Electronic Journal of Qualitative Theory of Differential Equations Spec. Ed. I, 2009 No. 13, 1–10; http://www.math.u-szeged.hu/ejqtde/

ON THE CRITICAL VALUES OF PARAMETRIC RESONANCE IN MEISSNER’S EQUATION BY THE

METHOD OF DIFFERENCE EQUATIONS

László Hatvani^{1} Bolyai Institute, University of Szeged Aradi vértanúk tere 1, Szeged, H-6720 Hungary e-mail:hatvani@math.u-szeged.hu

Honoring the Career of John Graef on the Occasion of His Sixty-Seventh Birthday

Abstract

In this paper the second order liner differential equation

\{ \begin{matrix}x′′ + a^{2}(t)x = 0, \\ a(t) = \{ \begin{matrix}π + ε, & if 2nT ≤ t < 2nT + T_{1}, \\ π − ε, & if 2nT + T_{1} ≤ t < 2nT + T_{1} + T_{2}, (n = 0, 1, 2, …), \end{matrix} \end{matrix}

is investigated, where T_{1} > 0, T_{2} > 0 (T := (T_{1} + T_{2})/2) and ε ∈ [0, π). We say that a parametric resonance occurs in this equation if for every ε > 0 sufficiently small there are T_{1}(ε), T_{2}(ε) such that the equation has solutions with amplitudes tending to ∞, as t → ∞. The period 2T_{∗} of the parametric excitation is called a critical value of the parametric resonance if T_{∗} = T_{1}(ε) + T_{2}(ε) with some T_{1}, T_{2} for all sufficiently small ε > 0. We give a new simple geometric proof for the fact that the critical values are the natural numbers. We apply our method also to find the most effective control destabilizing the equilibrium x = 0, x′ = 0, and to give a sufficient condition for the parametric resonance in the asymmetric case T_{1} ≠ T_{2}.

Key words and phrases: Parametric resonance, Meissner’s equation, impulsive differential equations, difference equations, problem of swinging. AMS (MOS) Subject Classifications: 34C11, 70L05

1 Introduction

Consider a system of differential equations

x′ = f(t, x) (f(t, 0) ≡ 0),  (1)

1Supported by the Hungarian National Foundation for Scientific Research (OTKA K75517) and Analysis and Stochastics Research Group of the Hungarian Academy of Sciences

with a right-hand side periodic with respect to t, i.e., f: [0, ∞) × ℝ^{n} → ℝ^{n}, f(t + P) ≡ f(t) for some P > 0. It can happen that the zero solutions of equations x′ = f(t_{0}, x) are stable in Lyapunov’s sense for every t_{0} ≥ 0, but the zero solution of equation (1) is unstable. This phenomenon is called parametric resonance. The conditions of this phenomenon can be studied [2, 5, 13, 15] by the method of period mapping (Poincaré mapping) x(P; 0, ·): ℝ^{n} → ℝ^{n}, where x(·; 0, x_{0}): [0, ∞) → ℝ^{n} denotes the solution of (1) satisfying the initial condition x(0; 0, x_{0}) = x_{0}. If (1) is linear, then any fundamental matrix X(t) of the system

x′ = A(t)x (A(t + P) ≡ A(t))  (2)

has a representation of the form

X(t) = Y (t)e^{Rt} (Y (t + P) ≡ Y (t)),

where R ∈ ℝ^{n×n} is called the monodromy matrix to (2) (Floquet theory [5]). Stability properties of the zero solution of (2) (among them conditions of the parametric resonance) are determined by the constant monodromy matrix R. However, to have matrix R one has to generate a set of n linearly independent solutions of (2).

A special case of (2) is Hill’s equation [9]

x′′ + a^{2}(t)x = 0 (x ∈ ℝ; a(t + P) ≡ a(t))  (3)

describing, e.g., the motion of the moon. The key step of generating two linearly independent solutions to (3) can be done by elementary functions in the case of Meissner’s equation [2, 10, 12, 13], when the coefficient a in (3) is a piece-wise constant function. This type of equation plays an important role in technical applications [16]. It has been found out that parametric resonance is possible only at certain critical values of the period P.

In this paper, using a purely elementary approach independent of Floquet’s theory, we deduce the critical values of parametric resonance for that general case of Meissner’s equation when the lengths in the time of the two pieces in the coefficient a in (3) can be different. D. R. Merkin [13, p. 263], proved that the critical values are independent of the relation between these lengths. We show, however, that the realization of parametric resonance at these critical values does depend on the difference between the lengths, and we give a condition (in term of the difference) sufficient for the parametric resonance. We apply our approach to the problem of swinging to get an effective state-dependent control of the lengths of the pieces in the coefficient a.

2 The Method

If a is a piece-wise constant function then equation (3) has the following form. Given two sequences of positive numbers {a_{k}}_{k=1}^{∞}, {t_{k}}_{k=1}^{∞}, which will be denoted simply by {a_{k}} and {t_{k}}, respectively, (t_{k} ≤ t_{k+1}, k ∈ ℕ), t_{0} := 0, consider the equation

x′′ + a^{2}(t)x = 0,  a(t) := a_{k} if t_{k−1} ≤ t < t_{k} (k ∈ ℕ).  (4)

Definition 2.1 A function x : [0, ∞) → ℝ is called a solution of (4) if it is continuously differentiable on [0, ∞), it is twice differentiable and solves the equation on every [t_{k−1}, t_{k}) for k ∈ ℕ.

At first we transform equation (4) into a system of impulsive differential equations, which will be replaced with a system of difference equations. The dynamics generated by this system can be described by elementary geometric transformations. This method was introduced in [6], and it was also used in [7, 8]; we reformulate it to make the present paper self-contained.

Introducing the new state variable y := x′/a_{k} on the interval [t_{k−1}, t_{k}), we can reexpress equation (4) in the form of a 2-dimensional system

x′ = a_{k}y, y′ = −a_{k}x,  if t_{k−1} ≤ t < t_{k},  (k ∈ ℕ).  (5)

Since we want to have a system of first order differential equations equivalent to (4), we ought to require additional “connectivity” conditions of solutions of (5) which follow from Definition 2.1: a function t ↦ (x(t), y(t)) is a solution of (5) on [0, ∞) if t ↦ (x(t), a(t)y(t)) is continuous on [0, ∞), and t ↦ (x(t), y(t)) is differentiable and solves system (5) on [t_{k−1}, t_{k}) for k ∈ ℕ.

Since the function t ↦ x′(t) = a(t)y(t) has to be continuous on [0, ∞), the function t ↦ y(t) is right-continuous for all t ≥ 0 and satisfies a_{k}y(t_{k} −0) = a_{k+1}y(t_{k}) for k ∈ ℕ, where y(t_{k} − 0) denotes the left-hand side limit of y at t_{k}. Accordingly, the system of first order differential equations equivalent to (4) is

\{ \begin{matrix}x′ = a_{k}y, y′ = −a_{k}x, if t_{k−1} ≤ t < t_{k}, \\ y(t_{k}) = \frac{a_{k}}{a_{k+1}} y(t_{k} − 0),  (k ∈ ℕ). \end{matrix}  (6)

This is a so-called impulsive differential equation (see [3, 4, 11, 14] and the references therein): the evolution of (x(t), y(t)) is governed by a differential equation for t ≠ t_{k} (k ∈ ℕ), and y(t) makes jumps at t = t_{k}. Due to its special form, the impulsive differential equation (6) can be represented as a discrete dynamical system on the plane in the following way. Introducing the polar coordinates r, φ by the formulae x = r cos φ, y = r sin φ (r > 0, −∞ < φ < ∞), we can transform system (5) into

r′ = 0, φ′ = −a_{k}, (t_{k−1} ≤ t < t_{k}, k ∈ ℕ).

Therefore, during the evolution governed by (6) the points of the plane revolve uniformly around the origin for t ∈ [t_{k−1}, t_{k}), then a contraction of size a_{k}/a_{k+1} along the y-axis occurs at t = t_{k}. Now introduce the following notation:

τ_{k} := t_{k} − t_{k−1}, φ_{k} := a_{k}τ_{k}, d_{k} := \frac{a_{k}}{a_{k+1}} ,

M_{0} := ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) ,

M_{k} = M_{k}(d_{k}, φ_{k}) := ( \begin{matrix}1 & 0 \\ 0 & d_{k} \end{matrix} ) ( \begin{matrix}cos φ_{k} & sin φ_{k} \\ − sin φ_{k} & cos φ_{k} \end{matrix} ) , (k ∈ ℕ).

(7)

Then from (6) we have

z_{k} := ( \begin{matrix}x(t_{k}) \\ y(t_{k}) \end{matrix} ) = M_{k}M_{k−1} ⋯ M_{1}M_{0} ( \begin{matrix}x(0) \\ y(0) \end{matrix} ) ∈ ℝ^{2},  (k = 0, 1, 2, …). (8)

Since the discrete dynamical system (8) has the same stability properties as our original system (6), in the remaining part of the paper we shall investigate (8).

3 Parametric Resonance

For given T_{1} > 0, T_{2} > 0 (T := (T_{1} + T_{2})/2) and ε ∈ [0, π) consider the second order differential equation

\{ \begin{matrix}x′′ + a^{2}(t)x = 0, \\ a(t) = \{ \begin{matrix}π + ε, & if 2nT ≤ t < 2nT + T_{1}, \\ π − ε, & if 2nT + T_{1} ≤ t < 2nT + T_{1} + T_{2}, (n = 0, 1, 2, …), \end{matrix} \end{matrix}  (9)

which is a Meissner’s equation.

Definition 3.1 We say that a parametric resonance occurs in equation (9) if for arbitrarily small ε > 0 there exists a non-empty open set S(ε) such that if 2T ∈ S(ε), then there are T_{1} > 0, T_{2} > 0 such that T_{1} + T_{2} = 2T and equation (9) has solutions with the property

lim_{n→∞} (max{|x(t)| : 2nT ≤ t ≤ 2(n + 1)T }) = ∞,

i.e., solutions with amplitudes tending to ∞, as t → ∞.

C ⊂ [0, ∞) is called the set of the critical values of the parametric resonance if 2T ∈ C implies 2T ∈ S(ε) for all sufficiently small ε > 0.

It is known [13] that the critical values of the parametric resonance are the multiples of the half of the own period of the harmonic oscillator obtained from (9) by the substitution ε = 0, i.e.,

C = ∪_{m=1}^{∞} ( ∩_{0<ε< \frac{1}{m}} S(ε)) = \{(\frac{1}{2} \frac{2π}{π} ) j : j ∈ ℕ\} = ℕ.

Now we prove this assertion by the method described in Section 2. Equation (9) is a special case of (4). Setting

a_{2n−1} = π + ε, a_{2n} = π − ε, d_{2n−1} = \frac{π + ε}{π − ε}, d_{2n} = \frac{π − ε}{π + ε},

τ_{2n−1} = T_{1}, τ_{2n} = T_{2} (n ∈ ℕ),

and introducing the notations

φ_{1} := (π + ε)T_{1}, φ_{2} := (π − ε)T_{2},  κ := \frac{π + ε}{π − ε},  (10)

for matrices (7) we get

M_{+} := M_{2n−1} = ( \begin{matrix}1 & 0 \\ 0 & κ \end{matrix} ) ( \begin{matrix}cos φ_{1} & sin φ_{1} \\ − sin φ_{1} & cos φ_{1} \end{matrix} ) ,
M_{−} := M_{2n} = ( \begin{matrix}1 & 0 \\ 0 & \frac{1}{κ} \end{matrix} ) ( \begin{matrix}cos φ_{2} & sin φ_{2} \\ − sin φ_{2} & cos φ_{2} \end{matrix} ) , (n ∈ ℕ).

Instead of (8) we may study the system of difference equations

w_{n} := ( \begin{matrix}x(2nT) \\ y(2nT) \end{matrix} ) = (M_{−}M_{+})^{n} ( \begin{matrix}x(0) \\ y(0) \end{matrix} ) , (n = 0, 1, 2, …).  (11)

A simple computation shows that M(ε) := M_{−}M_{+}

= ( \begin{matrix}cos φ_{1} cos φ_{2} − κ sin φ_{1} sin φ_{2} & sin φ_{1} cos φ_{2} + κ cos φ_{1} sin φ_{2} \\ −\frac{1}{κ} cos φ_{1} sin φ_{2} − sin φ_{1} cos φ_{2} & −\frac{1}{κ} sin φ_{1} sin φ_{2} + cos φ_{1} cos φ_{2} \end{matrix} );

i.e., (11) can be rewritten into the form

w_{n} = ( \begin{matrix}x(2nT) \\ y(2nT) \end{matrix} ) = M(ε)^{n} ( \begin{matrix}x(0) \\ y(0) \end{matrix} ) , (n = 0, 1, 2, …).  (12)

Since det(M(ε)) = det(M_{+}) det(M_{−}) = 1, the eigenvalues λ_{1}, λ_{2} of M(ε) are determined by the equation

λ^{2} − Trace(M(ε))λ + 1 = 0.

If λ_{1}, λ_{2} are not real then they are located on the unit circle of the complex plane, so the trivial solution of (11) is stable. Therefore,

S(ε) ⊂ D(ε) := {2T > 0 : |Trace(M(ε))| ≧ 2 for some T_{1}, T_{2}}.

For the trace of M(ε) we have

Trace(M(ε)) = 2 cos φ_{1} cos φ_{2} − (κ + \frac{1}{κ}) sin φ_{1} sin φ_{2}

= cos(φ_{1} + φ_{2}) \{2 + (κ + \frac{1}{κ} − 2) \frac{1}{2}\} − cos(φ_{1} − φ_{2}) (κ + \frac{1}{κ} − 2) \frac{1}{2},

(13)

whence, introducing the notation γ = γ(ε) = (κ + 1/κ − 2)/2, and taking into account the identities

φ_{1} + φ_{2} = 2T π + (T_{1} − T_{2})ε, φ_{1} − φ_{2} = 2T ε + (T_{1} − T_{2})π,  (14)

we get

D(ε) = {2T > 0 :|[2 + γ] cos(2T π + (T_{1} − T_{2})ε) − γ cos(2T ε + (T_{1} − T_{2})π)|

≧ 2 for some T_{1}, T_{2}}. (15)

If 2T_{∗} is a critical value of the parametric resonance, then 2T_{∗} ∈ D(ε) for all ε > 0, so (15) with ε → 0 + 0 and lim_{ε→0} γ(ε) = 0 yields |2 cos(2T_{∗}π)| ≧ 2, i.e., 2T_{∗} ∈ ℕ, which means that C ⊂ ℕ.

To prove the reversed relation it is enough to show that for every j ∈ ℕ and sufficiently small ε > 0 there is a δ > 0 such that j + τ ∈ S(ε) for |τ| < δ with the choice T_{1} = T_{2}. This will be implied by the inequality |f(τ)| > 2, where f is defined by

f(τ) := [2 + γ] cos((j + τ)π) − γ cos((j + τ)ε) = [2 + γ](−1)^{j} cos τ − γ cos((j + τ)ε)

(see (15)). Since

lim_{τ→0} f(τ) = (−1)^{j}2 + γ ((−1)^{j} − cos jε)

and γ > 0, for ε < π/(4j) we have |lim_{τ→0} f(τ)| > 2, which implies the existence of the desired δ, and the proof is complete.

4 The Problem of Swinging

In the problem of swinging (see §25 in [1]) the swinger changes the height of their center of gravity periodically. Small oscillations of the equivalent mathematical pendulum are described by the equation

φ′′ + \frac{g}{ℓ(t)}φ = 0,

where ℓ(t) denotes the length of the thread, which is a step function; φ denotes the angle measured anticlockwise between the axis directed vertically downward and the thread; g is the constant of gravity. After choosing an appropriate unit of length, the equation of motion is of the form (9). The swinger (perhaps a child with small ε > 0) would like to destabilize the equilibrium position x = 0, so the problem is to find the critical values of parametric resonance. As was shown in the previous section, they are the natural numbers. The most natural choice for the swinger is the smallest critical value 2T_{∗} = 1, i.e., T_{∗} = 1/2. This means that the period of the behaviour of the swinger is approximately equal to the half of the own period of the swing, which is in accordance with the practice of swinging.

4.1 Realistic and Most Effective Control

In Arnold’s model [1, §25] only the case T_{1} = T_{2} is allowed. However, this assumption is not realistic because it is not the elapsed time t but the position x(t) of the swing that the swinger is observing and is using to control the swing.

To make the situation clearer, let us suppose that the swinger has chosen T_{∗} = 1/2. Without loss of the generality we may assume that y(0) = 0, i.e., |x(0)| ≠ 0 is the first amplitude of the motion. If the swinger keeps the rule T_{1} = T_{2} = 1/2, then φ_{1} = (π + ε)/2 > π/2 (see (10)), which means that the swinger changes the height of their center of gravity (makes the pendulum longer) first after crossing the equilibrium position x = 0. When the next change (shortening the pendulum) happens, then φ(1) = φ_{1} + φ_{2} = (π + ε)/2 + (π − ε)/2 = π, i.e., x′(1) = 0 and |x(t)| has a maximum (in other words, |x(1)| is the second amplitude). It is more realistic to assume that the swinger changes the height of their gravity exactly at the moments when |x(t)| has extreme values. Then φ_{1} = φ_{2} = π/2, and

T_{1} = \frac{π}{2(π + ε)} = \frac{1}{2} \frac{1}{1 + \frac{ε}{π}} < \frac{1}{2} \frac{1}{1 − \frac{ε}{π}} = \frac{π}{2(π − ε)} = T_{2}.

From (13) we get

Trace(M(ε)) = − \{2 + (κ + \frac{1}{κ} − 2) \frac{1}{2}\} − (κ + \frac{1}{κ} − 2) \frac{1}{2} = − (κ + \frac{1}{κ}) < −2,

which implies parametric resonance. Actually, this obviously follows from the geometry of our method: when contractions happen, then y = 0, so the points z_{2n} = (x(2nT), 0) of the trajectory remain on the circles during the contractions (see Figure 1. a)).

Figure 1: a) The most effective control; b) T_{∗} = 1, φ_{1} = φ_{2}: no parametric resonance

It is also clear that this is the most effective control. In addition, if A_{n} denotes the n-th amplitude of the motion (A_{n} = |x(2nT)|), then

A_{n} = (\frac{π + ε}{π − ε})^{n} , (n = 0, 1, 2, …),

because only the dilations have influence on the motion.

Now let us take the second critical value 2T_{∗} = 2, i.e., T_{∗} = 1, and assume φ_{1} = φ_{2} = π. Then neither the dilations nor the contractions have influence on the motion with y(0) = 0, x(0) ≠ 0, so the amplitudes are constant (see Figure 1. b)). This means that the choice

T_{1} = \frac{π}{π + ε} = \frac{1}{1 + \frac{ε}{π}} < 1, T_{2} = \frac{π}{π − ε} = \frac{1}{1 − \frac{ε}{π}} > 1

does not result in instability.

4.2 Asymmetric Swinging

Finally suppose that the swinger can keep the rule 2T = 2T_{∗} for a critical value of the parametric resonance, but T_{1} and T_{2} may be essentially different. We study how the occurrence of the instability depends on the difference T_{1} − T_{2}.

Consider the practically most important case T_{∗} = 1/2. By (14) we have

φ_{1} + φ_{2} = π + (T_{1} − T_{2})ε, φ_{1} − φ_{2} = ε + (T_{1} − T_{2})π;

moreover,

γ = (κ + \frac{1}{κ} − 2) \frac{1}{2} = \frac{2ε^{2}}{π^{2} − ε^{2}} = \frac{2}{π^{2}} ε^{2} (1 + \frac{ε^{2}}{π^{2}} + O(ε^{4})) , (0 < ε ≪ 1).

Consequently,

Trace(M(ε)) = [2 + γ] cos(π + (T_{1} − T_{2})ε) − γ cos(ε + (T_{1} − T_{2})ε)

= −[2 + γ] cos((T_{1} − T_{2})ε) − γ {cos ε cos((T_{1} − T_{2})π) − sin ε sin((T_{1} − T_{2})π)}

= −2 + \{(T_{1} − T_{2})^{2} − \frac{2}{π^{2}} (1 + cos((T_{1} − T_{2})π))\} ε^{2} + O(ε^{3}), (0 < ε ≪ 1).

Since 1 + cos 2α = 2 cos^{2} α, the inequality |Trace(M(ε))| > 2 will be satisfied for small ε > 0 if

\frac{π}{2} |T_{1} − T_{2}| < cos (\frac{π}{2} |T_{1} − T_{2}|) .

So we have proved the following theorem:

Theorem 4.1 Let α_{∗}(≈ 0.739085) denote the root of the equation cos α = α (α ∈ ℝ). Then at the critical value 2T_{∗} = 1, i.e., at T_{∗} = 1/2 there occurs a parametric resonance in equation (9) if

2T = T_{1} + T_{2} = 1, |T_{1} − T_{2}| < α_{∗}.  (16)

In other words, for sufficiently small ε > 0, conditions (16) imply the existence of solutions to equation (9) whose amplitudes tend to ∞ as t → ∞.

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