin ω).

Since

Ω(iω) > τ(ω^{2} + a^{2})ω^{2} − (τ + ε)B^{2} − \frac{aBω^{2}}{2} + aB − ω^{2}B

the frequency domain inequality of Theorem 2.2 will be fulfilled if for all ω ∈ [0, +∞)

τω^{2}(ω^{2} + a^{2}) − τB^{2} − \frac{aBω^{2}}{2} + aB − ω^{2}B > 0.

That is why all the hypotheses of Theorem 2.2 are true if the polynomial

P (y) = τy^{2} + (τa^{2} − \frac{aB}{2} − B) y + (aB − τB^{2})

has no real zeros in the right half plane. It is not difficult to see that this requirement is fulfilled if

B < \frac{\sqrt{2} a \sqrt{(a + 1)^{2} + 1}}{(a + 2)}.  (29)

Let us compare this analytical estimate and the results of the numerical investigations of the equation

σ̈(t) + \frac{a}{h} σ̇(t) + \frac{b}{h} sin(σ(t − h)) = 0

which can be derived from (28) [19].

In the article [19] the results of numerical investigation in the case of a = 4.8, b = 0.186 are described. It is asserted there that if h < 35 then any solution of the equivalent system

\begin{matrix}σ̇ = −η, \\ η̇ = −(a/h)η − (b/h) sin(σ(t − h)) \end{matrix}  (30)

tends to (0, 0) as t → +∞. If h ∈ (35, 80) then the system (30) has a periodic solution. Our estimate (29) guarantees in this case that for h < 31.588 any solution of (30) tends to a certain equilibrium σ = πn, n ∈ Z. More subtle analysis of the term outside the integral and of the initial function, similar to the argument used in the proof of Theorem 2.3 shows that any solution of (30) tends to (0, 0).

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