# Preface

This section discusses the problem when second order differential equations have bounded solutions.

# Boundedness of solutions

It is well known that the longtime behaviour of a timedependent nonlineare differential equation

$\ddot{x} + f(t,x) =0 ,$
with periodic in t function f can be very intricate. For example, there are equations having unbounded solutions but with infinitely many zeroes and with nearby unbounded solutions having randomly prescribed numbers of zeroes and also periodic solutions (see V.M. Alekseev, K. Sitnikov, and J. Moser). In contrast, one can look for generalized Duffing equation
$\frac{{\text d}^2 x}{{\text d}t^2} + x^{2n+1} + \sum_{i=0}^{2n} p_i (t)\, x^{i} = f(t) ,$
which allow to conclude that all the solutions of the equation are bounded. This result, prompted by Littlewood, is due to G.R. Morris, Matkus, Moser, You, and Yuan.

Theorem: Every solution to the homogeneous generalized Duffing equation
$\frac{{\text d}^2 x}{{\text d}t^2} + x^{2n+1} + \sum_{i=0}^{2n} p_i (t)\, x^{i} = 0 ,$
where n is a positive integer, with periodic coefficients, pi(t+1) = pi(t) and piC, is bounded.    ▣
Another approach is to use Lyapunov's function (see Raffoul)

Theorem: Let Ω be a subset of ℝn. Suppose there exist a continuously differentiable Lyapunov function V : ℝ+ × Ω → ℝ+ that satisfies
$\lambda_1 \| x \|^p \le V(t,x) \le \lambda_2 \| x \|^q ,$
and
$V'(t,x) \le - \lambda_3 \| x \|^r + L$
for some positive constants λ1, λ2, λ3, p, q, r, and L. Moreover, if for some nonnegative constant γ the inequality
$V(t,x) \le - V^{r/q} (t,x) \le \gamma$
holds, then all solutions of $$\dot{\bf x} = {\bf f}(t, {\bf x})$$ that stay in Ω are uniformly bounded.    ▣

Next, we present some examples demonstrating the topic and utilizing compter software.

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