Year: 2026
Type: book-chapter
Abstract: Rocard’s oscillator is a third-order nonlinear equation given by $$ \dddot{z}+\varepsilon \omega \ddot{z}+\omega ^{2}\dot{z} +\omega ^{3}\Bigl [\varepsilon +\eta \Bigl (1-z^{2}-\frac{\dot{z}^{2}}{\omega ^{2}}\Bigr )\Bigr ]z=0, $$ z ⃛ + ε ω z ¨ + ω 2 z ˙ + ω 3 [ ε + η ( 1 - z 2 - z ˙ 2 ω 2 ) ] z = 0 , depending on the parameters $$\varepsilon $$ ε , $$\eta $$ η and $$\omega \ne 0$$ ω ≠ 0 , where $$\omega $$ ω sets a characteristic time scale and is treated as fixed throughout the analysis. We prove that, for $$\eta = 0$$ η = 0 , the equilibrium at the origin exhibits center-type behavior on a two-dimensional center manifold and admits a local analytic first integral. This explains the vanishing of the first Lyapunov coefficient and rules out a Hopf bifurcation at this parameter value. We then analyze the conservative limit $$\varepsilon = 0$$ ε = 0 , showing that the system is reversible and volume-preserving, in a setting compatible with the nonintegrability results of Hu and Tang (2025). Finally, we discuss these nonintegrability results within a geometric framework connecting the center-type dynamics at $$\eta =0$$ η = 0 with the conservative reversible structure arising in the dissipation-free regime.
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,390
OpenAlex ID: W7166709662