Unstable Saddle Point Research Articles

Bifurcation analysis and control of improved traffic flow model considering the effect of strong winds.

In recent years, the problem of traffic congestion has become increasingly serious, and research on traffic system control has become a new hotspot. Studying the bifurcation characteristics of traffic flow systems and designing control schemes for unstable support points can alleviate traffic congestion from a new perspective. This article improves the full speed differential model considering strong wind models from the perspective of bifurcation control to adjust traffic flow. This article theoretically proves the existence conditions of Hopf bifurcation and saddle node bifurcation in the model and finds the stability mutation point of the transportation system stability. A nonlinear system feedback controller was designed for unstable bifurcation points using Chebyshev polynomial approximation and random feedback control methods. Without changing the system equilibrium point, the advance, delay, and elimination of Hopf bifurcation were achieved, and the abrupt behavior of the transportation system was controlled, thereby alleviating traffic congestion. This article explains the changes in the stability of complex transportation systems from the perspective of bifurcation analysis, which can better capture the characteristics of traffic flow. By adjusting the control parameters in the feedback controller, the influence of boundary conditions on the stability of the transportation system is fully described, and the influence of unstable focal points and saddle points on the system is suppressed, thereby slowing down the traffic flow. In addition, unstable bifurcation points can be eliminated, and the Hopf bifurcation can be controlled to advance, delay, and disappear, thereby achieving control over the stable behavior of the transportation system. This helps alleviate traffic congestion and also helps describe actual traffic phenomena.

Particle-based and continuum models for confined nematics in two dimensions.

We use the particle-based stochastic multi-particle collision dynamics (N-MPCD) algorithm to simulate confined nematic liquid crystals in regular two-dimensional polygons such as squares, pentagons and hexagons. We consider a range of values of the nematicities, U, and simulation domain sizes, R, that canvass nano-sized polygons to micron-sized polygons. We use closure arguments to define mappings between the N-MPCD parameters and the parameters in the continuum deterministic Landau-de Gennes framework. The averaged N-MPCD configurations agree with those predicted by Landau-de Gennes theory, at least for large polygons. We study relaxation dynamics or the non-equilibrium dynamics of confined nematics in polygons, in the N-MPCD framework, and the kinetic traps bear strong resemblance to the unstable saddle points in the Landau-de Gennes framework. Finally, we study nematic defect dynamics inside the polygons in the N-MPCD framework and the finite-size effects slow down the defects and attract them to polygon vertices. Our work is a comprehensive comparison between particle-based stochastic N-MPCD methods and deterministic/continuum Landau-de Gennes methods, and this comparison is essential for new-age multiscale theories.

Nonlinear dynamics of 3D superconducting maglevs with six degrees of freedom: dependence on magnetization strategy and disturbance type

Based on the heat diffusion equation, Maxwell’s equations, and translational and rotational dynamic equations, we establish and theoretically validate an electromagnetic-thermal-mechanical coupling model to analyze the levitation performance during normal operation and the nonlinear dynamic behavior under disturbance for 3D maglev systems composed of a six-degree-of-freedom bulk superconductor (SC) and a Halbach-type guideway of permanent magnets (PMs) with different magnetization strategies and different types of disturbances, as well as the change rules of magnetic force and torque during translational or rotational cycle movement. In order to ensure the system security, we propose a generalized electromagnetic restoring force model to theoretically analyze the stability of the SC moving along the directions of various degrees of freedom. The results show that after being disturbed, the SC vibrates along the direction of each degree of freedom, and the vibration center, i.e. equilibrium position, will drift along each vibration direction. With time increasing, the equilibrium position will appear periodically on both sides of the working position. Compared to zero-field cooling magnetization, field cooling magnetization enables the SC to trap more flux in its interior to alleviate the drift phenomenon and reduce the energy loss. This advantage can be further enhanced by adding an extra step of preloading treatment. For the lateral motion, the system has one stable focus point and two unstable saddle points. Whether the system at these saddle points is stable depends on the direction of disturbance-induced velocity. For the rotational motion, the system has only one stable focus point, which means that regardless of the type of disturbance, the SC will finally come back to its stable equilibrium position. Besides, the stability is related to the axis around which the SC rotates, and rotating around the longitudinal axis is more likely to generate larger magnetic force, torque and local temperature rise. Either field cooling magnetization or preloading treatment can effectively improve system stability.

Bifurcation control analysis based on continuum model with lateral offset compensation

The study of traffic flow bifurcation control analysis is of great significance for understanding the essential evolution law of traffic flow, controlling traffic flow with practical means and alleviating traffic congestion. This paper describes the influence of lateral distance compensation in the process of lane change, and adjusts the traffic flow from the perspective of bifurcation control, which can capture the characteristics of traffic flow better, so as to describe the practical significance of the actual traffic phenomenon better. For example, the green ratio of traffic lights in the road, the planning and design of traffic signs, and the guidance of traffic condition prediction in ITS. In addition, the traffic flow is introduced and applied to the traffic model from the perspective of random function, and the bifurcation control is carried out according to the bifurcation behavior in traffic. That is, the bifurcation characteristics of the system are modified by the designed feedback controller, and the appearance of the equilibrium point of the system is adjusted to make it move forward, backward or disappear, so as to prevent or alleviate traffic congestion. There are few researches on bifurcation control of traffic flow model with lateral distance compensation. This paper introduces stochastic function based on lateral compensation model to study Hopf bifurcation phenomenon and Hopf bifurcation control. Firstly, the existence condition of Hopf bifurcation at the equilibrium point in the model is proved theoretically. Then, a feedback controller is designed to control the amplitudes of the Hopf bifurcation and the limit cycles formed by the Hopf bifurcation. Finally, the theoretical results are verified by numerical simulation. The research shows that by adjusting the control parameters in the feedback controller, the influence of boundary conditions on the stability of the traffic system is fully described, the effect of unstable focus and saddle point on the system is inhibited, and the traffic flow is slowed down. In addition, the unstable bifurcation points can be eliminated, and the amplitude of the limit cycle formed by Hopf bifurcation can be adjusted, so as to achieve the stability of the control system.

Out-of-plane chaotic motion and suppression for tethered tug-debris systems with thrust perturbation

This paper investigates the characterization and suppression of chaotic motion in an out-of-plane tethered tug-debris system that is towed by a constant thrust with time-varying micro-perturbations at a fixed non-zero in-plane libration angle. These perturbations arise due to imperfections in the thruster’s mechanical system on the tug. Static equilibrium solutions and homoclinic orbits of the corresponding Hamiltonian system are derived, and a subcritical fork bifurcation is identified and presented. By the Melnikov method, it is found that the micro-fluctuation in the thrust could induce chaos in the system near unstable saddle points. Then, a chaotic parameter region is produced to identify chaos efficiently. The pure out-of-plane chaotic motion would appear, provided system parameters are in the region. A sliding mode controller is designed to suppress the chaotic oscillation by deploying and retrieving the tether. Finally, the impact of the system parameters on the chaotic region is estimated numerically. Simulation results demonstrate that chaotic motions exist and can be identified efficiently by the chaotic parameter region and that the control method is effective in suppressing the chaotic motion.

Memristor synapse-coupled piecewise-linear simplified Hopfield neural network: Dynamics analysis and circuit implementation

Electromagnetic induction current is generated between the adjacent neurons in neural network caused by the existence of membrane potential difference. Memristor is the fourth fundamental electric element, which can mimic the behavior of neural synapses and simulate the electromagnetic induction effect. In this paper, in order to simplify practical implementation, the commonly used hyperbolic tangent activation function of Hopfield neural network (HNN) is replaced by a piecewise-linear function, and a simple bi-neuron-based memristor synapse-coupled HNN is proposed. Theoretical analysis and numerical simulation results illustrate that the memsirstive HNN possesses five equilibria including one unstable saddle-focus, two unstable saddle points, and two stable node points (or node-foci). Local attraction basins and phase plane plots show that the memristive HNN model behaves multistability of coexisting chaos, periodic limit cycles, and stable point attractors. Various system dynamical behaviors affected by parameters of the piecewise-linear activation function, memristor coupling strength, and initial conditions of the neurons are investigated by numerical simulations. Furthermore, an analog circuit of the memristive HNN model is simply designed, which is easier for hardware implementation as the piecewise-linear activation function can be implemented by simple op-amp-based module. Finally, the experimental results verify the correctness of the design and analyses.

Abstract 12800: Mathematical Modeling of Myocardial Transthyretin Amyloid Deposition and Clearance

Introduction: In vitro transthyretin amyloid (ATTR) fibrillation kinetics are well studied, however, the in vivo interplay between ATTR fibrillation and clearance causing cardiac amyloidosis (TTR-CA) is underexplored. We investigated these interactions using a mathematical model of ATTR deposition. Methods: We divided the mechanisms of ATTR deposition from tetramer to clearance by clusterin (an extracellular chaperone) into individual reactions (Fig. 1A) with 7 parameters. A system of ordinary differential equations (ODEs) describes the change in ATTR monomer, fibril, inhibitor, and complex deposition. Parameter values were determined using published experimental data and equilibrium analyses. We model an FDA-approved stabilizer, Tafamadis, by modifying S(t), the source term for production and dissociation of tetramers, as a step function downward at therapy initiation. Alternatively, a linear function S(t) models the hypothesis that senile TTR-CA is triggered by age-related wtTTR instability from oxidation. Results: The ODEs simulated a biologically meaningful model of TTR-CA. Equilibrium analyses found three phenotypes. (1) One stable, low equilibrium point representing normal myocardium (low S, high Ki). (2) Three equilibrium points, with 2 stable points (low and high) and a middle, unstable saddle point, indicating susceptibility to perturbation into a state with significant ATTR (low S/Kc2, high Ki/Kc). (3) No equilibrium points causing unbounded ATTR (high S, low Ki). In the Tafamadis model, ATTR fibril reduction was proportional to drug efficacy (Fig. 1B). Senile TTR-CA model showed rapid progression of cardiac ATTR deposition with onset dependent on slope of S (Fig. 1C). Conclusions: This mathematical model predicts physiologic ATTR clearance and pathological ATTR deposition with a limited number of parameters. It is a hypothesis generating model that can elucidate mechanisms of TTR-CA pathogenesis and investigate future therapies.

Research on System Risks of “Internet + Supply Chain Finance” Based on SNA, Dynamic Evolutionary Game, and Bayesian Learning Principle Simulation

Under the background of "Internet +", the supply chain financial model and process have undergone profound changes. Firstly, through the social network analysis, the correlation between the participants in the Internet + supply chain finance is directly visualized. Secondly, the dynamic risk evolution model of the system is constructed based on the different functions between the participants. Unstable solution and saddle points of the system be calculated ; on this basis, Bayesian learning principles are used to build an Internet + supply chain financial credit default risk simulation model, and the simulation model is encapsulated. Finally, a numerical example is used to verify the simulation model operation Convenience, efficiency and reliability.

Krylov complexity in saddle-dominated scrambling

In semi-classical systems, the exponential growth of the out-of-time-order correlator (OTOC) is believed to be the hallmark of quantum chaos. However, on several occasions, it has been argued that, even in integrable systems, OTOC can grow exponentially due to the presence of unstable saddle points in the phase space. In this work, we probe such an integrable system exhibiting saddle-dominated scrambling through Krylov complexity and the associated Lanczos coefficients. In the realm of the universal operator growth hypothesis, we demonstrate that the Lanczos coefficients follow the linear growth, which ensures the exponential behavior of Krylov complexity at early times. The linear growth arises entirely due to the saddle, which dominates other phase-space points even away from itself. Our results reveal that the exponential growth of Krylov complexity can be observed in integrable systems with saddle-dominated scrambling and thus need not be associated with the presence of chaos.

Cavity expansion in nonlinear viscoelastic solids: A nonlinear dynamic study

Dynamic cavity expansion has been studied for hyperelastic materials since the work of Knowles and Jakub in 1965. It has recently drawn new attention for potential applications in measuring the mechanical properties of soft polymer or biological tissues. These soft materials, however, are intrinsically rate-dependent, and the study of dynamic cavitation in nonlinear viscoelastic solid is relatively scarce. In this article, we utilize the established two-potential nonlinear viscoelastic framework to study the cavity behavior in infinite solid. Differently, we further introduced the surface energy into account and pay extra attention to the oscillation of the cavity surface based on nonlinear dynamics. The cavity behavior and stress change under a monotonically increasing pressure, creep experiment, and relaxation experiment are investigated. Considering surface effects, we notice the cavity surface has an oscillation process of expansion and shrink before infinitely large expansion before rupture. The oscillation amplitude gradually diminishes with time due to the energy consumption of viscosity. Regarding the viscoelastic material as a system with diminishing shear modulus, we analyze the stability of the cavity surface and find it can be either a stable center point or an unstable saddle point. The results of this study are helpful for understanding the cavitation phenomenon in viscoelastic solids caused by dynamic loading.

Stability analysis of circular orbits around a traversable wormhole with massless conformally coupled scalar field

We study the stability of circular orbits in the background of a traversable wormhole (TWH) spacetime obtained as a solution of Einstein’s field equations coupled conformally to a massless scalar field. The Lyapunov stability approach is employed to determine the stability of circular orbits (timelike and null) of non-spinning test particles around a TWH spacetime. In the case of timelike geodesics, the particle is confined to move in four different types of effective potentials depending on various values of the angular momentum tilde{L} with both centrifugal and gravitational part. The effective potential for null geodesics consists of only a centrifugal part. Further, we characterize each fixed point according to its Lyapunov stability, and thus classify the circular orbits at the fixed point into stable center and unstable saddle points by depicting the corresponding phase-portraits.

Coupled effects of surface interaction and damping on electromechanical stability of functionally graded nanotubes reinforced torsional micromirror actuator

This paper demonstrated the coupled surface effects of thermal Casimir force and squeeze film damping (SFD) on size-dependent electromechanical stability and bifurcation of torsion micromirror actuator. The governing equations of micromirror system are derived, and the pull-in voltage and critical tilting angle are obtained. Also, the twisting deformation of torsion nanobeam can be tuned by functionally graded carbon nanotubes reinforced composites (FG-CNTRC). A finite element analysis (FEA) model is established on the COMSOL Multiphysics platform, and the simulation of the effect of thermal Casimir force on pull-in instability is utilized to verify the present analytical model. The results indicate that the numerical results well agree with the theoretical results in this work and experimental data in the literature. Further, the influences of volume fraction and geometrical distribution of CNTs, thermal Casimir force, nonlocal parameter, and squeeze film damping on electrically actuated instability and free-standing behavior are detailedly discussed. Besides, the evolution of equilibrium states of micromirror system is investigated, and bifurcation diagrams and phase portraits including the periodic, homoclinic, and heteroclinic orbits are described as well. The results demonstrated that the amplitude of the tilting angle for FGX-CNTRC type micromirror attenuates slower than for FGO-CNTRC type, and the increment of CNTs volume ratio slows down the attenuation due to the stiffening effect. When considering squeeze film damping, the stable center point evolves into one focus point with homoclinic orbits, and the dynamic system maintains two unstable saddle points with the heteroclinic orbits due to the effect of thermal Casimir force.

On Equilibria Stability in an Epidemiological SIR Model with Recovery-dependent Infection Rate

We consider an epidemiological SIR model with an infection rate depending on the recovered population. We establish sufficient conditions for existence, uniqueness, and stability (local and global) of endemic equilibria and consider also the stability of the disease-free equilibrium. We show that, in contrast with classical SIR models, a system with a recovery-dependent infection rate can have multiple endemic stable equilibria (multistability) and multiple stable and unstable saddle points of equilibria. We establish conditions for the occurrence of these phenomena and illustrate the results with some examples.

Viscoelastic bistable tape spring behavior modeled by a two-dimensional system with rigid links, springs and dashpots

Bistable composite tape springs are often used as deployable structures in aerospace applications because of their ability to be tailored for compact size, deployed lengths and stiffness. Analytical studies of the coupling between stress-strain and moment-curvature of thin curved shells have been used to estimate the total stored strain energy in different tape spring configurations. The strain energy as function of the curvatures reveals the fundamental stability behavior of the tape springs in the different configurations. The present study investigates the minimum strain energy path between two stable equilibrium points and the passage through the unstable saddle point along this path. For a coiled fiber-reinforced composite tape spring, the strain energy level for the local minimum and the saddle points will change with respect to time due to viscoelastic effects. To explain the fundamental mechanical behavior of viscoelastic bistable tape springs, a two-dimensional (2D) model composed of rigid links, elastic springs and viscous dampers was proposed. It was shown how the 2D model can be made strain energy-equivalent to a shell model with respect to the minimum strain energy paths. The 2D model was able to capture the fundamental behavior of viscoelastic tape springs during both snap-through and snap-back paths.

FPGA-based Experimental Validations of Electrical Activities in Two Adjacent FitzHugh–Nagumo Neurons Coupled by Memristive Electromagnetic Induction

ABSTRACT With the consideration of electromagnetic induction generated by the difference of membrane potentials between two adjacent neurons, a five-dimensional (5-D) memristor-coupled FitzHugh–Nagumo (FHN) bi-neuron model derived from connecting two FHN neurons with the memristive electromagnetic induction is presented. Then, with the 5-D memristor-coupled FHN bi-neuron model, the equilibrium states and their stabilities are explored by theoretical and numerical analyses. It is indicated that the proposed model has two non-zero equilibrium states with the same stability of unstable saddle-focus or unstable saddle point associating to the considered model parameters. The parameter-associated electrical activities of periodic, quasi-periodic, and chaotic behaviors are disclosed by bifurcation plots, dynamical maps, phase plane plots and further confirmed by 0–1 tests. Besides, the phenomena of initial-associated bi-stability with the coexistence of chaotic and periodic behaviors, as well as the riddled attraction basin, are revealed by local attraction basins and phase plane plots. Furthermore, based on field programmable gate array (FPGA), a digitally implemented electronic circuit is made, from which the electrical activities are experimentally captured to verify the numerical ones.

Bifurcation Analysis of an Electro-Statically Actuated Nano-beam Based on the Nonlocal Theory considering Centrifugal Forces

Abstract A nonlocal elasticity theory is a popular growing technique for mechanical analysis of the micro- and nanoscale structures which captures the small-size effects. In this paper, a comprehensive study was carried out to investigate the influence of the nonlocal parameter on the bifurcation behavior of a capacitive clamped-clamped nano-beam in the presence of the electrostatic and centrifugal forces. By using Eringen’s nonlocal elasticity theory, the nonlocal equation of the dynamic motion for a nano-beam has been derived using Euler–Bernoulli beam assumptions. The governing static equation of motion has been linearized using step by step linearization method; then, a Galerkin based reduced order model have been used to solve the linearized equation. In order to study the bifurcation behavior of the nano-beam, the static non-linear equation is changed to a one degree of freedom model using a one term Galerkin weighted residual method. So, by using a direct method, the equilibrium points of the system, including stable center points, unstable saddle points and singular points have been obtained. The stability of the fixed points has been investigated drawing motion trajectories in phase portraits and basins of attraction set and repulsion have been illustrated. The obtained results have been verified using the results of the prior studies for some cases and a good agreement has been observed. Moreover, the effects of the different values of the nonlocal parameter, angular velocity and van der Waals force on the fixed points have been studied using the phase portraits of the system for different initial conditions. Also, the influence of the nonlocal beam theory and centrifugal forces on the dynamic pull-in behavior have been investigated using time histories and phase portraits for different values of the nonlocal parameter.

Dynamics of an ecological system

AbstractIn this paper, we investigate the deterministic and stochastic prey-predator system with refuge. The basic local stability results for the deterministic model have been performed. It is found that all the equilibrium points except the positive coexisting equilibrium point of the deterministic model are independent of the prey refuge. The trivial equilibrium point, predator free equilibrium point and prey free equilibrium point are always unstable (saddle point). The existence and local stability of the coexisting equilibrium point is related to the prey refuge. The permanence and extinction conditions of the proposed biological model have been studied rigourously. It is observed that the stochastic effect may be seen in the form of decaying of the species. The numerical simulations for different values of the refuge values have also been included for understanding the behavior of the model graphically.

Exact stationary solutions of the parametrically driven and damped nonlinear Dirac equation.

Two exact stationary soliton solutions are found in the parametrically driven and damped nonlinear Dirac equation. The parametric force considered is a complex ac force. The solutions appear when their frequencies are locked to half the frequency of the parametric force, and their phases satisfy certain conditions depending on the force amplitude and on the damping coefficient. Explicit expressions for the charge, the energy, and the momentum of these solutions are provided. Their stability is studied via a variational method using an ansatz with only two collective coordinates. Numerical simulations confirm that one of the solutions is stable, while the other is an unstable saddle point. Consequently, the stabilization of damped Dirac solitons can be achieved via time-periodic parametric excitations.

Non-ideal memristor synapse-coupled bi-neuron Hopfield neural network: Numerical simulations and breadboard experiments

This paper presents a third-order non-ideal memristor synapse-coupled bi-neuron Hopfield neural network (HNN), in which a non-ideal memristor synapse is employed to generate an electromagnetic induction current caused by the potential difference between two neurons. The memristive HNN model possesses an odd number of equilibrium points including one (or three, dependent on the memristor coupling strength) unstable saddle point, two unstable index-2 saddle-foci, and two (or four) stable node-foci. Bifurcation plots, local attraction basins, and phase plane plots show that the memristive HNN model behaves bi-stability of coexisting chaotic and stable point attractors, which is perfectly validated by breadboard experiments based on discrete components.

Analysis and Control of Fractional Order Generalized Lorenz Chaotic System by Using Finite Time Synchronization

In this paper, we present a corresponding fractional order three-dimensional autonomous chaotic system based on a new class of integer order chaotic systems. We found that the fractional order chaotic system belongs to the generalized Lorenz system family by analyzing its linear term and topological structure. We also found that the equilibrium point generated by the fractional order system belongs to the unstable saddle point through the prediction correction method and the fractional order stability theory. The complexity of fractional order chaotic system is given by spectral entropy algorithm andC0algorithm. We concluded that the fractional order chaotic system has a higher complexity. The fractional order system can generate rich dynamic behavior phenomenon with the values of the parameters and the order changed. We applied the finite time stability theory to design the finite time synchronous controller between drive system and corresponding system. The numerical simulations demonstrate that the controller provides fast and efficient method in the synchronization process.