Consider a second order gradient-like system with linear damping
$$\ddot{x}+\dot{x}+\nabla f(x)=0, \quad x(0)=x_0,\quad\dot{x}(0)=0$$
Suppose $f\in C^2(\mathbb{R}^n)$ and the solution $x:[0,\infty)\rightarrow \mathbb{R}^n$ is bounded, i.e., $\lVert x(t)\rVert\leq c$ for all $t\in [0,\infty)$, where $c$ is a constant. 

If we change the so called "gravity constant" (see [this paper][1]) from $1$ to any positive constant $g$, i.e.,
$$\ddot{x}+\dot{x}+g\nabla f(x)=0, \quad x(0)=x_0,\quad\dot{x}(0)=0$$
Is the solution to this new equation still bounded?



[1]: https://www.researchgate.net/publication/263878249_THE_HEAVY_BALL_WITH_FRICTION_METHOD_I_THE_CONTINUOUS_DYNAMICAL_SYSTEM_GLOBAL_EXPLORATION_OF_THE_LOCAL_MINIMA_OF_A_REAL-VALUED_FUNCTION_BY_ASYMPTOTIC_ANALYSIS_OF_A_DISSIPATIVE_DYNAMICAL_SYSTEM