Bifurcation Behavior Research Articles

Invariant solutions, lie symmetry analysis, bifurcations and nonlinear dynamics of the Kraenkel-Manna-Merle system with and without damping effect

This work investigates the Kraenkel-Manna-Merle (KMM) system, which models the nonlinear propagation of short waves in saturated ferromagnetic materials subjected to an external magnetic field, despite the absence of electrical conductivity. The study aims to explore and derive new solitary wave solutions for this system using two distinct methodological approaches. In the first approach, the KMM system is transformed into a system of nonlinear ordinary differential equations (ODEs) via Lie group transformation. The resulting ODEs are then solved analytically using a similarity invariant approach, leading to the discovery of various types of solitary wave solutions, including bright, dark, and exponential solitons. The second approach involves applying wave and Galilean transformations to reduce the KMM system to a system of two ODEs, both with and without damping effects. This reduced system is further analyzed to investigate its bifurcation behavior, sensitivity to initial conditions, and chaotic dynamics. The analysis reveals the presence of strange multi-scroll chaotic dynamics in the presence of damping and off-boosting dynamics without damping. In addition to these approaches, the study also applies the planar dynamical theory to obtain further new soliton solutions of the KMM system. These solitons include bright, kink, dark, and periodic solutions, each of which has been visualized through 3D and 2D graphs. The results of this research provide new insights into the dynamics of the KMM system, with potential applications in magnetic data storage, magnonic devices, material science, and spintronics.

Dynamic Modeling and Configuration Transformation of Origami with Soft Creases

Origami has attracted much attention from scientists and engineers in recent years. Classic origami only permits rotations around creases, while origami-like structures in nature can also undergo a certain degree of extension at creases. In this paper, an earwig-inspired origami with rotational symmetry is proposed and analyzed by introducing soft creases. Such a soft crease is equivalent to a combination of a rotational spring and an extensional spring. The motion of the earwig-inspired origami is described using spherical triangles. Based on Lagrange's Equation, the nonlinear dynamic equation of the system is formulated. It is then solved by using the fourth-order Runge-Kutta method. The results of theoretical calculations are consistent with those of simulations in ADAMS. With the established framework, the bifurcation behaviors of the equilibria of the proposed system, including supercritical pitchfork and saddle-node bifurcations, are investigated. Such origami can realize both mono-stable and bi-stable mechanisms, while the corresponding design parameters are demonstrated in the design map. In particular, the properties of the bi-stable origami can vary with different equilibria. The configuration transformation could be achieved using a continuous excitation. However, it is sometimes sensitive to the initial conditions. Inspired by the working mechanism of earwig wings, a simple control approach for origami configuration transformation is proposed. The key is to stop exciting the origami when its deformation crosses an energy barrier. This work lays a foundation for the design and study of novel multi-stable and morphing structures and provides an efficient approach for their configuration transformation.

Nonlinear vibrations and static bifurcation of a hyperelastic pipe conveying fluid

This study investigates the free and forced nonlinear vibrations, along with the static bifurcation behavior, of a fluid transmission hyperelastic pipe supported by clamped ends. Both geometrical and material nonlinearities are taken into account. The mathematical model is grounded in Euler–Bernoulli beam theory, incorporating von Kármán nonlinearity to capture the complexities of the system. For the hyperelastic materials, the strain energy function proposed by Yeoh is employed. Considering that the lowest critical velocity of the fluid flow is associated with the first mode, after deriving the governing equation using Hamilton’s principle, Galerkin discretization is applied to obtain the reduced order model in the first mode. Analysis of static bifurcation and stability is performed and critical velocity values of fluid flow are determined. The analysis of free and forced vibrations of the hyperelastic pipe is conducted using the method of multiple time scales, specifically focusing on the sub-critical velocity range of fluid flow, which corresponds to the unbuckled state of the pipe. This approach allows for a comprehensive examination of how fluid flow velocity influences the frequency characteristics of free vibrations, as well as the frequency response in the vicinity of the main resonance. The results obtained from this study can significantly enhance our understanding of the design and application of hyperelastic pipes in various fluid transfer scenarios. By elucidating the dynamic behavior and critical parameters associated with these pipes, the findings can inform engineering practices and contribute to the optimization of systems that utilize hyperelastic materials for fluid conveyance.

Dynamic analysis of a modified Leslie-Gower model with Michaelis-Menten prey harvesting and additive allee effect on predator population

In the current study, the dynamics of a Leslie-Gower model have been studied considering the Allee effect in the growth of the predator and the Michaelis-Menten type harvesting effect on the prey from biological and theoretical perspectives. Key findings highlight how these effects influence the existence of positive equilibria, with local stability properties and potential bifurcation behaviors examined using the center manifold theorem and linearization. Criteria for saddle-node bifurcation are established through Sotomayor’s theorem, while the direction and stability of Hopf bifurcation are explored via the first Lyapunov exponent. An optimal control model is developed to promote sustainable ecosystem development, utilizing the harvesting rate as a control parameter. Further, a few numerical simulations validate our significant findings using phase portrait diagrams.

Analytical Solutions of the Fractional Hirota–Satsuma Coupled KdV Equation along with Analysis of Bifurcation, Sensitivity and Chaotic Behaviors

Analytical Solutions of the Fractional Hirota–Satsuma Coupled KdV Equation along with Analysis of Bifurcation, Sensitivity and Chaotic Behaviors

Investigation on nonlinear flutter modes competition of a long-span suspension bridge based on full-bridge aeroelastic model tests

Nonlinear flutter has attracted wide attention due to the bottleneck caused by linear flutter theory on flutter-resistant design of super-long-span bridges. The longer the span, the more closely spaced the natural modes, which may cause competition among flutter-modes in a nonlinear flutter, but it is rarely reported so far. To this end, this study experimentally investigates the 3D nonlinear flutter characteristics of a long-span suspension bridge with closely spaced natural modes based on full-bridge aeroelastic model wind tunnel tests. Since there are two unstable flutter-modes within the interested post-critical regime and thus various complex but interesting flutter-modes competition processes were observed, such as continuous modes competition where the vibration amplitude evolution exhibits various different types of “sawtooth” shapes during stable vibration stages. Meanwhile, the complex nonlinear bifurcation behaviors caused by modes competition were also observed. The evolutionary characteristics of flutter-modes competition are analyzed in detail and relevant mechanisms are discussed. As the wind speed increases, the flutter of the studied bridge mainly undergoes a transition from being dominated by the symmetric mode to being dominated by the antisymmetric mode, accompanied by a continuous modes competition zone in between as a transitional zone. The results show that the continuous modes competition will result in a decrease in the maximum amplitude RMS of the full-span, which is beneficial for the structure. But it may also lead to a transient increase in the maximum amplitude of the full-span due to the coupling vibration shape formed by the two significant flutter-modes, which may be bad for the structure.

The bifurcation, chaotic behavior and exact solutions of the fractional stochastic Jimbo–Miwa equations

The Jimbo–Miwa equation is the second equation in the KP hierarchy of integrable systems. In this paper, this equation is extended and introduced with the stochastic process and fractional derivatives. Firstly, the phase portrait of the Hamiltonian system generated by it is studied to understand its bifurcation behavior. Additionally, non-periodic and periodic perturbation terms are added to this system. Different values are assigned to the parameters in the perturbation terms to analyze its sensitivity and the resulting chaos is obtained. Finally, through integration techniques, the expression of the solution of this equation is obtained. These solutions are related to rational functions, trigonometric functions, exponential functions and Jacobi elliptic functions. To observe the form of the solutions more intuitively, 3D and 2D numerical simulations are conducted on the solutions and the solution images of the stochastic fractional differential equation are given by Matlab software. Compared with the existing literature, the research on the stochastic fractional equation of this equation is relatively rare and the analysis of the phase portrait is even scarcer. Our solution method is quite different from that in the previous literature. Therefore, this paper is novel. The conclusion of this paper will be of great help for the practical application of this equation.

The experimental research of which different types of impact signals affect the bifurcation chaos behavior of three-degree-of-freedom collision system

Abstract Collision is a faulty behavior in mechanical equipment, which greatly affects the safety of equipment. In this paper, aiming at the collision failures that are prone to occur in mechanical equipment, the nonlinear fault collision behavior of a class of three-degree-of-freedom collision systems is studied. A three-degree-of-freedom piecewise non-smooth collision experiment platform system has been established to facilitate further analysis of the bifurcations and chaos inherent to the bearings system. This analysis is conducted through the use of Poincaré mapping on the collision plane. The excitation signals employed, which include sinusoidal, triangular, and rectangular signals, result in the emergence of specific types and orders of bifurcation behaviours within the system, as evidenced by the establishment of a Poincaré mapping at the collision surface, and this is observed to occur as the vibration frequency increases. The characteristics of impact signals generated by force sensors and the influence of impact signal type on the system are investigated. The nonlinear impact behavior of three-degree-of-freedom collision systems provides a reliable foundation for the design, fault diagnosis, and theoretical guidance of large-scale high-speed rotating machinery, as well as technical support for safe and stable operation of high-speed rotating machinery, which is essential for failure analysis and vibration reduction in equipment.

Theoretical Analysis of Viscoelastic Friction System Characteristics of Robotic Arm Brake Based on Fractional Differential Theory

In engineering practice, the nonlinear vibration effect can easily lead to chaos in the system, which will not only reduce the performance of the system but also lead to premature fatigue of components, control failure, and increased safety risks. In view of the core position of the robotic arm in modern industry, this study relies on the robotic arm brake system to explore the theoretical basis of integrated viscoelastic materials as a vibration isolation layer. By analyzing the dynamic characteristics of the friction braking system with fractional differential terms, it aims to provide a new perspective for understanding and controlling the chaotic phenomena of a class of nonlinear friction systems. Firstly, we construct a model of a friction system and analyze its dynamic characteristics in detail. The self-excited vibration of the system under disturbance is studied. The relationship between amplitude and frequency is calculated by a nonlinear approximate analytical algorithm, and the accuracy of this relationship is verified by a numerical algorithm. Then, we compare the differences between non-fractional systems and fractional systems. It is found that with the increase in the fractional order term, the vibration amplitude of the system decreases significantly, which helps to reduce the nonlinear characteristics generated by the friction system and narrow the range of unstable solutions. Secondly, we also study the influence of parameter coefficients on the amplitude–frequency characteristics and analyze the local static bifurcation characteristics through singularity theory. Finally, we study the dynamic bifurcation behavior under different parameter perturbations and find that the change in system parameters will lead to the alternation of periodic motion and chaotic motion.

Optical solitons, dynamics of bifurcation, and chaos in the generalized integrable (2+1)-dimensional nonlinear conformable Schrödinger equations using a new Kudryashov technique

In the present paper, the new Kudryashov approach is utilized to construct several novel optical soliton solutions for the generalized integrable (2 + 1)-dimensional nonlinear Schrödinger system with conformable derivative. Additionally, the dynamics of bifurcation behavior and chaos analysis in this system are investigated. We applied bifurcation and chaos theories to enhance our understanding of the planar dynamical system derived from the current model while we obtained and illustrated the chaotic solutions for the perturbed dynamical system using graphs. The study yields a class of new optical soliton solutions, including bell-shaped, wave, dark, dark-bright, dark, multi-dark, and singular soliton solutions. Three-dimensional, two-dimensional, and contour plots are presented to visually demonstrate the physical implications and dynamic characteristics of the current conformable equation system. Further, an analysis is discussed on how the conformable derivative parameter and the parameter of time impact the present optical solutions, demonstrating the system’s importance. It is believed that the solutions analyzed in this study are entirely new and have not been previously reported. These discoveries have the potential to significantly enhance our understanding of nonlinear physical phenomena, especially in nonlinear optics and traffic signaling effects with optical dromion transmission.

Non-smooth self-excited vibration of a novel dynamical model for a disc brake

This paper proposes a new dynamic model for the study of friction-induced self-excited vibration of a disc brake system, where the pad’s motions in both radial and circumferential/tangential directions are included, which is in stark contrast to the previous studies that normally consider the pad’s motion in the tangential/circumferential direction only. The non-smooth dynamics of the system including three different states of motion, i.e., stick, slip and separation, is investigated. Both the linear stability analysis and the transient dynamic analysis are performed. The numerical results in the linear stability analysis indicate that the inclusion of pad’s radial motion in the present brake model significantly expands the ranges of operating parameters for dynamic instability than the brake model with only circumferential/tangential motion for the pad. For the transient dynamic analysis, two different methods, i.e., the time integration method and the shooting method, are employed for the calculation of steady-state response. The accuracy and efficiency of the shooting method are subsequently examined. The numerical results show rich bifurcation behaviours of the steady-state response in the present brake model with the variations of brake pressure and disc speed , and (the stiffness of the inclined spring in the radial direction) is a key parameter for controlling the occurrence of chaotic vibration in the system. Conflict of Interest Statement The authors declare no conflicts of interest.

The role of dynamic friction in the appearance of periodic oscillations in mechanical systems

This article investigates the appearance of periodic mechanical oscillations associated with the transition between static and dynamic friction regimes. The study employs a mechanical system with one degree of freedom and a friction model recently proposed by Brown and McPhee, whose continuity and differentiability properties make it particularly appropriate for an analytical treatment of the equations. A bifurcation study of the system, including stability analysis, transformation to normal form and numerical continuation techniques, reveals that stable periodic orbits can be created either by a supercritical Hopf bifurcation or by a saddle-node bifurcation of limit cycles. The influence of all system parameters on the appearance of periodic oscillations is investigated in detail. In particular, the effect of the friction model parameters (static-to-dynamic friction ratio and transition speed between the static and dynamic regimes) on the bifurcation behavior of the system is addressed.

Nodal solutions for a nonlocal fourth order equation of Kirchhoff type

We study the bifurcation behavior of nodal solutions for the Kirchhoff type beam equation u′′′′−M(∫01|u′|2dx)u′′=λf(x,u),x∈(0,1),u(0)=u(1)=u′′(0)=u′′(1)=0, where λ∈R is a parameter, M:[0,∞)→[0,∞) and f:[0,1]×R→R are smooth functions. We obtain the existence of nodal solutions under some suitable conditions. The proof of our main result is based upon bifurcation techniques.

Bifurcation and Stability of a Haptotaxis Mathematical Model for Complex MAP

This paper studies the bifurcation and stability of a parabolic–ordinary-parabolic system to a model with three reactive species: molecular Anchors [Formula: see text], Nanoparticulates [Formula: see text] and Matrix constituents [Formula: see text] ([Formula: see text]). Global asymptotic stability is studied for both the ODE system and the reaction–diffusion system. The results show that the positive equilibrium is globally asymptotically stable in both the ODE and the corresponding reaction–diffusion systems. Furthermore, by selecting the haptotaxis coefficient as a bifurcation factor, we analyze the existence, direction, and stability of the Hopf bifurcation. The results also reveal that haptotaxis is crucial in determining the stability and bifurcation behavior of the model, suggesting that it has a destabilizing effect and can undergo Hopf bifurcation. Moreover, numerical simulations are presented to verify and illustrate the theoretical results.

Modulation instability, bifurcation and chaotic behaviors for a generalized (2+1)-dimensional nonlinear wave equation in a fluid or solid

In this paper, we investigate a generalized (2+1)-dimensional nonlinear wave equation characterizing nonlinear waves in a fluid or solid. We use the Painlevé analysis to test the integrability of that equation. In order to research the modulation instability (MI) of that equation, we obtain the one-soliton solutions of that equation through the Hirota bilinear method. The propagation velocity formula and characteristic line of the one soliton are derived. Then, we perform the MI to that equation through the standard linear stability. We study the distribution of the MI gain under the parameters K, Ω and R, which are the perturbation wave numbers in x, t directions and the initial amplitude, respectively. We find that the amplitude, bandwidth and distance to the line Ω=0 or R=0 of the MI gain vary as the K or R value changes. Finally, we analyze the bifurcation behavior of the system using direction field maps and investigate its chaotic behavior under the influence of a periodic external force using phase portraits.

Double flame dynamics in hotspot ignition

Hotspot ignition is a key problem in fundamental flame initiation, combustion control and prevention of abnormal combustion phenomena. Depending on the characteristics of the hotspot and the thermochemical condition of the reactive mixture, subsequent reaction front initiated from a hotspot can vary substantially. The current paper focus on the dynamics and chemistry for one of the most complicated scenarios - hotspot induced double flames, which include a cool and a hot flame segment. This work adopts a recently developed compressible reacting flow simulation code COGNAC to investigate the initiation and propagation of double flame dynamics from hotspot ignition in a one-dimensional spherical coordinate. The results have shown that under atmospheric pressure, double flame initiation is not feasible for sufficiently small hotspot. Under elevated pressures, double flame can be initiated within a much wider hotspot window, exhibiting complex flame dynamics, such as variations in cool and hot flame bifurcation behaviors and their initiation locations. Analysis on thermochemical structure and dynamics of double flame has shown that the interaction of cool and hot flames in a double flame structure is inherently intermutual, in that the leading cool flame enhances trailing hot flame speed through reform of the unburnt mixture, while the trailing hot flame initiation affects the leading cool flame via thermal expansion effects.

Role of inertial nonlinearity and coupling stiffness on a series of coupled harvesters

This study investigates the effect of coupling configuration on a state-of-the-art harvesting system consisting of a series of spring-coupled piezoelectric nonlinear monostable tip-loaded cantilever-based vibration energy harvesters (VEHs). The investigation finds favorable coupling configurations that exploit complex nonlinear interaction for improving system performance. The system model incorporates the beam's geometric nonlinearities in the governing equation using inertial and stiffness nonlinear terms. The study first analytically investigates the effect of grounding stiffness on a single harvester's performance. Then, it numerically investigates the effect of increasing array size and changing stiffness distribution. The study discovers a novel hardening/softening transition phenomenon at a critical grounding stiffness value induced by the inertial nonlinear effect. This critical event also maximizes peak output power near the largest eigenfrequency at an optimum grounding stiffness value. At larger arrays, the peak power shows a complex bifurcation behavior, hinting at the presence of multiple attractors. The numerical study of different stiffness configurations and linearized eigenanalysis reveals the necessity of a multilevel optimization strategy focused on both the stiffness ratio between the coupling springs' and their overall stiffness level. This study bridges the gap between previous studies on spring-coupled linear and bistable arrays, offering new insights into coupled nonlinear interactions and proposing novel optimization strategies.

Research on Stability and Bifurcation for Two-Dimensional Two-Parameter Squared Discrete Dynamical Systems

This study investigates a class of two-dimensional, two-parameter squared discrete dynamical systems. It determines the conditions for local stability at the fixed points for these proposed systems. Theoretical and numerical analyses are conducted to examine the bifurcation behavior of the proposed systems. Conditions for the existence of Naimark–Sacker bifurcation, transcritical bifurcation, and flip bifurcation are derived using center manifold theorem and bifurcation theory. Results of the theoretical analyses are validated by numerical simulation studies. Numerical simulations also reveal the complex bifurcation behaviors exhibited by the proposed systems and their advantage in image encryption.

Unraveling Chaos, Transition and Dynamical Complexities in a Generalized Predator–Prey Model with Cooperation-Induced Fear and Gestation Delay

This research investigates the interaction between a generalist predator and a prey, where the predator exhibits cooperative behavior during hunting, inducing fear into the prey population. Additionally, both the prey and predator populations are subject to harvesting. The study establishes the positivity and boundedness of the model’s solutions, ensuring the existence of the population. Analyzing the system, we explore its feasible steady states and their stability, along with various types of bifurcations, including Hopf with direction of stability, Saddle-node, Transcritical, Homoclinic, Bogdanov–Takens, and Cusp bifurcation. We also demonstrate the stability and bifurcation behavior of a delayed system. These findings are verified through one-parameter and two-parameter bifurcation structures, complemented by respective phase portraits. Notably, the system displays transition between different equilibria and bistability. Furthermore, numerical investigations reveal the impact of gestation delay, indicating chaotic behavior in the system due to this time delay.

Bifurcation Mechanism at a Sustain Point of a Long Narrow Economy

In this paper, we investigate population agglomeration in a long narrow economy, in which an odd number of places are evenly distributed over a line segment. The bifurcation analysis of this economy elucidates the mechanism of the emergence of twin cities around the central city. The validity and usefulness of this analysis are confirmed using several well-known economic geography models that display various kinds of bifurcation behaviors. By this analysis, we investigate the historical change in the population distribution in a chain of cities on Japan’s Main Island.