Preface
This section discusses the problem when second order differential equations have bounded solutions.
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Glossary
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
Another approach is to use Lyapunov's function (see Raffoul)
\[
\frac{{\text d}^2 x}{{\text d}t^2} + x^{2n+1} + \sum_{i=0}^{2n} p_i (t)\, x^{i} = 0 ,
\]
where n is a positive integer, with periodic coefficients, p_{i}(t+1) = p_{i}(t) and p_{i} ∈ C^{∞}, is bounded.
▣
Theorem:
Let Ω be a subset of ℝ^{n}. 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 λ_{1}, λ_{2}, λ_{3}, 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.
- Alekseev, V.M., Quasirandom dynamic systems, I, II, III, Math USSR Sb, 1868, Vol. 5, pp. 73--128, Vol. 6, pp. 505--560, 1969, pp. 1--43.
- Chen, Y.M. and Liu, J.K., A new method based on the harmonic balance method for nonlinear oscillators, Physics Letters A, 2007, Volume 368, Issue 5, pp. 371-378
- Dickerhoff, R. and Zehnder, E., Boundedness for solutions via twist theorem, Annali della Scuola Normale Superiore di Pisa, 1987, Vol. 14, No. 1, pp. 79--95.
- Lakshmikantham, V., Leela, S., and Martynyuk, A., Stability Analysis of Nonlinear Systems, 1989, Marcel DEkker, New York.
- Littlewood, J.,
- Littlewood, J.,
- Markus, L., Differential dynamic systems, 1969, American Mathematical Society, Providence, RI.
- Morris, G., A case of the boundedness in Littlewood's problem on oscillatory differential equations, Bulletine Austral Mathematical Society, 1976, Vol. 14, pp. 71--93.
- Moser, J., Stable and random motions in dynamic systems, Ann Math Studies, 1973, Vol. 77, Princeton, NJ.
- Raffoul, Y., Boundedness in nonlinear differential equations.
- Sitnikov, K., Existence of oscillating motions for three-body problemDokladu Akademii Nayk, SSSR, 1960, Vol. 133, No 2, pp. 303--306.
- Yoshizawa, T., Stability Theory by Lyapunov Second Method, 1966, The Math Society of Japan, Tokyo.
- You, J.-G., Boundedmess for solutions of super-linear duffing equations via the Twist theorem, Science in China, Series A: Mathematics, Physics, Astronamy & Technological Science, 1992, Vol.35No. 4, pp. 399--412.
- Yuan, X., Boundedness of solutions for Duffing-type equation, Science in China, SEries A: Mathematics, 1998, Vol. 41, No. 6, pp. 595--605.
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