We construct the large B-type and D-type universal character hierarchies (LBUC and LDUC) by enlarging the fermionic Fock space. We introduce the corresponding D-type and B-type Lie algebras gD and gB, together with their spin groups GD and GB. Their fermionic representations lead to the bilinear relations of the LBUC and LDUC hierarchies. We also prove that these bilinear relations characterize the corresponding group orbits under the product Cartan maps. In particular, the four parity sectors of the fermionic Fock space yield the decomposition LBUC=LDUC+mLDUC, where mLDUC consists of three types of mixed relations. Using the boson-fermion correspondence, we derive the Hirota bilinear equations of the LBUC hierarchy. The same decomposition is also proved at the Hirota level. Furthermore, we construct tau-functions of the LBUC hierarchy and obtain their expansions in universal characters with Pfaffian coefficients. Finally, we prove that these LBUC tau-functions can also be represented as shifted compositions of two large BKP (LBKP) tau-functions.
We study a reduced biochemical oscillator associated with lymphangion activation. The model describes the interaction of calcium ions (Ca2+) and nitric oxide (NO), motivated by previous models of lymphatic regulation. Under constant-radius, prescribed-shear and coincident-threshold assumptions, we analyze a two-dimensional piecewise-smooth system. We classify its crossing and sliding regions, analyze its equilibria and boundary-equilibrium bifurcations, and identify sufficient parameter conditions yielding the existence of a periodic orbit. We express these conditions in terms of the original model parameters to explain the balance between calcium input, calcium clearance and NO-mediated inhibition. Numerical examples illustrate the dynamical regimes. A formal derivation of reduced vessel-flow equations is retained as mechanical background in an appendix. The present results concern the biochemical system with prescribed mechanical inputs.
We have studied the mean field dynamics of heterogenous quadratic integrate-and-fire neurons in presence of spike frequency adaptation (SFA) by considering the exact formulation in terms of the next generation neural mass models (NG-NMMs). In particular, we have examined the emergence of slow relaxation oscillations in purely excitatory and excitatory-inhibitory NG-NMMs with SFA acting only on the excitatory population. Relaxation oscillations (ROs) connect low firing state (Down state) to high firing state (Up state) via the slow adaptation. In the examined cases, the orbit relaxes towards the Up State via a sequence of collective damped oscillations (peaks of activity), thus revealing population bursting dynamics. The slower is the adaptation time scale (the lower is the adaptation strength) the higher is the complexity (number of peaks) displayed by the relaxation oscillations. The increase in the number of peaks is due to spike-adding sequences controlled by the SFA parameters. The increase in the adaptation time scale leads to a chaos-induced spike-adding mechanism where the spike-adding occurs via intermediate chaotic transition regions. Quite peculiarly these chaotic regimes can be of two different types: Population Spiking and Population Bursting Chaos. The emergence of ROs with an increasing number of peaks, controlled by the adaptation strength, occurs via a different spike-adding mechanism. The ROs with (n+1) peaks emerge through a saddle-node bifurcation, while coexisting, in a very narrow parameter region, with the solutions with n peaks, that looses stability via a sub-critical period doubling transition. In this case chaotic dynamics is not observable at the spike-adding bifurcations, but only in proximity of the population spiking-bursting transition. The considered NG-NMMs with SFA could be employed to interpret the results of neuro-physiological experiments on populations of neurons where the concentration of carbachol can be varied. Indeed, the increase (decrease) of carbachol can be interpreted as effectively reducing (increasing) the strength of SFA. As a matter of fact, the increase of the adaptation strength leads to shorter (longer) Up (Down) state durations somehow mimicking the effect of charbachol in in vitro experiments, where spontaneous slow waves are observed. Furthermore, the scenario depicted in [1], where an increase of the concentration of carbachol induces a transition from anesthesia-like to sleep-like dynamics is consistent with our results based on the variation of the adaptation strength.
Aiming at the problems of screen clogging, frequent fatigue failure and excitation force during drilling fluid recovery, a novel multifactor-coupled system with four-motor, frequency-doubling, and dual-rigid-body is proposed to improve the mechanical performance and safety reliability of vibrating screens. The electromechanical coupling behavior and nonlinear vibration stability are analyzed adopting small parameter averaging method and Lyapunov's first approximation theory. Furthermore, the amplitude ratio, adaptive range and synchronization state are investigated via numerical analysis. Combined with experiments and computer simulations, the accuracy of theoretical and numerical results is verified. Theoretically, the concepts of identical-frequency load torque (IFLT) and different-frequency load torque (DFLT) are defined, thereby further clarifying the synchronization characteristics of frequency-doubling systems. Specifically, the DFLT exerts a significant effect on synchronization characteristics in system resonance region, while the effect can be neglected in non-resonance region. Experimental results confirm that the amplitude ratio remains at 0.685 with negligible fluctuation, which ensures stable operation with a constant amplitude ratio under actual working conditions. The proposed theory provides an efficient and reliable mathematical solution for frequency-doubling systems. Meanwhile, the multifactor-coupled model exhibits significant practical application value.
The stability of discrete breather waves in mass-in-mass (MiM) Fermi–Pasta–Ulam–Tsingou (FPUT) chains has been investigated for the asymmetric cubic nonlinearity of the α-FPUT model. However, the stability of optical breather waves remains insufficiently understood. This study presents a systematic analysis of the Floquet stability of optical breather waves in one-dimensional α-FPUT MiM chains. We examine parameter configurations defined by two carrier wavenumbers (k=0.1,3.1), two mass ratios (κ=3,1/3), and two linear coupling strengths (ρ=3,1/3). We treat internal cubic asymmetry (α ≠ 0, a=0) and inter-site cubic asymmetry (a ≠ 0, α=0) separately to isolate their effects on breather wave stability.The two types of asymmetry produce qualitatively different stability behavior, with internal asymmetry having the stronger destabilizing effect. Nevertheless, internal cubic asymmetry acts as a conditional stabilizer under strong coupling (ρ=3), for which the spectral growth rate remains low, with γ reaching values as small as approximately 2×10−5 per period. Under weak coupling (ρ=1/3) and a small mass ratio (κ=1/3), however, γ exceeds 2 per period, corresponding to doubling times shorter than one carrier period. In contrast, inter-site cubic asymmetry produces nearly stable optical breathers across all parameter combinations studied at k=0.1. The Floquet multipliers deviate from the unit circle only under weak coupling and a large mass ratio, and even then the deviation remains very small, approximately 6×10−3 per period. The linear coupling strength ρ therefore emerges as a key control parameter, whereas the influence of the mass ratio κ depends on the carrier wavenumber. These quantitative results enable control of nonlinear energy localization in asymmetric MiM FPUT chains.
