On a numerical bifurcation analysis of a particle reaction-diffusion model for a motion of two self-propelled disks

Theoretical analysis using mathematical models is often used to understand a mechanism of collective motion in a self-propelled system. In the experimental system using camphor disks, several kinds of characteristic motions have been observed due to the interaction of two camphor disks. In this paper, we understand the emergence mechanism of the motions caused by the interaction of two self-propelled bodies by analyzing the global bifurcation structure using the numerical bifurcation method for a mathematical model. Finally, it is also shown that the irregular motion, which is one of the characteristic motions, is chaotic motion and that it arises from periodic bifurcation phenomena and quasi-periodic motions due to torus bifurcation.

Bifurcation analysis of the predator–prey model with the Allee effect in the predator

The use of predator–prey models in theoretical ecology has a long history, and the model equations have largely evolved since the original Lotka–Volterra system towards more realistic descriptions of the processes of predation, reproduction and mortality. One important aspect is the recognition of the fact that the growth of a population can be subject to an Allee effect, where the per capita growth rate increases with the population density. Including an Allee effect has been shown to fundamentally change predator–prey dynamics and strongly impact species persistence, but previous studies mostly focused on scenarios of an Allee effect in the prey population. Here we explore a predator–prey model with an ecologically important case of the Allee effect in the predator population where it occurs in the numerical response of predator without affecting its functional response. Biologically, this can result from various scenarios such as a lack of mating partners, sperm limitation and cooperative breeding mechanisms, among others. Unlike previous studies, we consider here a generic mathematical formulation of the Allee effect without specifying a concrete parameterisation of the functional form, and analyse the possible local bifurcations in the system. Further, we explore the global bifurcation structure of the model and its possible dynamical regimes for three different concrete parameterisations of the Allee effect. The model possesses a complex bifurcation structure: there can be multiple coexistence states including two stable limit cycles. Inclusion of the Allee effect in the predator generally has a destabilising effect on the coexistence equilibrium. We also show that regardless of the parametrisation of the Allee effect, enrichment of the environment will eventually result in extinction of the predator population.

Center Bifurcation in the Lozi Map

We consider the well-known Lozi map, which is a 2D piecewise linear map depending on two parameters. This map can be considered as a piecewise linear analog of the Hénon map, allowing to simplify the rigorous proof of the existence of a chaotic attractor. The related parameter values belong to a part of the parameter plane where the map has two saddle fixed points. In the present paper, we investigate a different part of the parameter plane, namely, the vicinity of the curve related to a center bifurcation of the fixed point. A distinguishing property of the Lozi map is that it is conservative at the parameter value corresponding to this bifurcation. As a result, the bifurcation structure close to the center bifurcation curve is quite complicated. In particular, an attracting fixed point (focus) can coexist with various attracting cycles, as well as with chaotic attractors, and the number of coexisting attractors increases as the parameter point approaches the center bifurcation curve. The main result of the present paper is related to the rigorous description of this bifurcation structure. Specifically, we obtain, in explicit form, the boundaries of the main periodicity regions associated with the pairs of complementary cycles with rotation number [Formula: see text]. Similar approach can be applied to other periodicity regions. Our study contributes also to the border collision bifurcation theory since the Lozi map is a particular case of the 2D border collision normal form.

Fractal structure and hydration-driven shape memory of duck down in the dry–wet state

Duck down, as a natural keratin material, has been widely used as a filling material. The multilevel bifurcation structure of down has been observed and characterized through scanning electron microscopy. The structure is a complex fractal structure composed of four-level self-similar structures including five units, that is, the calamus, main barb, barb, barbule, and node or prong. The differential friction effect of the dynamic friction coefficients of the barb was reduced from 0.4 (dry state) to 0.23 (wet state), namely a decrease of 42.5%. The friction locking effect decreases due to the swelling of the fiber diameter. The down is zero gravity in water, and under the action of vibration and internal stress, down that has been subjected to friction or heat setting treatment can quickly return to its original shape in water. This shape memory mechanism was further confirmed, in which down after heat setting can restore its shape to the natural state by shaking it quickly and vigorously. This research provides inspiration to investigate more complicated functions of natural materials and encourages the creation of very intelligent synthetic polymers.

Stabilizing nanolasers via polarization lifetime tuning

We investigate the emission dynamics of mutually coupled nanolasers and predict ways to optimize their stability, i.e., maximize their locking range. We find that tuning the cavity lifetime to the same order of magnitude as the dephasing time of the microscopic polarization yields optimal operation conditions, which allow for wider tuning ranges than usually observed in conventional semiconductor lasers. The lasers are modeled by Maxwell–Bloch type class-C equations. For our analysis, we analytically determine the steady state solutions, analyze the symmetries of the system and numerically characterize the emission dynamics via the underlying bifurcation structure. The polarization lifetime is found to be a crucial parameter, which impacts the observed dynamics in the parameter space spanned by frequency detuning, coupling strength and coupling phase.

Bifurcation structure of coexistence states for a prey–predator model with large population flux by attractive transition

This paper is concerned with a prey–predator model with population flux by attractive transition. Our previous paper (Oeda and Kuto, 2018, Nonlinear Anal. RWA, 44, 589–615) obtained a bifurcation branch (connected set) of coexistence steady states which connects two semitrivial solutions. In Oeda and Kuto (2018, Nonlinear Anal. RWA, 44, 589–615), we also showed that any positive steady-state approaches a positive solution of either of two limiting systems, and moreover, one of the limiting systems is an equal diffusive competition model. This paper obtains the bifurcation structure of positive solutions to the other limiting system. Moreover, this paper implies that the global bifurcation branch of coexistence states consists of two parts, one of which is a simple curve running in a tubular domain near the set of positive solutions to the equal diffusive competition model, the other of which is a connected set characterized by positive solutions to the other limiting system.

Stabilizing Nanolasers via Polarization Lifetime Tuning

We investigate the emission dynamics of mutually coupled nanolasers and predict ways to optimize their stability i.e. maximize their locking range. We find that tuning the cavity lifetime to the same order of magnitude as the dephasing time of the microscopic polarization yields optimal operation conditions which allow for wider tuning ranges than usually observed in conventional semiconductor lasers. The lasers are modeled by a Maxwell-Bloch type class-C laser model. For our analysis we analytically determine the steady state solutions, analyze the symmetries of the system and numerically characterize the emission dynamics via the underlying bifurcation structure. The polarization lifetime is found to be a crucial parameter which impacts the observed dynamics in the parameter space spanned by frequency detuning, coupling strength and coupling phase.

Influence of the pressure-dependent resonance frequency on the bifurcation structure and backscattered pressure of ultrasound contrast agents: A numerical investigation

The bifurcation structure of the oscillations of ultrasound contrast agents (UCAs) was studied as a function of the driving pressure for excitation frequencies that were determined using the UCAs pressure-dependent resonances (fs)(fs). It was shown that when excited by the (fs)(fs), the UCA can undergo a saddle-node bifurcation (SNB) to higher amplitude oscillations. The driving pressure at which the SNB occurs is controllable and depends on the (fs)(fs) magnitude. Utilizing the appropriate (fs)(fs), the scattering cross section of the UCAs can significantly be enhanced (e.g., ∼∼ninefold) while at the same time avoiding potential UCA destruction (by limiting the radial expansion ratio <<2). This offers significant advantages for optimizing UCA-mediated imaging and therapeutic ultrasound applications.

Influence of the pressure-dependent resonance frequency on the bifurcation structure and backscattered pressure of ultrasound contrast agents: A numerical investigation

The bifurcation structure of the oscillations of ultrasound contrast agents (UCAs) was studied as a function of the driving pressure for excitation frequencies that were determined using the UCAs pressure-dependent resonances (fs)(fs). It was shown that when excited by the (fs)(fs), the UCA can undergo a saddle-node bifurcation (SNB) to higher amplitude oscillations. The driving pressure at which the SNB occurs is controllable and depends on the (fs)(fs) magnitude. Utilizing the appropriate (fs)(fs), the scattering cross section of the UCAs can significantly be enhanced (e.g., ∼∼ninefold) while at the same time avoiding potential UCA destruction (by limiting the radial expansion ratio <<2). This offers significant advantages for optimizing UCA-mediated imaging and therapeutic ultrasound applications.