Simple flows on tori with uncommon chaos
The author studies flows \(\varphi_t:\mathbb{T}^n\longrightarrow\mathbb{T}^n\) on the \(n\)-dimensional torus induced by the family of ODE's \[ \frac{d\mathbf{x}}{dt}=\Omega f_{\gamma}(\mathbf{x}), \] where \(f_{\gamma}(\mathbf{x})=f(\mathbf{x})^{\gamma/2}\) with \(f(\mathbf{x})=\sum_{i=1}^n f_i(x_i)\) and \(f_i(x_i)=\sin^2(\pi x_i)\) and \(\Omega=(\omega_1,\omega_2,\ldots,\omega_n)^T\) is a normalized vector whose components are positive and satisfy \(\omega_i<\omega_{j}\) for \(i< j\) and a nonresonance condition, namely \((\mathbf{k},\Omega)\neq 0\) for all \(\mathbf{k}\in\mathbb{Z}^n\setminus\{0\}\). The flows induced by these equations have a unique fixed point of parabolic type and neither homoclinic nor heteroclinic points. However, as it is shown in Theorem 1, these flows are {chaotic} in the sense that they are topologically transitive and present sensitive dependence on initial conditions. The author also studies bounds for the Lyapunov exponents in the case \(\gamma>1\). The main result in this direction is Theorem 2 that states that the maximal Lyapunov exponent whenever \(\gamma>1\) and \(\Omega\) is nonresonant is zero. The case \(\gamma=1\) is also treated for the \(2\)-dimensional torus showing that there are choices of \(\Omega\) that ensure that the maximal Lyapunov exponent is positive. In addition, some examples of maps with parabolic points and positive Lyapunov exponent are presented.
scientific article; zbMATH DE number 7189264