Gain modulation unlocks phase plasticity thus allowing for malleable nonlinear dynamics

The paper defines two Rössler variants with f1(x,z)=tanh(x+z) and f2(x,z)=tanh(xz), introduces the instantaneous gain G as ∂f/∂z evaluated along the trajectory, and argues that linear modulation collapses the I/O/G representation because the gain is functionally tied to the output, whereas gain modulation does not, producing a phase-dependent relationship and greater phase plasticity. These claims are clearly stated and illustrated (system and variants; definition of G and I/O/G; 2D collapse vs 3D; phase-plasticity via polar plots and parameter perturbation) . The model’s solution provides a concise, formal calculus-based derivation: for f1, G1=1−O1^2 so the I/O/G triples lie on the surface G=1−O^2; for f2, G2=x(1−O2^2), so G cannot generically be written as a function of O alone. It further formalizes ‘phase plasticity’ via an event equation E=dG/dt and the implicit function theorem to compare how extremum times depend on parameters. Substantively, both reach the same conclusions; the paper’s argument is qualitative, while the model provides a rigorous sketch. A minor imprecision in the paper is the phrase “one-to-one mapping” from output to instantaneous gain; the mapping is functional but not injective over (−1,1) for tanh. Overall, the paper’s claims are correct and the model’s derivations support them.