Continuous dependence of duffing oscillator solution on forcing term

Question

Consider the Duffing oscillator $\ddot{x} + 2 \gamma \rho \dot{x} + \rho^2 (\dot{x} + \alpha x^3) = F(t)$ for some forcing term $F(t)$. Are there any results that state that $x$ depends continuously on the forcing term $F(t)$? If so, how does one show this? Or are there any results I can cite?

Answer 1

For any differential equation initial value problem satisfying a local Lipschitz condition, the value at $t$ varies continuously as the equation is perturbed. For example, see Theorem V.3 in Birkhoff and Rota:

Let ${\bf x}(t)$ and ${\bf y}(t)$ satisfy the DE's $$ d{\bf x}/dt = {\bf X}({\bf x},t) \ \text{and}\ d{\bf y}/dt = {\bf Y}({\bf y},t),$$ respectively, on $a \le t \le b$. Further, let the functions $\bf X$ and $\bf Y$ be defined and continuous in a common domain $D \times [a,b]$, and let $$ |{\bf X}({\bf z},t) - {\bf Y}({\bf z},t)| \le \epsilon,\ a \le t \le b,\ {\bf z} \in D. $$ Finally, let ${\bf X}({\bf x},t)$ satisfy the Lipschitz condition $$ |{\bf X}({\bf x},t) - {\bf X}({\bf y},t)| \le L |{\bf x} - {\bf y}|,\ \text{if}\ {\bf x}, {\bf y} \in D,\; a \le t \le b $$ Then $$ |{\bf x}(t) - {\bf y}(t)| \le |{\bf x}(a) - {\bf y}(a)| e^{L(t-a)} + \frac{\epsilon}{L} \left( e^{L(t-a)}-1\right) $$