Differential Equations and Computational Simulations III J. Graef, R. Shivaji, B. Soni J. & Zhu (Editors) Electronic Journal of Differential Equations, Conference 01, 1997, pp. 97–108. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp 147.26.103.110 or 129.120.3.113 (login: ftp)

On Properties of Nonlinear Second Order Systems under Nonlinear Impulse Perturbations ^{∗}

John R. Graef & János Karsai

Abstract

In this paper, we consider the impulsive second order system

ẍ + f(x) = 0 (t ≠ t_{n}); ẋ(t_{n} + 0) = b_{n}ẋ(t_{n}) (t = t_{n})

where t_{n} = t_{0} + n p (p > 0, n = 1, 2 …). In a previous paper, the authors proved that if f(x) is strictly nonlinear, then this system has infinitely many periodic solutions. The impulses account for the main differences in the attractivity properties of the zero solution. Here, we prove that these periodic solutions are attractive in some sense, and we give good estimates for the attractivity region.

1 Introduction

Investigations of asymptotic stability problems for the intermittently damped second order differential equation

ẍ + g(t)ẋ + f(x) = 0  (1)

have led to asymptotic stability investigations of the impulsive system

ẍ + f(x) = 0, (t ≠ t_{n})
x(t_{n} + 0) = x(t_{n}),  (2)
ẋ(t_{n} + 0) = b_{n}ẋ(t_{n}),

where t_{n} → ∞ (n → ∞), xf(x) > 0 (x ≠ 0), and f is continuous (x ∈ ℝ) (see [3, 7, 8]). Although there are analogies between the systems (1) and (2) in the case 0 ≤ b_{n} ≤ 1, system (2) has unexpected properties due to the instantaneous effects. In addition, if b_{n} < 0 ([3, 5]), there are some new beating phenomena, and the beating impulses can stabilize the oscillatory behavior of the system (see [5]). In particular, in both the positive and negative impulse cases, if t_{n} = t_{0} + n p (p > 0), there can exist nonzero periodic solutions, which are small or large depending on the nonlinearity of the function f(x). The existence of such solutions can destroy the global nature of the attractivity, or the attractivity itself, of the zero solution. In this paper, we investigate the attractivity properties of these periodic solutions. We show that the periodic solutions are attractive in some sense, and we describe the attractivity regions as well.

∗1991 Mathematics Subject Classifications: 34D05, 34D20, 34C15.

Key words and phrases: Asymptotic stability, attractrivity of periodic solutions,

impulsive systems, nonlinear equations, second order systems.

\begin{matrix}c \\ ◯ \end{matrix}1998 Southwest Texas State University and University of North Texas.

Published November 12, 1998.

2 Definitions and Preliminaries

For the system (2), we use the following assumptions: t_{n} = t_{0} + n p (p > 0), f(x) is continuous, xf(x) > 0 (x ≠ 0), and for the sake of simplicity, we assume that f is an odd function, i.e., f(−x) = −f(x).

We say that the zero solution of (2) or (1) is stable if for any ε > 0 there exists δ > 0 such that |x(0)| + |ẋ(0)| < δ implies |x(t)| + |ẋ(t)| < ε (t ≥ 0). The zero solution is asymptotically stable (a.s.) if it is stable and there exists δ > 0 such that |x(0)|+|ẋ(0)| < δ implies lim_{t→∞}(x(t), ẋ(t)) = (0, 0). The asymptotic stability is global (g.a.s.) if δ = ∞.

We investigate the solutions of the equations with the aid of the energy function

V (x, y) = y^{2} + 2 ∫_{0}^{x} f =: y^{2} + F (x),  (3)

and often use the notation V (t) = V (x(t), ẋ(t)) for the solutions of system (2). Furthermore, without future reference, we assume that

lim_{x→±∞} F (x) = ∞.

This condition allows us to obtain boundedness of the solutions from the boundedness of the energy.

Let us consider the undamped equation

ü + f(u) = 0.  (4)

All solutions are periodic, and the energy is constant along each solution. The distance between the extremal points is given by (see [9])

Δ(r) = ∫_{−F^{−1}(r)}^{F^{−1}(r)} \frac{dx}{\sqrt{r − F (x)}},  (5)

where F^{−1} is the inverse of the positive part of F (x). Calculations yield the following expressions for Δ(r) in the case where f(x) = |x|^{α}sgn (x):

a) α = 1, Δ(r) = π,  (6)

b) α ≠ 1, Δ(r) = Ar^{β} with β = \frac{1 − α}{2(α + 1)} and

A = 2 (\frac{α + 1}{2})^{\frac{1}{α+1}} \frac{\sqrt{π}Γ (\frac{1}{α+1})}{(α + 1)Γ (\frac{3+α}{2(1+α)})},

where Γ(·) denotes Euler’s Γ function. In the linear case, we obtain the known value π provided β < 0 for α > 1 or 0 < β < 1/2 for 0 < α < 1.

Consider the system (2). Since lim_{n→∞} t_{n} = ∞, every solution can be continued to ∞. In addition, the solutions are differentiable, and ẋ(t) is piecewise continuous and continuous from the left at every t > 0.

The variation of the energy along the solutions of (2) is given by

V (t_{n+1}) − V (t_{n}) = V (t_{n} + 0) − V (t_{n})
= ẋ^{2}(t_{n} + 0) + F (x(t_{n} + 0)) − ẋ^{2}(t_{n}) − F (x(t_{n})) (7)
= b_{n}^{2}ẋ^{2}(t_{n}) − ẋ^{2}(t_{n}) = −ẋ^{2}(t_{n})(1 − b_{n}^{2}) = −a_{n}ẋ^{2}(t_{n}),

where a_{n} = 1 − b_{n}^{2} is the n-th energy-quantum. The energy is nonincreasing if b_{n}^{2} ≤ 1, independent of the sign of b_{n}, and it is constant between any t_{n} and t_{n+1}. In case b_{n} = 0, the solutions of initial value problems are not unique in the backwards direction. In this case, there can be solutions which are identically zero on [t_{n}, ∞).

Consider the energy along the solutions of (2) on the interval [0, t]. Using (7) repeatedly easily yields

V (t) = V (0) − ∑_{t_{n}<t} a_{n}ẋ^{2}(t_{n})  (8)

along solutions of (2). From inequality (8), it is easy to prove the following lemma.

Lemma 1 If |b_{n}| ≤ 1 for every n = 1, 2, …, then V (t) is nonincreasing along every solution. Moreover, every solution is bounded, and the zero solution of (2) is stable.

3 Asymptotic Stability and Existence of Periodic Solutions

The following theorem guarantees the existence of periodic solutions; it is based on Theorem 20 in [2].

Theorem 1 Suppose 0 ≤ b_{n} ≤ 1, p > 0, and t_{n} = t_{0} + n p, and let D_{0} = {r : Δ(r) = p/k, k = 1, 2 …}. The solutions of (2) with initial conditions satisfying F (x(t_{i})) = r ∈ D_{0}, ẋ(t_{i}) = 0 (i = 0, 1, …) are periodic and satisfy equation

This theorem assures the existence of periodic solutions in both the superlinear and sublinear cases. If lim sup_{r→∞} Δ(r) = 0, then the set D_{0} is unbounded, i.e., there are infinitely many periodic solutions in some set {V (x, y) > r_{0}}. This is the case if f(x) = |x|^{α} sgn x (α > 1). On the other hand, if lim inf_{r→0} Δ(r) = 0, then D_{0} has no positive infimum, i.e., there are arbitrarily small periodic solutions (e.g., f(x) = |x|^{α} sgn x (0 < α < 1)). If lim_{r→0} Δ(r) = ∞, D_{0} has a positive minimal element.

As consequence of the above statements, we obtain that the zero solution of the system (2) cannot be globally asymptotically stable in the strictly superlinear case (lim sup_{r→∞} Δ(r) = 0), and it cannot be asymptotically stable in the strictly sublinear case (lim inf_{r→0} Δ(r) = 0). More precisely, a simplified version of Corollary 3.3 in [5] is the following.

Theorem 2 Let t_{n} = t_{0} + n p and |b_{n}| ≤ 1, and assume that there exists a sequence of integers {n_{k}} such that

∑_{k} min(a_{n_{k}}, a_{n_{k}+1}) = ∞.  (9)

Then: Case (a): 0 < inf_{r>0} Δ(r). If p < inf_{r>0} Δ(r), the zero solution is globally asymptotically stable. If p > inf_{r>0} Δ(r), the behavior depends on the shape of Δ(r). Case (b): lim_{r→0} Δ(r) = ∞ and lim_{r→∞} Δ(r) = 0. The zero solution is asymptotically (but not globally) stable. Case (c): lim_{r→0} Δ(r) = 0. The zero solution is not asymptotically stable.

4 Attractivity of the Periodic Solutions

Following Theorem 1, let u_{p} denote the periodic solution of equation (4) (it is also a solution of system (2)) such that r_{p} = V (u_{p}(t_{0}), u̇_{p}(t_{0})) = Δ^{−1}(p) and u̇_{p}(t_{0}) = 0 (i.e., F (u_{p}(t_{0})) = Δ^{−1}(p)). First, we consider which attractivity properties might reasonably be expected. Let x(t) be another solution of (2) for which r = V (x(t_{0}), ẋ(t_{0})) < r_{p}. Since V (u_{p}(t), u̇_{p}(t)) is constant and V (x(t), ẋ(t)) is nonincreasing, x(t) − u_{p}(t) cannot tend to zero as t → ∞. Consequently, the relation lim_{t→∞}(x(t) − u_{p}(t)) = 0 can hold only if V (x(t_{0}), ẋ(t_{0})) ≥ r_{p}.

To formulate our results, we need some additional concepts. We assume that Δ(r) is monotone on some interval I_{p} of form [r_{p}, r_{p} + ε] or in [r_{p}, ∞). This is a reasonable assumption since it is satisfied, for example, if f(x) is an odd power of x. More generally, if f(x)/x is monotone, then Δ(F (x)) is monotone in the opposite direction ([15; Theorem 3.1.6]).

For a solution u(t) of (4), let U_{0} = (u(0), u̇(0)) ∈ ℝ^{2} and U(t; U_{0}) = (u(t), u̇(t)), and for a solution x(t) of the impulsive system (2), let X_{0} = (x(0+0), ẋ(0+0)) ∈ ℝ^{2} and X(t; X_{0}) = (x(t), ẋ(t)). To simplify the notation, let u_{p} > 0 be such that F (u_{p}) = r_{p}, and define τ(p, r) = Δ(r)−Δ(r_{p}) = Δ(r)−p. If

Figure 1: The mapping δ_{−} → γ_{+} → {(−x_{p}, y)}

Δ(r) is increasing on the interval I_{p}, then τ(p, r) > 0, and if Δ(r) is decreasing, then τ(p, r) < 0; moreover, |τ(p, r)| is increasing for x ≥ u_{0}.

Let γ_{+} and γ_{−} be the curves which are mapped to the sets {(−u_{p}, y) : y ∈ ℝ} and {(u_{p}, y) : y ∈ ℝ}, respectively, by the mapping U(p; ·). It is easy to see that this is equivalent to γ_{+} = {U(τ(p, r); U_{0}) : U_{0} = (u_{p}, y)} and γ_{−} = {U(τ(p, r); U_{0}) : U_{0} = (−u_{p}, y)} where r = V (u_{p}, y) is the energy of U(t; U_{0}). From the symmetry of f(x), we see that γ_{−} = {(x, y) : (−x, −y) ∈ γ_{+}}. The mapping δ_{−} → γ_{+} → {(−x_{p}, y)} is shown in Figure 1.

The monotonicity of Δ(r) and the continuous dependence of solutions on initial conditions imply that the curve γ_{+} is a graph of a continuous function of one variable, and so it can be written in the form u̇(τ(p, r); U_{0}) = γ_{+}(u(τ(p, r); U_{0})). To see this, assume the contrary. Let (x, y_{1}), (x, y_{2}) ∈ γ_{+}, |y_{1}| < |y_{2}|, and let r_{1} = V (x, y_{1}) < r_{2} = V (x, y_{2}). Then

|τ(p, r_{1})| = ∫_{u_{p}}^{x} \frac{ds}{\sqrt{r_{1} − F (s)}} > ∫_{u_{p}}^{x} \frac{ds}{\sqrt{r_{2} − F (s)}} = |τ(p, r_{2})|,

which contradicts the monotonicity of |τ(r, p)|. The case y_{1} = −y_{2} cannot happen because of the uniqueness of solutions to initial value problems for equation (4).

Note that γ_{+}(x) is positive (negative) if Δ(r) is increasing (decreasing), x − u_{p} is small enough, and x > u_{p}. Let x̂ be the first zero of γ_{+} if F (x̂) ∈ I_{p}, let x̄ = min{x̂, sup I_{p}}, and let r̄ = F (x̄). We also use the notation γ_{+}^{−} = {(x, y) : (x, −y) ∈ γ_{+}} and γ_{−}^{−} = {(x, y) : (x, −y) ∈ γ_{−}}. Now, we can define the following closed sets:

G_{+} := \{ \begin{matrix}\{(x, y) : x ≥ x_{p}, V (x, y) ≤ r̄, y ≤ γ_{+}(x)\}, & if Δ(r) is increasing, \\ \{(x, y) : x ≥ x_{p}, V (x, y) ≤ r̄, y ≥ γ_{+}(x)\}, & if Δ(r) is decreasing, \end{matrix}

G_{−} := {(x, y) : (−x, −y) ∈ G_{+}},
G_{+}^{−} := {(x, y) : (x, −y) ∈ G_{+}},
G_{−}^{−} := {(x, y) : (x, −y) ∈ G_{−}}.

Next, we consider the impulsive system (2), and assume that X(t_{n−1} + 0; X_{0}) ∈ G_{+} (G_{−}) and −1 ≤ b_{n} ≤ 0 for an impulse at t_{n−1}. Then X(t_{n} − 0; X_{0}) ∈ G_{−}^{−} (G_{+}^{−}) and X(t_{n} + 0; X_{0}) ∈ G_{−} (G_{+}). The first relation follows immediately from the properties of the sets G_{i}^{j} (i, j ∈ {+, −}). For the second one, we only have to prove that (x, y) ∈ G_{+} implies (x, b y) ∈ G_{+} for every 0 ≤ b ≤ 1. But this immediately follows from the fact that γ_{+} is a function of x.

Applying the above arguments, we can formulate the basic attractivity theorem for the beating impulses.

Theorem 3 Assume that Δ(r) is monotone on an interval [r_{p}, r_{p}+ε] or [r_{p}, ∞), and −1 < b_{n} ≤ 0 for every n = 1, 2, . . .. If a solution x(t) of (2) satisfies (x(t_{0} +0), ẋ(t_{0} +0)) = X_{0} ∈ G_{+} (G_{−}), then (x(t_{0} +2np+0), ẋ(t_{0} +2np+0)) = X_{0} ∈ G_{+} (G_{−}) for every n = 1, 2, . . .. Since V (x(t), ẋ(t)) is nonincreasing, the solution u_{p}(t) is conditionally stable with respect to the set G_{+}. If, in addition, condition (9) holds, then lim_{t→∞} V (x(t), ẋ(t)) = r_{p}, i.e., lim_{t→∞}(x(t)−u_{p}(t)) = 0.

Proof We only have to prove the second part. This proof is analogous to the proof of Theorem 19 in [2]. Consider the case where Δ(r) increasing; the decreasing case can be proved analogously. To the contrary, suppose x(t) is a solution of (2) such that (x(t_{0} + 2np + 0), ẋ(t_{0} + np + 0)) ∈ G_{+} ∪ G_{−} and lim_{t→∞} V (x(t), ẋ(t)) = r > r_{p}. Then Δ(V (x(t_{n}), ẋ(t_{n}))) > p_{1} > p for n > N. Now, it is easy to see that there exist a positive number μ, independent of n, such that

max(ẋ^{2}(t_{n}), ẋ^{2}(t_{n+1})) > μ

for n > N. Consequently,

V (t) = V (t_{0}) − ∑_{t_{n}<t} a_{n}ẋ^{2}(t_{n})
≤ V (0) − ∑_{t_{n_{k}+1}<t} min(a_{n_{k}} + a_{n_{k}+1})(ẋ^{2}(t_{n_{k}}) + ẋ^{2}(t_{n_{k}+1})
≤ V (0) − ∑_{t_{n_{k}+1}<t} μ min(a_{n_{k}} + a_{n_{k}+1}),

which tends to −∞ as k → ∞. This contradiction completes the proof of the theorem. ♢ Figure 2: The sets G_{+}, G_{−}, H_{+}, and H_{−}, and the trajectory of u_{p}(t)

The left hand side of Figure 3 shows the attractivity of u_{p} for negative impulses.

Remark. As was the case with the asymptotic stability theorems in [8], condition (9) can not be replaced by the weaker condition ∑_{n} a_{n} = ∞.

To see this, let p > 0, t_{n} = np. b_{2n} = 0, and b_{2n−1} = −1, (n = 1, 2, …). Let x(t) be any solution with ẋ(t_{2}) = 0. This solution is periodic but does not satisfy the equation ẍ + f(x) = 0.

We can observe that the conditional stability of the solution u_{p} is satisfied independently of the specific values of the impulse constants b_{n}. The negative sign guarantees that the sets G_{+} and G_{−} map into each other by the mapping X(t_{n−1} + p + 0; ·). The case of positive impulses is different. If b_{n} is positive, such invariance will hold for much narrower sets under stronger conditions on b_{n}.

To formulate results for the case 0 ≤ b_{n} ≤ 1, we need some additional definitions. Let δ_{+} and δ_{−} be the curves that are mapped respectively to γ_{−} and γ_{+} by the mapping U(p; ·). Obviously, these curves can be defined analogously to the γ curves above, that is, δ_{+} = {U(2τ(p, r); U_{0}) : U_{0} = (u_{p}, y)} and δ_{−} = {U(2τ(p, r); U_{0}) : U_{0} = (−u_{p}, y)} = {(x, y) : (−x, −y) ∈ δ_{+}}.

The curves δ_{+} and δ_{−} also represent graphs of continuous functions of one variable, and can be written in the form δ_{+}(x) and δ_{−}(x). Similarly, δ_{+}(x) is positive (negative) if Δ(r) is increasing (decreasing), x − u_{p} is small enough, and x > u_{p}. Let x̌ be the first zero of δ_{+} if F (x̌) ∈ I_{p}, let x̃ = min{x̌, sup I_{p}}, Figure 3: The attractivity of u_{p}(t), f(x) = x^{1/3}, b_{n} = 0.7 (l.h.s) and b_{n} = 0.6 (r.h.s)

and let r̃ = F (x̃). We can then define the following closed sets:

H_{+} := \{ \begin{matrix}\{(x, y) : x ≥ x_{p}, V (x, y) ≤ r̃, y ≤ δ_{+}(x)\}, & if Δ(r) is increasing, \\ \{(x, y) : x ≥ x_{p}, V (x, y) ≤ r̃, y ≥ δ_{+}(x)\}, & if Δ(r) is decreasing, \end{matrix}

H_{−} := {(x, y) : (−x, −y) ∈ H_{+}}.

The sets G_{+}, G_{−}, H_{+}, and H_{−} are shown in Figure 2.

It follows immediately from the definition of the sets that the mapping U(p; ·) maps the sets H_{+} and H_{−} into G_{−} ∩ {V (x, y) ≤ r̃} and G_{+} ∩ {V (x, y) ≤ r̃}, respectively. Now let x(t) be a solution of (2) such that X(t_{n−1} + 0; X_{0}) ∈ H_{+} (H_{−}) and 0 ≤ b_{n} ≤ 1. Then X(t_{n} − 0; X_{0}) ∈ G_{−} ∩ {V (x, y) ≤ r̃} (G_{+} ∩ {V (x, y) ≤ r̃}). To guarantee that X(t_{n} + 0; X_{0}) ∈ H_{−} (X(t_{n} + 0; X_{0}) ∈ H_{+}), we need an additional condition on b_{n}, such as b_{n} ≤ sup{δ_{+}(x)/γ_{+}(x) : x ∈ (u_{p}, x(t_{n} − 0))}. The following theorem then holds.

Theorem 4 Assume that Δ(r) is monotonic on an interval [r_{p}, r_{p} + ε] or [r_{p}, ∞). Let r_{0} ≤ r̃, and assume that

0 ≤ b_{n} ≤ sup{δ_{+}(x)/γ_{+}(x) : x ∈ (u_{p}, F^{−1}(r_{0}))}, n = 1, 2, … (10)

If a solution x(t) of (2) satisfies (x(t_{0} + 0), ẋ(t_{0} + 0)) = X_{0} ∈ H_{+} ∩ {V (x, y) ≤ r_{0}} (H_{−} ∩ {V (x, y) ≤ r_{0}}), then (x(t_{0} + 2n p + 0), ẋ(t_{0} + 2n p + 0)) ∈ H_{+} (H_{−}). In addition, lim_{t→∞} V (x(t), ẋ(t)) = r_{p}, i.e., lim_{t→∞}(x(t) − u_{p}(t)) = 0.

For the proof of the last statement, we have only to note that condition (10) is stronger than (9) since the supremum in (10) is smaller than 1. The right hand side of Figure 3 shows the attractivity of u_{p} for positive impulses.

To illustrate our theorem, in Figure 4 we show the values of the mappings X(t_{0}+np; ·) for b_{n} = 1, 0.9, 0.8, 0.7, 0.6, 0.5. Later, we will return to the question of the sharpness of the estimates of the fraction δ_{+}(x)/γ_{+}(x).

Combining the arguments for negative and positive impulses, we can formulate the following more general theorem.

Figure 4: Mappings X(t_{0}+np; ·), f(x) = x^{1/3}, p = 3, b_{n} = 1, 0.9, 0.8, 0.7, 0.6, 0.5

Theorem 5 Assume that Δ(r) is monotone on an interval [r_{p}, r_{p}+ε] or [r_{p}, ∞), r_{0} ≤ r̃, and assume that conditions (9) and

−1 ≤ b_{n} ≤ sup{δ_{+}(x)/γ_{+}(x) : x ∈ (u_{p}, F^{−1}(r_{0}))} n = 1, 2, … (11)

hold. If a solution x(t) of (2) satisfies (x(t_{0} + 0), ẋ(t_{0} + 0)) = X_{0} ∈ H_{+} ∩ {V (x, y) ≤ r_{0} ≤ r̃} (H_{−} ∩ {V (x, y) ≤ r_{0} ≤ r̃}), then lim_{t→∞} V (x(t), ẋ(t)) = r_{p}, i.e., lim_{t→∞}(x(t) − u_{p}(t)) = 0.

The key to the applicability of our results is to either find the curves γ_{+} and δ_{+} analytically or to approximate them numerically. In either case, computer algebra programs are very useful. The curves in Figures 1 and 2 are obtained from the definitions of γ_{+} and δ_{+}, interpolating the points {U(p, U_{0}) : U_{0} ∈ {(u_{0}, i d), i = 1, …}} and {U(2p, U_{0}) : U_{0} ∈ {(u_{0}, i d), i = 1, …}}, respectively, where the step size d is small enough. This approach is quite fast and good enough (although not analytically certain) to verify that a point (x, y) is in G_{+} (H_{+}), but it is not applicable to estimate the quotient δ_{+}(x)/γ_{+}(x) if x is close to u_{p} since lim_{x→u_{0}+0} δ_{+}(x)/γ_{+}(x) is of form 0/0.

First, let us give estimates for the sets G_{+} and H_{+}; for simplicity, we assume that f is monotonic. From the definition, we have

γ_{+}(x) = y_{p} − ∫_{0}^{τ(p,r)}f(u(s; (u_{p}, y_{p}))) ds.

Let r_{p} < r_{0} ≤ r̃ be given, and assume that x > u_{p} and V (x, γ_{+}(x)) = r ≤ r_{0}. Since γ_{+}(x) > 0 (< 0) and τ(p, r) > 0 (< 0) for x > u_{0}, r > r_{p}, Δ(r) is increasing (decreasing), and f(x) is monotonic, we have

\sqrt{|r − r_{p}|}−f(F^{−1}(r_{p}))|Δ(r)−p| ≥ |γ_{+}(x)| ≥ \sqrt{|r − r_{p}|}−f(F^{−1}(r_{0}))|Δ(r)−p|, where |y_{p}| = \sqrt{|r − r_{p}|} = \sqrt{|r − Δ^{−1}(p)|}. If, in addition, Δ(r) is differentiable and |τ(p, r)| is concave up for r_{p} < r ≤ r_{0}, then

\sqrt{|r − r_{p}|} − f(F^{−1}(r_{p}))|Δ′(r_{p})||r − r_{p}|
≥ |γ_{+}(x)|
≥ \sqrt{|r − r_{p}|} − f(F^{−1}(r_{0}))|Δ′(r_{0})||r − r_{p}|.

From the above estimates, we can easily give sufficient conditions for (x, y) to be in H_{+}. Let (x, y) be such that r_{p} ≤ r = V (x, y) ≤ r_{0}, x ≥ u_{0}, and y ≥ 0 if Δ(r) is increasing and y ≤ 0 if Δ(r) is decreasing. Then (x, y) ∈ H_{+}, if and only if x = u(t_{1}; (u_{p}, y_{p})) and t_{1} ≥ τ(p, r). Obviously, for u_{p} ≤ x̄, the inequality t_{1} ≥ τ(p, r) is equivalent to |y| ≤ |u̇(τ(p, r))| = |γ_{+}(F^{−1}(r))|. If the inequality

|y| ≤ \sqrt{|r − r_{p}|} − f(F^{−1}(r_{0}))|Δ′(r_{0})||r − r_{p}|  (12)

holds, then (x, y) ∈ G_{+} since

|y| ≤ \sqrt{|r − r_{p}|} − f(F^{−1}(r))|Δ(r) − p| ≤ |y_{p}| − ∫_{0}^{|τ(p,r)|}f(u(s; (u_{p}, y_{p}))) ds
= |γ_{+}(F^{−1}(r))| .

Similarly, for δ_{+}, we obtain

\sqrt{|r − r_{p}|} − 2f(F^{−1}(r_{p}))|Δ′(r_{p})||r − r_{p}|
≥ |γ_{+}(x)|
≥ \sqrt{|r − r_{p}|} − 2f(F^{−1}(r_{0}))|Δ′(r_{0})||r − r_{p}|.

A sufficient condition for (x, y) to be in H_{+} for x ≥ u_{p} and r_{p} ≤ V (x, y) = r ≤ r_{0} is

|y| ≤ \sqrt{|r − r_{p}|} − 2f(F^{−1}(r_{0}))|Δ′(r_{0})||r − r_{p}|.  (13)

On the basis of the above estimations, we can define the curves \underline{γ}_{+} and \underline{δ}_{+} as the set of points (x, y) satisfying

y = sgn(τ(p, V (x, y)))(\sqrt{|V (x, y) − r_{p}|} − f(F^{−1}(r_{0}))|Δ′(r_{0})||V (x, y) − r_{p}|)

and

y = sgn(τ(p, V (x, y)))(\sqrt{|V (x, y) − r_{p}|} − 2f(F^{−1}(r_{0}))|Δ′(r_{0})||V (x, y) − r_{p}|),

respectively. The above equations can be solved for y, and the curves can be written in the form \underline{γ}_{+}(x) and \underline{δ}_{+}(x). The expressions are very complicated, and Figure 5: The curves \underline{δ}_{+}(x), δ_{+}(x), δ̄_{+}(x), \underline{γ}_{+}(x), γ_{+}(x), γ̄_{+}(x); the fraction

\frac{\underline{δ}_{+}(x)}{γ̄_{+}(x)} (f(x) = x^{1/3}, p = 3, u_{p} = 1.06813, x_{0} = 1.3)

we omit the details here. (We suggest that the reader perform the calculations with the aid of a computer algebra system such as Mathematica.) Using the curves \underline{γ}_{+} and \underline{δ}_{+} instead of γ_{+} and δ_{+}, we can define the sets \underline{G}_{+}, \underline{G}_{−}, \underline{H}_{+} and \underline{H}_{−} as attractivity regions.

Finally, to increase the applicability of Theorems 4 and 5, we give a lower estimate for δ_{+}(x)/γ_{+}(x). Let γ̄_{+} be the set of points (x, y) satisfying

y = sgn(τ(p, V (x, y)))(\sqrt{|V (x, y) − r_{p}|} − f(F^{−1}(r_{0}))|Δ′(r_{p})||V (x, y) − r_{p}|);

a similar expression holds for the curve y = δ̄_{+}. If x ≥ u_{p}, V (x, γ̄_{+}(x))) ≤ r_{0}, and \underline{δ}_{+}(x) has no zero on (u_{p}, F^{−1}(r_{0})), then

\frac{δ_{+}(x)}{γ_{+}(x)} ≥ \frac{\underline{δ}_{+}(x)}{γ̄_{+}(x)}.

Computer calculations with Mathematica resulted in Figure 5, which shows the curves \underline{δ}_{+}(x), δ_{+}(x), δ̄_{+}(x), \underline{γ}_{+}(x), γ_{+}(x), and γ̄_{+}(x), and the fraction \frac{\underline{δ}_{+}(x)}{γ̄_{+}(x)} for the case f(x) = x^{1/3}, p = 3, u_{p} = 1.06813, and x_{0} = 1.3. This explains why, in Figure 4, the attractivity properties change somewhere around 0.7.

5 Generalizations and Open Problems

In the theorems in the previous section, we considered only the periodic solution with energy r_{p} = Δ^{−1}(p), but in the proofs we did not use the fact that 2p is the smallest period. Thus, the theorems can also be formulated for the periodic solutions r_{p/k} = Δ^{−1}(p/k) by changing r_{p} to r_{p/k}. The definition of τ is τ(p/k, r) = k(Δ(r) − Δ(r_{p/k})) = kΔ(r) − p.

If lim_{r→0} Δ(r) = 0, then there are infinitely many periodic solutions, and they accumulate at the origin (this is the case for f(x) = x^{1/3}). The following question arises: are all the solutions trapped at some t_{n} and by some periodic trajectory? The answer is closely related to the classical problem on the existence of solutions that tend to zero as t → ∞. We note that this problem is still unsolved for the equation

ẍ + g(t)ẋ + f(x) = 0

if f(x) is a nonlinear function, e.g., a power function. For the linear case, see [6].

Acknowledgment Research is supported by the Mississippi State University Biological and Physical Sciences Research Institute (first author), by the Hungarian National Foundation for Scientific Research grant no. T-016367 (second author), and by the Hungarian Ministry of Education Grant No. 179/1996 (both authors).

References

[1] Elbert Á., On-off damping of linear oscillators, Acta Sci. Math., (to appear).

[2] Graef J. R., Karsai J., On the asymptotic behavior of solutions of impulsively damped nonlinear oscillator equations, J. Comput. Appl. Math. 71(1996), 147–162.

[3] Graef J. R., Karsai J., Asymptotic properties of nonoscillatory solutions of impulsively damped nonlinear oscillator equations, Dynamics Continuous Discrete Impulsive Systems 3(1997), 151–165.

[4] Graef J. R., Karsai J., On irregular growth and impulse effects in oscillator equations, in: Advances in Difference Equations, Proceedings of the 2nd International Conference on Difference Equations and Applications, Veszprém, 1995, Gordon and Breach, 1997, 253–262.

[5] J. R. Graef, J. Karsai, Intermittent and impulsive effects in second order systems, Nonlinear Anal. 30 (1997), 1561–1571.

[6] P. Hartman, On a theorem of Milloux, Amer. J. Math. 70(1948), 395–399.

[7] J. Karsai, Attractivity criteria for intermittently damped second order nonlinear differential equations, Differential Equations Dynam. Systems, to appear.

[8] J. Karsai, Asymptotic behavior and its visualization of second order nonlinear differential equations with intermittent and impulse damping, Appl. Math. Comp., to appear.

[9] R. Reissig, G. Sansone, R. Conti, Qualitative Theory of Nonlinear Differential Equations, Izd. Nauka, Moskow, 1974 (in Russian).

[10] P. Pucci, J. Serrin, Asymptotic stability for intermittently controlled nonlinear oscillators, SIAM J. Math. Anal. 25(1994), 815–835.

John R. Graef Department of Mathematics and Statistics Mississippi State University, Mississippi State, MS 39762 USA. Email address: graef@math.msstate.edu

János Karsai Department of Medical Informatics Albert Szent-Györgyi Medical University, Szeged, Korányi fasor 9, Hungary Email address: karsai@silver.szote.u-szeged.hu