Lyapunov stability of the Einstein-Friedmann dynamical equations of barotropic FRW cosmologies (Q6553018)

The authors consider the following singular equation\N\[\N\left( 2x\frac{{{\mathrm{d}}^{2}}x}{\mathrm{d}{{t}^{2}}}+{{\left( \frac{\mathrm{d}x}{\mathrm{d}t} \right)}^{2}}+k \right)/{{x}^{2}}-\Lambda =-8\pi G p(t),\N\]\Nwhere \(\Lambda \) is Einstein's cosmological constant, \(G\) is the universal gravitational constant, \(k\) is the curvature of the universe, \(p(t)\) is a continuous \(T\)-periodic function. The equation under consideration is a consequence of two Friedman equations. Based on the third-order approximation method, the upper-lower solutions method and some quantitative analysis the authors prove that the considered equation has an unstable positive \(T\)-periodic solution for \(k>0\), and also establish two sufficient conditions for the existence of stable positive \(T\)-periodic solutions of the equation for \(k<0\). Some of the obtained periodic solutions are of twist type [\textit{R. Ortega}, J. Differ. Equations 128, No. 2, 491--518 (1996; Zbl 0855.34058)] and can potentially lead to intricate dynamics, such as subharmonic solutions, stability islands and chaotic regions.