Search Results (42)

The weakly nonlinear stability analysis of a convective flow in a planar vertical fluid layer is performed in this paper. The base flow in the vertical direction is generated by internal heat sources distributed within the fluid. The system of Navier–Stokes equations under the Boussinesq approximation and small-Prandtl-number approximation is transformed to one equation containing a stream function. Linear stability calculations with and without a small-Prandtl-number approximation lead to the range of the Prantdl numbers for which the approximation is valid. The method of multiple scales in the neighborhood of the critical point is used to construct amplitude evolution equation for the most unstable mode. It is shown that the amplitude equation is the complex Ginzburg–Landau equation. The coefficients of the equation are expressed in terms of integrals containing the linear stability characteristics and the solutions of three boundary value problems for ordinary differential equations. The results of numerical calculations are presented. The type of bifurcation (supercritical bifurcation) predicted by weakly nonlinear calculations is in agreement with experimental data. Full article

Hydrogen peroxide (H₂O₂) plays a crucial role in cell signaling in response to physiological and environmental perturbations. H₂O₂ can oxidize typical 2-Cys peroxiredoxin (PRX) first into a sulfenic acid, which resolves into a disulfide that can be reduced by thioredoxin (TRX)/TRX reductase (TR). At high levels, H₂O₂ can also hyperoxidize sulfenylated PRX into a sulfinic acid that can be reduced by sulfiredoxin (SRX). Therefore, PRX, TRX, TR, and SRX (abbreviated as PTRS system here) constitute the coupled sulfenylation and sulfinylation cycle (CSSC), where certain oxidized PRX and TRX forms also function as redox signaling intermediates. Earlier studies have revealed that the PTRS system is capable of rich signaling dynamics, including linearity, ultrasensitivity/switch-like response, nonmonotonicity, circadian oscillation, and possibly, bistability. However, the origins of ultrasensitivity, which is fundamentally required for redox signal amplification, have not been adequately characterized, and their roles in enabling complex nonlinear dynamics of the PTRS system remain to be determined. Through in-depth mathematical modeling analyses, here we revealed multiple sources of ultrasensitivity that are intrinsic to the CSSC, including zero-order kinetic cycles, multistep H₂O₂ signaling, and a mechanism arising from diminished H₂O₂ removal at high PRX hyperoxidation state. The CSSC, structurally a positive feedback loop, is capable of bistability under certain parameter conditions, which requires embedding multiple sources of ultrasensitivity identified. Forming a negative feedback loop with cytosolic SRX as previously observed in energetically active cells, the mitochondrial PTRS system (where PRX3 is expressed) can produce sustained circadian oscillations through supercritical Hopf bifurcations. In conclusion, our study provided novel quantitative insights into the dynamical complexity of the PTRS system and improved appreciation of intracellular redox signaling. Full article

This study explores the use of a passive control technique to mitigate aeroelastic effects on a wing operating near the ground. An aeroelastic model, based on a typical airfoil section, equipped with a nonlinear tuned vibration absorber (NLTVA), is established to study the interactions between the airfoil’s dynamics, aerodynamics, and the nonlinear energy dissipation mechanisms. Geometric nonlinearity is incorporated into the airfoil’s dynamics to account for possible large wing deflection and rotation. The flow is modeled based on the nonlinear unsteady discrete vortex method with the ground effect simulated using the mirror image method. Stability analyses are conducted to study the influence of NLTVA parameters on flutter mitigation and the bifurcation behavior of the airfoil near the ground. The numerical results demonstrate that the NLTVA effectively delays the onset of flutter and promotes a supercritical bifurcation in the presence of ground effect. Optimally tuning the NLTVA’s linear parameters significantly increases flutter speed, while selecting the optimal nonlinear parameter is key to preventing subcritical behavior near the ground and reducing the amplitude of post-flutter limit cycle oscillations. Overall, this study highlights the potential of the NLTVA in enhancing the aeroelastic stability of flying vehicles with highly flexible wings, especially under the influence of ground effects during takeoff and landing. Full article

In this paper, we study the local, global, and bifurcation properties of a planar nonlinear asymmetric discrete model of Ricker type that is derived from a Darwinian evolution strategy based on evolutionary game theory. We make a change of variables to both reduce the number of parameters as well as bring symmetry to the isoclines of the mapping. With this new model, we demonstrate the existence of a forward invariant and globally attracting set where all the dynamics occur. In this set, the model possesses two symmetric fixed points: the origin, which is always a saddle fixed point, and an interior fixed point that may be globally asymptotically stable. Moreover, we observe the presence of a supercritical Neimark–Sacker bifurcation, a phenomenon that is not present in the original non-evolutionary model. Full article

Waddington envisioned stem cell differentiation as a marble rolling down a hill, passing through hierarchically branched valleys representing the cell’s temporal state. The terminal valleys at the bottom of the hill indicate the possible committed cells of the multicellular organism. Although originally proposed as a metaphor, Waddington’s hypothesis establishes the fundamental principles for characterizing the differentiation process as a dynamic system: the generated equilibrium points must exhibit hierarchical branching, robustness to perturbations (homeorhesis), and produce the appropriate number of cells for each cell type. This article aims to capture these characteristics using a mathematical model based on two fundamental hypotheses. First, it is assumed that the gene regulatory network consists of hierarchically coupled subnetworks of genes (modules), each modeled as a dynamical system exhibiting supercritical pitchfork or cusp bifurcation. Second, the gene modules are spatiotemporally regulated by feedback mechanisms originating from epigenetic factors. Analytical and numerical results show that the proposed model exhibits self-organized multistability with hierarchical branching. Moreover, these branches of equilibrium points are robust to perturbations, and the number of different cells produced can be determined by the system parameters. Full article

The achievement and control of desired motions in active machines often involves precise manipulation of artificial muscles in a distributed and sequential manner, which poses significant challenges. A novel motion control strategy based on self-oscillation in active machines offers distinctive benefits, such as direct energy harvesting from the ambient environment and the elimination of complex controllers. Drawing inspiration from automobiles, a self-moving automobile designed for operation under steady illumination is developed, comprising two wheels and a liquid crystal elastomer fiber. To explore the dynamic behavior of this self-moving automobile under steady illumination, a nonlinear theoretical model is proposed, integrating with the established dynamic liquid crystal elastomer model. Numerical simulations are conducted using the Runge-Kutta method based on MATLAB software, and it is observed that the automobile undergoes a supercritical Hopf bifurcation, transitioning from a static state to a self-moving state. The sustained periodic self-moving is facilitated by the interplay between light energy and damping dissipation. Furthermore, the conditions under which the Hopf bifurcation occurs are analyzed in detail. It is worth noting that increasing the light intensity or decreasing rolling resistance coefficient can improve the self-moving average velocity. The innovative design of the self-moving automobile offers advantages such as not requiring an independent power source, possessing a simple structure, and being sustainable. These characteristics make it highly promising for a range of applications including actuators, soft robotics, energy harvesting, and more. Full article

A dynamic model is established to investigate the shimmy instability of a landing gear system, considering the influence of nonlinear damping. The stability criterion is utilized to determine the critical speed at which the landing gear system becomes unstable. The central manifold theorem and canonical method are employed to simplify the dynamic model of the landing gear. The first Lyapunov coefficient of the system is theoretically derived and verified using numerical simulation. Further investigation on the Hopf bifurcation characteristics and stability of the shimmy in the landing gear system is conducted. The results indicate that above a certain threshold speed, with a tire stability distance greater than half the tire length in contact with the ground plus the slack length, the aircraft remains stable during taxiing. At critical speeds, a shimmy system with higher-order nonlinear damping will undergo supercritical Hopf bifurcation. Quantitative analysis suggests an increase in the linear damping coefficient within a range that ensures a stability margin to mitigate undesired oscillation, while the nonlinear damping coefficient should be designed within a reasonable range to decrease the amplitude of the limit cycle. Full article

We here investigate the dynamic behavior of continuous and discrete versions of a fractional-order predator–prey system with anti-predator behavior and a Holling type IV functional response. First, we establish the non-negativity, existence, uniqueness and boundedness of solutions to the system from a mathematical analysis perspective. Then, we analyze the stability of its equilibrium points and the possibility of bifurcations using stability analysis methods and bifurcation theory, demonstrating that, under specific parameter conditions, the continuous system exhibits a Hopf bifurcation, while the discrete version exhibits a Neimark–Sacker bifurcation and a period-doubling bifurcation. After providing numerical simulations to illustrate the theoretically derived conclusions and by summarizing the various analytical results obtained, we finally present four interesting conclusions that can contribute to better management and preservation of ecological systems. Full article

This article establishes a nonlinear flutter system for a long-span suspension bridge, aiming to analyze its supercritical flutter response under the influence of nonlinear aerodynamic self-excited force. By fitting the experimental discrete values of flutter derivatives using the least squares method, a polynomial function of flutter derivatives with respect to reduced wind speed is obtained. Flutter critical value is determined by the linear matrix eigenvalues of a state-space equation. The occurrence of a supercritical Hopf bifurcation in the nonlinear system is determined by the Jacobian matrix eigenvalues of the state-space equation and the system’s vibrational response at the critical state. The vibrational response of the supercritical state is obtained through Runge–Kutta integration, revealing the presence of stable limit cycle oscillation (LCO) and unstable limit cycle oscillation in the system, and through analyzing the relationship between the LCO amplitude and wind speed. Considering cubic nonlinear damping and stiffness, the effects of different factors on the nonlinear flutter system are analyzed. Full article

The influence of photovoltaic (PV) output with stochasticity and uncertainty on the grid-connected system’s voltage stability is worth further exploration. The long-term voltage stability of a 3-bus system with a large-scale PV power station considering the adjustment of an on-load tap changer (OLTC) was studied. In this typical system, two supercritical Hopf bifurcation (SHB) points are found using the bifurcation calculation. At the SHB point that appears first, a small sudden increase in reactive load power or a sudden increase in PV active power P_pv can eventually cause a voltage collapse after a long increasing oscillation. The long-term collapse phenomenon shows that SHB cannot be ignored in the PV grid-connected system. Meanwhile, the time constant of OLTC can affect the progress of long-term voltage collapse, but it has different effects under different disturbances. When P_pv drops suddenly at the SHB point, due to the adjustment of OLTC, the load bus voltage can recover to near the target value of OLTC after a long period of time. Similarly, the time constant of OLTC can affect the progress of long-term voltage recovery. To prevent the long-term voltage collapse when P_pv increases suddenly at the SHB point, a new locking-OLTC index I_lock, depending on the value of P_pv corresponding to the SHB point, and a locking OLTC method are proposed, and the voltage can be recovered to an acceptable stable value quickly. Compared with the system without OLTC, OLTC adjustment can effectively prevent long-term voltage oscillation instability and collapse, so that PV power can play a bigger role in power systems. Full article

Intense thermoacoustic oscillations may lead to severe deterioration due to the induced intolerable damage to combustors. A better understanding of unstable behaviors is important to prevent or suppress these oscillations. Active thermoacoustic coupling in practical combustors is caused primarily by two approaches: inherent turbulent fluctuations and the flame response to acoustic waves. Turbulent fluctuations are generally characterized by random noise. This paper experimentally expands on previous analytic studies regarding the influence of colored disturbances on the thermoacoustic response near the supercritical bifurcation point. Therein, a laboratory-scale Rijke-type thermoacoustic system is established, and both supercritical and subcritical bifurcations are observed. Then, Ornstein–Uhlenbeck (OU)-type external colored noise is introduced near the supercritical bifurcation point, and the effects of the corresponding correlation time ${\tau }_c$ and noise intensity D are studied. The experimental results show that these variables of the colored noise significantly influence the dynamics of thermoacoustic oscillations in terms of the most probable amplitude and autocorrelation properties. A resonance-like behavior is observed as the noise intensity or the autocorrelation time of the colored noise is continuously varied, which means that the coherent resonance occurs in the thermoacoustic system. Finally, when the system is configured closer to the stability boundary, the extent of the coherence motion is intensified in the stochastic system response. Meanwhile, the signal-to-noise ratios (SNRs) of the colored-noise-induced response are found to become more distinguished, the optimal colored noise intensity decreases, and the optimal autocorrelation time increases. These findings provide valuable guidance to predict the onset of thermoacoustic instabilities. Full article

A cylindrical plunge grinding process was modeled to investigate nonlinear regenerative chatter vibration. The rotating workpiece was a slender Euler–Bernoulli beam, and the grinding wheel was a rigid body moving towards the workpiece at a very low feed speed. A numerical method was proposed to provide the critical boundaries for chatter-free grinding. It was demonstrated that the intersection set surrounded by these critical boundaries was the chatter-free region for the considered parameters. When these parameters were outside of the chatter-free region, the stable grinding process underwent a supercritical Hopf bifurcation, resulting in the loss of the chatter-free behavior and the emergence of periodic chatter motions. Then, the periodic motions of both the grinding wheel and the workpiece were predicted analytically using the method of multiple scales, showing the effect of the regenerative force on the grinding process. We demonstrated that the analytical prediction was valid since it agreed with the numerical simulation. The results showed that there exist three kinds of nonlinear chatter motion, with different amplitudes and mode frequencies. Full article

Two nutrient–phytoplankton–zooplankton (NZP) models for a closed ecosystem that incorporates a delay in nutrient recycling, obtained using the gamma distribution function with one or two degrees of freedom, are analysed. The models are described by systems of ordinary differential equations of four and five dimensions. The purpose of this study is to investigate how the mean delay of the distribution and the total nutrients affect the stability of the equilibrium solutions. Local stability theory and bifurcation theory are used to determine the long-time dynamics of the models. It is found that both models exhibit comparable qualitative dynamics. There are a maximum of three equilibrium points in each of the two models, and at most one of them is locally asymptotically stable. The change of stability from one equilibrium to another takes place through a transcritical bifurcation. In some hypotheses on the functional response, the nutrient–phytoplankton–zooplankton equilibrium loses stability via a supercritical Hopf bifurcation, causing the apparition of a stable limit cycle. The way in which the results are consistent with prior research and how they extend them is discussed. Finally, various application-related consequences of the results of the theoretical study are deduced. Full article

This article presents a mathematical and experimental model of a neuronal oscillator with memristor-based nonlinearity. The mathematical model describes the dynamics of an electronic circuit implementing the FitzHugh–Nagumo neuron model. A nonlinear component of this circuit is the Au/Zr/ZrO₂(Y)/TiN/Ti memristive device. This device is fabricated on the oxidized silicon substrate using magnetron sputtering. The circuit with such nonlinearity is described by a three-dimensional ordinary differential equation system. The effect of the appearance of spontaneous self-oscillations is investigated. A bifurcation scenario based on supercritical Andronov–Hopf bifurcation is found. The dependence of the critical point on the system parameters, particularly on the size of the electrode area, is analyzed. The self-oscillating and excitable modes are experimentally demonstrated. Full article

This paper presents a two-dimensional simplified Hodgkin–Huxley model under exposure to electric fields. The Hopf bifurcations of the simplified Hodgkin–Huxley model are investigated through qualitative analysis and numerical simulations. A necessary and sufficient condition for the existence of Hopf bifurcations is derived, and the conditions for supercritical and subcritical Hopf bifurcations are obtained. Finally, bifurcation diagrams are given for two parameters, and numerical examples are presented to illustrate the effectiveness of the theoretical results. Full article