Unstable Saddle Research Articles

Influence of Controller’s Parameters on Static Bifurcation of Magnetic-Liquid Double Suspension Bearing

Magnetic-Liquid Double Suspension Bearing (MLDSB) is composed of an electromagnetic supporting and a hydrostatic supporting system. Due to greater supporting capacity and static stiffness, it is appropriate for occasions of middle speed, overloading, and frequent starting. Because of the complicated structure of the supporting system, the probability and degree of static bifurcation of MLDSB can be increased by the coupling of hydrostatic force and electromagnetism force, and then the supporting capacity and operation stability are reduced. As the key part of MLDSB, the controller makes an important impact on its supporting capacity, operation stability, and reliability. Firstly, the mathematical model of MLDSB is established in the paper. Secondly, the static bifurcation point of MLDSB is determined, and the influence of parameters of the controller on singular point characteristics is analyzed. Finally, the influence of parameters of the controller on phase trajectories and basin of attraction is analyzed. The result showed that the pitchfork bifurcation will occur as proportional feedback coefficient Kp increases, and the static bifurcation point is Kp = −60.55. When Kp < −60.55, the supporting system only has one stable node (0, 0). When Kp > −60.55, the supporting system has one unstable saddle (0, 0) and two stable non-null focuses or nodes. The shape of the basin of attraction changed greatly as Kp increases from −60.55 to 30, while the outline of the basin of attraction is basically fixed as Kp increases from 30 to 80. Differential feedback coefficient Kd has no effect on the static bifurcation of MLDSB. The rotor phase trajectory obtained from theoretical simulation and experimental tests are basically consistent, and the error is due to the leakage and damping effect of the hydrostatic system within the allowable range of the engineering. The research in the paper can provide theoretical reference for static bifurcation analysis of MLDSB.

Decoding complex state space trajectories for neural computing.

In biological neural circuits as well as in bio-inspired information processing systems, trajectories in high-dimensional state-space encode the solutions to computational tasks performed by complex dynamical systems. Due to the high state-space dimensionality and the number of possible encoding trajectories rapidly growing with input signal dimension, decoding these trajectories constitutes a major challenge on its own, in particular, as exponentially growing (space or time) requirements for decoding would render the original computational paradigm inefficient. Here, we suggest an approach to overcome this problem. We propose an efficient decoding scheme for trajectories emerging in spiking neural circuits that exhibit linear scaling with input signal dimensionality. We focus on the dynamics near a sequence of unstable saddle states that naturally emerge in a range of physical systems and provide a novel paradigm for analog computing, for instance, in the form of heteroclinic computing. Identifying simple measures of coordinated activity (synchrony) that are commonly applicable to all trajectories representing the same percept, we design robust readouts whose sizes and time requirements increase only linearly with the system size. These results move the conceptual boundary so far hindering the implementation of heteroclinic computing in hardware and may also catalyze efficient decoding strategies in spiking neural networks in general.

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.

A novel current-controlled memristor-based chaotic circuit

In this paper, a novel third-order autonomous memristor-based chaotic circuit is proposed. The circuit has simple topology and contains only four elements including one linear negative impedance converter-based resistor, one linear capacitor, one linear inductor, and one nonlinear current-controlled memristor. Firstly, the voltage-current characteristic analysis of the memristor emulator for different driving amplitudes and frequencies are presented. With dimensionless system, the symmetry, equilibrium point and its stability are analysed. It is shown that the system has two unstable saddle-foci and one unstable saddle. A set of typical parameters are chosen for the generation of chaotic attractor. Differing from the common period-doubling bifurcation route in smooth dynamical systems, this memristive system shows abrupt transition from the coexisting period-1 limit cycles to robust chaos when varying system parameters. Various dynamical behaviors are analysed using the numerical simulations and circuit verifications.

Sliding homoclinic bifurcations in a Lorenz-type system: Analytic proofs.

Non-smooth systems can generate dynamics and bifurcations that are drastically different from their smooth counterparts. In this paper, we study such homoclinic bifurcations in a piecewise-smooth analytically tractable Lorenz-type system that was recently introduced by Belykh et al. [Chaos 29, 103108 (2019)]. Through a rigorous analysis, we demonstrate that the emergence of sliding motions leads to novel bifurcation scenarios in which bifurcations of unstable homoclinic orbits of a saddle can yield stable limit cycles. These bifurcations are in sharp contrast with their smooth analogs that can generate only unstable (saddle) dynamics. We construct a Poincaré return map that accounts for the presence of sliding motions, thereby rigorously characterizing sliding homoclinic bifurcations that destroy a chaotic Lorenz-type attractor. In particular, we derive an explicit scaling factor for period-doubling bifurcations associated with sliding multi-loop homoclinic orbits and the formation of a quasi-attractor. Our analytical results lay the foundation for the development of non-classical global bifurcation theory in non-smooth flow systems.

Dynamics of a stage-structured SI model for food adulteration with media-induced response function

In this work, an eco-epidemic predator–prey model with media-induced response function for the interaction of humans with adulterated food is developed and studied. The human population is divided into two main compartments, namely, susceptible and infected. This system has three equilibria; trivial, disease-free and endemic. The trivial equilibrium is forever an unstable saddle position, while the disease-free state is locally asymptotically stable under a threshold of delay parameter tau as well as {mathcal {R}}_0<1. The sufficient conditions for the local stability of the endemic equilibrium point are further explored when min {{mathcal {R}}_0,{mathcal {R}}_0^*}>1 . The conditions for the occurrence of the stability switching are also determined by taking infection delay time as a critical parameter, which concludes that the delay can produce instability and small amplitude oscillations of population masses via Hopf bifurcations. Further, we study the stability and direction of the Hopf bifurcations using the center manifold argument. Furthermore, some numerical simulations are conducted to validate our analytical findings and discuss their biological inferences. Finally, the normalized forward sensitivity index is used to perform the sensitivity analysis of {mathcal {R}}_0 and {mathcal {R}}_0^*.

Dynamical instantons and activated processes in mean-field glass models

We focus on the energy landscape of a simple mean-field model of glasses and analyze activated barrier-crossing by combining the Kac-Rice method for high-dimensional Gaussian landscapes with dynamical field theory. In particular, we consider Langevin dynamics at low temperature in the energy landscape of the pure spherical p-spin model. We select as initial condition for the dynamics one of the many unstable index-1 saddles in the vicinity of a reference local minimum. We show that the associated dynamical mean-field equations admit two solutions: one corresponds to falling back to the original reference minimum, and the other to reaching a new minimum past the barrier. By varying the saddle we scan and characterize the properties of such minima reachable by activated barrier-crossing. Finally, using time-reversal transformations, we construct the two-point function dynamical instanton of the corresponding activated process.

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.

Riddled Attraction Basin and Multistability in Three-Element-Based Memristive Circuit

By coupling a diode bridge-based second-order memristor and an active voltage-controlled memristor with a capacitor, a three-element-based memristive circuit is synthesized and its system model is then built. The boundedness of the three-element-based memristive circuit is theoretically proved by employing the contraction mapping principle. Besides, the stability distributions of equilibrium points are theoretically and numerically expounded in a 2D parameter plane. The results imply the memristive circuit has a zero unstable saddle focus and a pair of nonzero stable node-foci or unstable saddle-foci depending on the considered parameters. The dynamical behaviors include point attractor, period, chaos, coexisting bifurcation mode, period-doubling bifurcation route, and crisis scenarios, which are explored using some common dynamical methods. Of particular concern, riddled attraction basins and multistability are uncovered under two sets of specified model parameters nearing the tiny neighborhood of crisis scenarios by local attraction basins and phase plane plots. The riddled attraction basins with island-like structure demonstrate that their dynamical behaviors are extremely sensitive to the initial conditions, resulting in the coexistence of limit cycles with period-2 and period-6, as well as the coexistence of period-1 limit cycles and single-scroll chaotic attractors. Moreover, a feasible on-breadboard hardware circuit is manually made and the experimental measurements are executed, upon which phase plane trajectories for some discrete model parameters are captured to further confirm the numerically simulated ones.

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.

Remerging Feigenbaum Trees, Coexisting Behaviors and Bursting Oscillations in a Novel 3D Generalized Hopfield Neural Network

This paper, a novel 3D generalized Hopfield neural network is proposed and investigated in order to generate and highlight some unknown behaviors related to such type of neural network. This generalized model is constructed by exploiting the effect of an external stimulus on the dynamics of a simplest 3D autonomous Hopfield neural network reported to date. The stability of the model around its ac-equilibrium point is studied. We note that the model has three types of stability depending on the value of the external stimulus. The stable node-focus, the unstable saddle focus, and the stable node characterize the stability of the equilibrium point of the model. Traditional nonlinear analyses tools are used to highlight and support several complex phenomena such as remerging Feigenbaum trees, the coexistence of up to two, four and six disconnected stable states when the amplitude of the external stimulus is set to zero. Furthermore, for some values of the frequency and amplitudes of the external stimulus, bursting oscillations occur. This latter behavior is characterized by the fact that oscillations switch between quiescent states and spiking states, repetitively. The transformed phase portraits are used to support the bursting mode of oscillations found. Finally, PSpice simulations enable to support the results of the theoretical studies.

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.

Parametrically forced stably stratified cavity flow: complicated nonlinear dynamics near the onset of instability

The dynamics of a fluid-filled square cavity with stable thermal stratification subjected to harmonic vertical oscillations is investigated numerically. The nonlinear responses to this parametric excitation are studied over a comprehensive range of forcing frequencies up to two and a half times the buoyancy frequency. The nonlinear results are in general agreement with the Floquet analysis, indicating the presence of nested resonance tongues corresponding to the intrinsic $m:n$ eigenmodes of the stratified cavity. For the lowest-order subharmonic $1:1$ tongue, the responses are analysed in great detail, with complex dynamics identified near onset, most of which involves interactions with unstable saddle states of a homoclinic or heteroclinic nature.

No-Boundary Proposal as a Path Integral with Robin Boundary Conditions.

Realizing the no-boundary proposal of Hartle and Hawking as a consistent gravitational path integral has been a long-standing puzzle. In particular, it was demonstrated by Feldbrugge, Lehners, and Turok that the sum over all universes starting from a zero size results in an unstable saddle point geometry. Here we show that, in the context of gravity with a positive cosmological constant, path integrals with a specific family of Robin boundary conditions overcome this problem. These path integrals are manifestly convergent and are approximated by stable Hartle-Hawking saddle point geometries. The price to pay is that the off-shell geometries do not start at a zero size. The Robin boundary conditions may be interpreted as an initial state with Euclidean momentum, with the quantum uncertainty shared between the initial size and momentum.

Memristor initial boosting behaviors in a two-memristor-based hyperchaotic system

Abstract By substituting two resistive couplings with two memristive couplings, a two-memristor-based hyperchaotic system is proposed. This hyperchaotic system has a plane equilibrium with two zero eigenvalues and three nonzero ones, and the equilibrium plane is divided into three regions with different stabilities, including unstable saddle, stable node, and stable node-focus. Through using the local attraction basin, bifurcation diagrams and Lyapunov exponent spectra, dynamical behaviors are analyzed in the unstable saddle region, from which the bi-stability phenomenon is observed in such a hyperchaotic system. More particularly, the memristor initial boosting behaviors are found, which indicates that the attractor offset boosting is controlled by the memristor initial values. The memristor initial boosting is multi-dimension, nonlinearity and non-monotonic, completely different from the variable offset boosting. Subsequently, PSIM circuit simulations are performed to verify the numerical results.