Nonlinear dynamics and bifurcation analyses of non-Gaussian Belousov–Zhabotinsky gels

Self-oscillating polymer gels, notably Belousov-Zhabotinsky (BZ) gels, have shown immense potential to replicate biological features at different length and time scales due to their ability to undergo chemo-mechanical transduction. Experimental studies reveal that such hydrogels demonstrate swelling behavior, governed by polymer chains with both infinite extensibility (Gaussian) and limited extensibility (non-Gaussian). In this work, our objective is to employ normal from analyses for non-Gaussian BZ gels to delineate their associated dynamics at different bifurcations. In particular, we construct a detailed bifurcation diagram, followed by the determination of the amplitude and frequency of chemo-mechanical oscillations in BZ gels. Furthermore, we determine the conservative and non-conservative contributions to these oscillations using the generalized Hamiltonian function. To do so, we compute higher-order Lyapunov and frequency coefficients to identify limit cycles (LCs) such as stable and unstable LCs, their coexistence, and semi-stable LC at the limit point of cycle arising due to the emergence of Bautin bifurcation. Our investigations reveal that BZ gels with limited chain extensibility exhibit significant differences in the characteristics of chemo-mechanical oscillations when compared to their Gaussian counterparts. Our normal form-based bifurcation analysis framework can be extended to other active nonlinear dynamical systems and, thus, play a key role in harnessing their potential for designing active stimuli-responsive smart devices.