Monotonicity of topological entropy in continuous-time dynamical systems

Determine whether the topological entropy, viewed as a function of a system parameter, is monotonically increasing in continuous-time dynamical systems (ordinary differential equations), analogous to the monotonicity proven for the quadratic (logistic) family of interval maps; identify conditions under which such monotonicity holds in flows.

Background

The paper reviews the one-dimensional sine-map scenario in which changes in parameter lead to global changes in the invariant set and growth in topological entropy. It notes the well-known result by Milnor and Thurston that the entropy function is monotonically increasing for the quadratic (logistic) family of interval maps, highlighting a discrete-time benchmark.

Motivated by this, the authors raise the question of whether an analogous monotonicity property for topological entropy can be established in continuous-time systems (flows). They later paper the Rössler system to demonstrate an algorithmic approach for proving growth of lower bounds of entropy via saddle-node bifurcations, but they do not provide a general monotonicity theorem for continuous-time systems.