Search Results (395)

Adaptive feedback can destabilize a loop that would be stable under any fixed gain, so the speed at which the gain adapts is itself a design parameter. We study this effect in a minimal three-dimensional model motivated by the photocatalytic degradation of microplastics: a pollutant concentration is driven toward a setpoint by an ultraviolet (UV) actuator whose gain adapts online. The model has a single bilinear nonlinearity, so the local analysis can be carried out in closed form. Under an explicit feasibility condition, the system has a unique positive equilibrium. The Routh–Hurwitz criterion shows that this equilibrium is locally asymptotically stable below an explicit critical adaptation speed ${\kappa }_c$ and unstable above it. At $\kappa ={\kappa }_c$, a purely imaginary eigenvalue pair crosses the imaginary axis transversally, and a Hopf bifurcation occurs, with an explicit onset frequency. The first Lyapunov coefficient is computed in closed form; it separates a supercritical onset, for well-damped actuators, from a subcritical onset with hysteresis, for weakly damped actuators. Numerical experiments confirm the predicted limit cycle and the classification. All the stability results established here are local. Full article

In this paper, a delayed fractional-order three-trophic-level food chain model with Allee effect and prey shelter is proposed and analyzed. The model adopts the Caputo fractional derivative to describe the memory effect, and introduces two discrete time delays which represent the gestation and response delays of predators, respectively. We establish the positivity, boundedness, existence, uniqueness and continuity of solutions. By using Jacobian matrix analysis and fractional stability theory, we investigate the equilibrium points and their local stability. The corrected characteristic equation is derived, and sufficient conditions for a Hopf-type stability switch induced by the delays are obtained. In the integer-order case $\alpha =1$, the critical delay and transversality condition are verified numerically, supporting a classical Hopf-bifurcation conclusion subject to the usual nondegeneracy assumptions. Numerical simulations verify the theoretical results and illustrate the combined effects of time delays, fractional memory, prey shelter and Allee effect on system dynamics. The results show that time delays may destabilize the system and produce persistent oscillatory numerical behaviour, while stronger memory effects, shelter strength and Allee effects enhance system stability and suppress oscillatory behaviors. Full article

This paper develops a new generalized model to describe the complex dynamical behavior of a financial system through three state variables, namely the interest rate, investment demand and price index. The proposed model extends and improves numerous financial models available in the literature by incorporating two time delays. The first delay accounts for the time lag in price adjustment, whereas the second captures the delayed feedback effect on investment demand. For the first time in the context of financial systems, a novel threshold parameter is introduced to characterize the existence of equilibria. The dynamical properties of the proposed model, including the stability and occurrence of a Hopf bifurcation, are rigorously analyzed. Furthermore, sensitivity analysis and numerical simulations are conducted to investigate the influence of model parameters on the dynamics of the financial system and to illustrate the analytical results. Full article

This paper investigates the effects of diffusion and two time delays on the dynamics of an HIV infection model in one-dimensional domain. The Galerkin method is employed to derived theoretical equations. A condition is established for identifying exact Hopf bifurcation points, followed by an in-depth discussion on stability analysis maps. Bifurcation maps are constructed to illustrate the system dynamics under three distinct delay scenarios, highlighting the stable and unstable areas. When the delay parameter ${\tau }_i>0$, there are two distinct stability areas, whereas in the absence of delay, only one stable region exists. The findings indicate that the time delays and the diffusion rate significantly influence the stability regions of the system. Bifurcation maps are presented to illustrate selected examples of 3D periodic oscillations to validate all theoretical outputs shown in this paper. Full article

We formulate and study a predator–prey model with pursuit-evasion spatial behaviour, incorporating the fear effect in a sexually-reproducing prey population. The pursuit-evasion movements are described as indirect taxis: populations respond to diffusively dispersed and decaying kairomonal cues of their antagonists. The prey-emitted kairomone attracts predators, while the predator-emitted kairomone repels prey and locally reduces prey reproduction rate, mimicking a physiological fear response. To isolate the net effect of predator’s prey-taxis, we assume predator birth/death rates are negligible, implying a constant predator abundance. Linear stability analysis yields a condition for taxis-driven oscillatory instability of the homogeneous steady state. Numerical simulations reveal spatially heterogeneous dynamics, coexistence of periodic travelling waves, and transitions to spatiotemporal chaos. The results highlight interrelations between fear responses, spatial movements, spatiotemporal heterogeneity, and viability of the trophic system. Full article

This paper investigates the semi-analytical solutions of the Schnakenberg reaction system in a circularly symmetric reaction–diffusion annulus. The Galerkin method is used to approximate spatial concentration profiles of both the autocatalyst and reactant. This method transforms governing partial differential equations into a system of ordinary differential equations, allowing a semi-analytical investigation of this special geometric domain. Singularity theory is then used to determine the regions of the parameter space where different steady states are obtained, and a degenerate Hopf bifurcation analysis is used to determine the region where the Hopf bifurcation points are located. A novel result is that the width of the annulus significantly influences both static and dynamic multiplicity; specifically, a thicker annulus expands the region of parameter space where multiple steady-state solutions and Hopf bifurcations exist. Finally, the reliability and precision of these semi-analytical results are confirmed through direct comparison with numerical solutions of the original partial differential equations. Full article

In this paper, we present a new fractional mathematical model to describe the dynamics and the interaction between plants and water in arid and semi-arid environments with and without slope. By the Caputo fractional operator, the model allows for simulating the phenomena related to the vegetation migration, which occur in domains with different slopes. The new fractional model represents a connection between the Klausmeier model, where water advection occurs, to the Klausmeier–Gray–Scott model, where water diffuses. The proposed model describes an anomalous physical phenomenon that changes as the fractional parameter changes, modeling an anomalous water advection. We obtain the Hopf bifurcation of migration speed as function of the fractional parameter. The oscillatory solutions and the vegetation pattern formation, obtained by an explicit first-order numerical method, validate the analytical results and confirm the reliability and efficiency of the fractional formulation of the considered model. Full article

Modeling and stability analysis of a PV–energy storage AC/DC integrated three-port grid-connected power electronic device is investigated in this paper for low-voltage single-phase renewable energy applications. The device consists of a PV Boost converter port, a battery-side bidirectional DC-DC converter port, and a single-phase full-bridge grid-connected inverter. To analyze the coupling-induced stability characteristics of this multi-converter system, mathematical models of the PV array, battery, and AC/DC integrated device are established. Considering the periodic time-varying nature introduced by the single-phase grid voltage, a phase-angle-based simplified discrete model is developed to transform the system into a discrete model evaluated at a fixed grid-voltage phase angle. Based on this model, eigenvalue sensitivity, eigenvalue trajectories, and bifurcation diagrams are used to identify the influence of key control parameters on system stability. The results show that excessive proportional gains in the inverter current loop and energy storage control loop reduce the stability margin and may lead to period-doubling bifurcation, Hopf bifurcation, or unstable grid current operation. The period-doubling and Hopf stability boundaries are identified at k_p4 ≈ 1.40 and k_p2 ≈ 1.75, respectively. In simulation, the grid-current THD increases from 2.07% to 3.60% as k_p4 rises from 1.3 to 1.6 and from 2.07% to 2.27% as k_p2 rises from 1.7 to 1.8. MATLAB/Simulink simulations and hardware-in-the-loop experiments further verify that the identified stability boundaries are consistent with the degradation of grid current quality and the increase in total harmonic distortion. The proposed modeling and analysis method provides a reference for parameter tuning and stable operation of single-phase PV–energy storage three-port grid-connected devices. Full article

To overcome the limitation of conventional nose landing gear shimmy damper design, which is mainly based on the critical damping for torsional shimmy while neglecting the high-damping-side instability boundary, this study establishes a nonlinear shimmy dynamic model of a nose landing gear equipped with a hydraulic damper and an inerter-based suppression system. Hopf bifurcation analysis, numerical continuation, and energy-evolution analysis are employed to investigate the effects of damper, structural, and inerter parameters on the zero-shimmy region over the full speed range. The results show that the zero-shimmy damping interval is not governed by a single torsional critical damping value but is jointly bounded by the tire-induced torsional shimmy boundary on the low-damping side and the lateral or structural torsional shimmy boundary on the high-damping side. Thus, increasing damper damping is not always beneficial. Among the structural parameters, trail determines the existence of the zero-shimmy region, strut torsional stiffness mainly regulates the high-damping-side boundary, and rake angle provides local correction. With proper inertance and tuning-stiffness matching, the inerter-based system raises the upper critical damping by about 210% and improves low-speed shimmy suppression by transferring vibration energy to the damping branch through inertial coupling. Full article

Dissipative structures are open dynamical systems that sustain coherent macroscopic organization by continuously exchanging energy and matter with their environment and generating entropy. A recent thermodynamic analysis of the paradigmatic Malkus–Lorenz waterwheel interpreted the Lorenz system as an engine, deriving an exact formula for its thermodynamic efficiency and showing that efficiency tends to increase as the system is driven far from equilibrium while displaying sharp drops near the Hopf subcritical bifurcation to chaos. Here, we extend that single-engine framework to coupled dissipative structures. We introduce two canonical couplings—master–slave coupling (series) and symmetric diffusive coupling (parallel)—and prove two fundamental association laws allowing us to reduce the composite systems to an equivalent engine with a specified efficiency. We then apply these abstract results to coupled Lorenz waterwheels, deriving efficiency formulas consistent with the underlying power balance. We perform numerical simulations confirming that (a) series coupling induces an increase in thermodynamic efficiency, (b) parallel coupling averages the efficiency of engines and increases total energy flow, (c) synchronization is typically neutral or beneficial for efficiency except in narrow parameter regions, and (d) coupling modifies the curvature of entropy-generation trends. Our theorems suggest a mathematically rigorous and transparent route to define and compute thermodynamic efficiency for generalized flow networks, with potential application to complex systems energetics. Full article

This paper revisits a delayed tumor–immune model by replacing the discrete delay with weak and strong Gamma distributed memories, reflecting the realistic spread of immune-response times. Because the Gamma kernels are normalized, the biologically relevant equilibria of the reference model are preserved. The local stability problem, however, changes substantially: a careful linearization shows that the characteristic equation contains both the first and the second power of the memory transfer function, since delayed immune and tumor variables enter coupled feedback terms. Consequently, the weak Gamma chain leads to a quintic characteristic polynomial, whereas the strong Gamma chain leads to a seventh-degree polynomial. Routh–Hurwitz conditions and explicit Hopf bifurcation tests are derived for both memory structures, including simplicity and transversality requirements. Numerical simulations performed with the parameter sets of the reference study show that distributed memory reproduces the main biological regimes while shifting stability thresholds and modifying transient oscillations. The results indicate that not only the mean immune-response time but also the shape of its distribution can influence tumor–immune dynamics. Full article

This research explores the influence of delayed species interactions on the stability of ecological communities. A generalized food-chain framework is considered, consisting of plants, herbivores, carnivores, and apex predators. Consumer growth in the model depends on resource availability at earlier times rather than instantaneously. By applying delay differential equations, we demonstrate that increasing time delays can destabilize an initially stable equilibrium and give rise to sustained oscillatory dynamics through a Hopf bifurcation. The analysis shows that once the delay exceeds a critical value, the equilibrium loses stability and periodic population cycles emerge. To analyze these dynamic transitions, techniques such as linear stability analysis, center manifold reduction, and normal form theory are employed. The calculation of key bifurcation coefficients determines the nature of the bifurcation, indicating that stable limit cycles typically form beyond the critical delay point. These findings provide a theoretical basis for understanding population fluctuations in natural systems, where time delays are inevitably due to resource tracking or developmental lags. The framework explains how ecological cycles can be driven by delayed interactions without the need for outside environmental forcing. Also, the proposed research supports Life on Land, Life Below Water, and Climate Action, by helping to provide understanding of ecological dynamics. Full article

Complex nonlinear dynamical systems, often associated with high-entropy time series, have been widely employed to describe and predict intricate dynamic phenomena in real-world systems. Motivated by the need to better understand such complex dynamics in network-based epidemic processes, this paper investigates bifurcation dynamics in a fractional-order extension of the classical Susceptible–Latent–Breaking–Out model for computer virus propagation. The proposed framework incorporates two distinct transmission-related time delays and employs Caputo fractional derivatives of incommensurate orders, with the delays associated with infection rate and latent period selected as the primary bifurcation parameters. Due to the combined influence of multiple delays and incommensurate fractional exponents, the resulting system exhibits a complexity that goes beyond most existing models in the literature. By linearizing the model around its endemic equilibrium and analyzing the associated characteristic roots, we characterize how the system’s qualitative behavior depends on the magnitudes of the time delays, and establish explicit sufficient conditions for bifurcation to occur. In particular, the endemic equilibrium remains asymptotically stable as long as each delay stays below a certain critical value; once any delay exceeds its threshold, the system undergoes a Hopf bifurcation, leading to sustained periodic oscillations in virus prevalence. Numerical simulations are provided to support the analytical results, and they show strong agreement between predicted and observed system responses. These findings enhance theoretical insight into bifurcation mechanisms in fractional-order delay models of epidemic dynamics on networks, and may offer useful guidance for designing containment strategies in large-scale interconnected systems. Full article

In this article, we formulated a mathematical model to describe SARS-CoV-2 development in humans, accounting for the dynamics of susceptible and infected epithelial cells, viral particles, ACE2 receptors, cytotoxic T lymphocytes (CTLs), and antibodies. The basic reproduction number and equilibrium points are derived, with stability analysis showing that the disease-free equilibrium is maintained when ${\mathcal{R}}_0<1$, while an endemic equilibrium arises for ${\mathcal{R}}_0>1$. Additionally, Hopf bifurcating periodic solutions are observed under elevated viral replication and infection rates. To capture therapeutic intervention, an impulsive control framework based on antibody-mediated drug administration is introduced. The existence and stability of a disease-free periodic orbit are established through the impulsive reproduction number ${\mathcal{R}}_0^{imp}$, with stability ensured when ${\mathcal{R}}_0^{imp}<1$. The findings from numerical simulations support the analytical outcomes, proving the efficacy of impulsive control in suppressing viral persistence. The current research work offers important knowledge on the interaction between immune system and impulsive control mechanisms, which serves as a basis to develop therapies against SARS-CoV-2. Full article

This paper generalizes a fractional Cournot–Bertrand duopoly model with delayed self-feedback by replacing discrete memory effects with equal-mean, distributed Gamma memories. The resulting framework preserves the original economic structure while distinguishing the effect of the shape of the memory distribution from that of its average length. We derive the equilibria, the linearized characteristic equations, and real–imaginary crossing conditions for point, weak-Gamma and strong-Gamma memories. For the reference output-feedback configuration, the point-delay benchmark has the reported Hopf threshold ${\tau }_0=3.6355$ at frequency ${\psi }_0=0.5947$, whereas the two Gamma kernels remain separated from an imaginary-axis crossing over the tested equal-mean interval. At the reference crossing frequency, their feedback gains are approximately $0.42$ and $0.46$, respectively, compared with unit gain for the point delay. Numerical root tracking, stability diagnostics, parameter scans, and solver-convergence checks support the conclusion that distributed aggregation can materially enlarge the practically stable operating region. Full article