Unstable Equilibrium Point Research Articles

Dynamical stability in a delayed neural network with reaction–diffusion and coupling

In this paper, a delayed neural network with reaction–diffusion and coupling is considered. The network consists of two sub-networks each with two neurons. In the first instance, some parameter regions are identified by employing partial functional differential equation theory. Moreover, sufficient conditions of stationary bifurcation and Bogdanov–Takens bifurcation are also derived. Further, analytical results and illustrations are proved for the case where the unstable trivial equilibrium point becomes stable in the presence of reaction–diffusion terms with appropriate values. We emphasize that the non-trivial role of diffusions is enlarging the stability region in the system described by PDE, comparing with the corresponding system described by DDE. Finally, numerical simulations are carried out to verify the efficiency of the theoretical analysis and provide comparisons with some existing literature.

THEORIES OF COLLECTIVE BEHAVIOR AND RESOURCE MOBILIZATION: ELABORATION ON THE CONCEPT OF POLITICAL PROTEST

Organized rallies are gaining more influence in socio-political processes in Russia and Europe. Modern protest is taking on new forms and is using new technical capabilities to mobilize participants. Determining potential capacity of a rally, its “unstable equilibrium points” and gauging the public’s possible negative reactions to the political elite’s decision making is becoming an important goal in contemporary political management. The article examines the heuristic potential of two approaches to studying political protest: theory of collective action and theory of mobilization of political opportunities. The first approach stems from the idea of dominance of symbolic values in modern political consciousness, while the second looks in more detail at the “balance of power” and resources that lend to the effectiveness of a protest. Does the environment created by the “new methods” have an impact on the efficacy of a protest? How important are institutional defects in order for citizens to want to organize a rally? Studying these aspects can help gain insight into the public’s real protest potential at regional and local levels in Russia.

Bifurcation Gaps in Asymmetric and High-Dimensional Hypercycles

Hypercycles are catalytic systems with cyclic architecture. These systems have been suggested to play a key role in the maintenance and increase of information in prebiotic replicators. It is known that for a large enough number of hypercycle species ([Formula: see text]) the coexistence of all hypercycle members is governed by a stable periodic orbit. Previous research has characterized saddle-node (s-n) bifurcations involving abrupt transitions from stable hypercycles to extinction of all hypercycle members, or, alternatively, involving the outcompetition of the hypercycle by so-called mutant sequences or parasites. Recently, the presence of a bifurcation gap between a s-n bifurcation of periodic orbits and a s-n of fixed points has been described for symmetric five-member hypercycles. This gap was found between the value of the replication quality factor [Formula: see text] from which the periodic orbit vanishes ([Formula: see text]) and the value where two unstable (nonzero) equilibrium points collide ([Formula: see text]). Here, we explore the persistence of this gap considering asymmetries in replication rates in five-member hypercycles as well as considering symmetric, larger hypercycles. Our results indicate that both the asymmetry in Malthusian replication constants and the increase in hypercycle members enlarge the size of this gap. The implications of this phenomenon are discussed in the context of delayed transitions associated to the so-called saddle remnants.

Complicated dynamics and control of a hyperchaotic complex nonlinear autonomous Lü model with complex parameters

In this work, we consider the dynamics and complicated properties of an independent Lu model with complex nonlinear conditions and parameters. We study the influence of complex parameters on the dynamics and behaviors of nonlinear hyperchaotic models. The complex parameters such as generalized Hamiltonian, symmetry, dispersal, equilibria and their stability, Lyapunov exponents, Lyapunov dimension, bifurcation graphs, and hyperchaotic satisfaction are considered. Furthermore, we analyze the stability of the trivial points and limit the conditions under which the complex nonlinear conditions with complex parameters have negative, zero, or positive Lyapunov exponents; we also focus on the chaos, hyperchaos, periodic, and quasiperiodic attractors for an extensive range of parameter values. Moreover, we verify the control of hyperchaotic arrangements of the autonomous Lu model with complex nonlinear conditions and complex parameters. We propose a method to transform the model from its hyperchaotic state to an unstable equilibrium point using a Lyapunov stability hypothesis. Finally, with the use of complex periodic driving, we demonstrate that the model can be transformed from hyperchaotic to quasiperiodic motions, thus resulting in a correct periodic arrangement in which the amplitude and frequency depend on the parameters of the model. Because the obtained arrangement is steady for an extensive range of parameter values, it may be used to control the model by entraining it with the connected periodic compelling term.

Predictive Feedback Control Method for Stabilization of Continuous Time Systems

Recent researchers have studied the chaotic phenomenon and introduced methods to control chaos. One of these methods is the Predictive Feedback Control (PFC). PFC method has been used to stabilize only the discrete chaotic maps. In this paper, we generalize the PFC method by extending it to stabilize continuous time systems. The numerical simulations are carried out to solve the system using Euler method for its simplicity, and then the PFC method is applied to the discretized system. The controlled trajectory converges to an unstable equilibrium point via small control action. The stability analysis is shown compared to that of the continuous system in details. Also the choice of the controlling input of the PFC method and its properties are discussed. Lorenz system and R\ossler system are the well known chosen chaotic examples. The PFC method is applied to each of them, showing stability at each of their equilibrium point. With a very small control, the behaviour of the system is completely changed from chaos to stable.

Coexistence between silent and bursting states in a biophysical Hodgkin-Huxley-type of model.

Classification of the dynamical mechanisms that support bistability between bursting oscillations and silence has not yet been clarified in detail. The purpose of this paper is to demonstrate that the coexistence of a stable equilibrium point with a state of continuous bursting can occur in a slightly modified, biophysical model that describe the dynamics of pancreatic beta-cells. To realize this form of coexistence, we have introduced an additional voltage-dependent potassium current that is activated in the region around the original, unstable equilibrium point. It is interesting to note that this modification also leads the model to display a blue-sky catastrophe in the transition region between chaotic and bursting states.

New approach of controlling cardiac alternans

The alternans of the cardiac action potential duration is a pathological rhythm. It is considered to be relating to the onset of ventricular fibrillation and sudden cardiac death. It is well known that, the predictive control is among the control methods that use the chaos to stabilize the unstable fixed point. Firstly, we show that alternans (or period-2 orbit) can be suppressed temporally by the predictive control of the periodic state of the system. Secondly, we determine an estimation of the size of a restricted attraction's basin of the unstable equilibrium point representing the unstable regular rhythm stabilized by the control. This result allows the application of predictive control after one beat of alternans. In particular, using predictive control of periodic dynamics, we can delay the onset of bifurcations and direct a trajectory to a desired target stationary state. Examples of the numerical results showing the stabilization of the unstable normal rhythm are given.

Real-Time Swing-up and Stabilization Control of a Cart-Pendulum System with Constrained Cart Movement

Abstract The cart-pendulum system is a typical benchmark problem in the control field. It is a fully underactuated system having one control input for two degrees-of-freedom (DOFs) system. It has highly nonlinear structure, which can be used to validate different nonlinear and linear controllers and has wide range of real-time (realistic) applications like rockets propeller, tank missile launcher, self-balancing robot, stabilization of ships, design of earthquake resistant buildings, etc. In this work, modeling, simulation and real-time control of a cart-pendulum system is performed. The mathematical model of the system is developed using Euler-Lagrange approach. In order to achieve a more realistic model, the actuator dynamics is considered in the mathematical model. The main aim of this work is to investigate the performance of two different control strategies- first to swing-up the pendulum to near unstable equilibrium region and second to stabilize the pendulum at unstable equilibrium point. The swing-up problem is addressed by using energy controller in which cart is accelerated by providing a force to the cart with a AC servo motor with the help of timing pulley arrangement. The initial velocity of the cart is taken into account to confirm swing-up in the restricted track length. The cart displacement in the restricted track length is verified by simulation and experimental test-run. The regulation problem of stabilization of pendulum is addressed by developing the controller using Pole Placement Controller (PPC) and LQR Controller (LQRC). Both the control strategies are performed analytically and experimentally using the Googoltech Linear Inverted Pendulum (GLIP) setup. The analytical results, simulated in MATLAB and SIMULINK environment, are found in close agreement with the experimental results. In order to demonstrate the effect of both the stabilizing controllers on the performance of the system, comparison of the experimental results is reported in this work. It is demonstrated experimentally that LQR controller outperforms the Pole Placement controller, in terms of reduction in the oscillations of the inverted pendulum (56 %), as well as the magnitude of maximum control input (66.7 %). Further, robustness of the closed-loop system is investigated by providing external disturbances.

Nonlinear vibration of permanent magnet synchronous motors in electric vehicles influenced by static angle eccentricity

An additional transverse electromagnetic moment caused by the inclination of the rotor of the permanent magnet synchronous motors in electric vehicles is modeled based on Maxwell stress tensor method. The model of this moment is suitable for the case of pole-pair number larger than 3 and is independent of time; its nonlinearity results in multiple equilibrium points of the conservative generalized force: an isolated equilibrium point and a continuum of equilibrium points. The continuum is symmetric with respect to the isolated equilibrium point; a pitchfork bifurcation occurs as the system stiffness of the rotor in the non-inclined state passes through zero. When static angle eccentricity is introduced, the symmetry is lost and the continuum degrades into two unstable equilibrium points. This parameter leads to a generic bifurcation with defect. However, considering it as a bifurcation parameter, a pair of saddle-node bifurcations appears. Eigenvalue-based stability analysis and center manifold theorem are employed to determine the stabilities of the multiple equilibrium points. The frequency characteristics of the forced response caused by the static angle eccentricity are developed by harmonic balance method, and the stability of the steady-state solution is studied with Routh–Hurwitz criterion in a rotating system. The frequency response curve generally passes through all the equilibrium points. For a disk-shaped rotor, the frequency characteristics exhibit hardening and the solution has the same stability as the equilibrium point in the same branch. For a cylindrical rotor, the characteristics show softening and only the solution running near the equilibrium point has the same stability as the equilibrium point in the same branch. Furthermore, there is a frequency band in which the forced response is globally unstable. However, for comparatively large damping, large inertia ratio, small stiffness ratio and small static angle eccentricity, the frequency characteristics of a cylindrical rotor are similar to those of a disk-shaped rotor. The globally unstable frequency band diminishes and then vanishes. All of the solutions have the same stability in a branch.

Stable periodic orbits for spacecraft around minor celestial bodies

We are interested in stable periodic orbits for spacecrafts in the gravitational field of minor celestial bodies. The stable periodic orbits around minor celestial bodies are useful not only for the mission design of the deep space exploration, but also for studying the long-time stability of small satellites in the large-size-ratio binary asteroids. The irregular shapes and gravitational fields of the minor celestial bodies are modeled by the polyhedral model. Using the topological classifications of periodic orbits and the grid search method, the stable periodic orbits can be calculated and the topological cases can be determined. Furthermore, we find five different types of stable periodic orbits around minor celestial bodies: A) Stable periodic orbits generated from the stable equilibrium points outside the minor celestial body, B) Stable periodic orbits continued from the unstable periodic orbits around the unstable equilibrium points, C) Retrograde and nearly circular periodic orbits with zero-inclination around minor celestial bodies, D) Resonance periodic orbits, E) Near-surface inclined periodic orbits. We take asteroid 243 Ida, 433 Eros, 6489 Golevka, 101955 Bennu, and the comet 1P/Halley for examples.

Design of a robust fault-tolerant control algorithm based on failure mode effects criticality analysis for a three-axis satellite

This paper proposes a robust fault-tolerant control algorithm for a three-axis satellite. In this regard, an adaptive sliding attitude control algorithm is suggested, which has the capability of fault estimation in the satellite actuators and correction of their effects. For this, the disturbances due to environmental effects and actuator failures and also the satellite unknown parameters are estimated by the adaptive updating law; the sliding mode algorithm compensates the errors due to estimation process. In the suggested design process, the sliding surface is selected so that the unwinding and singularity problems are solved, and also a compensator part is included to remove unstable equilibrium points. In this paper, the failure mode effects criticality analysis have been done to classify different failure modes of reaction wheel according to their severity and probability of occurrence. Accordingly, the critical failure modes and their effects at the control system level are derived. It is shown that the derived critical failures lead to small or severe variations in the generated torques of reaction wheels for which a supervision level will be proposed to correct their effects. Finally, different simulations are conducted to validate expected performance of the suggested algorithms.

Matrix measure based exponential stabilization for complex-valued inertial neural networks with time-varying delays using impulsive control

In this paper, the problem on the exponential stabilization of complex-valued inertial neural networks with time-varying delays via impulsive control is studied. By virtue of an appropriate variable transformation, the original inertial neural network is transformed into the first order complex-valued differential system. Based on matrix measure and applying impulsive differential inequality, some easily verifiable algebraic criteria on delay-dependent conditions are derived to ensure the global exponential stabilization for the addressed neural networks using impulsive control. Moreover, the different unstable equilibrium point can also be exponentially stabilized by using the different impulsive controllers and the exponential convergence rate index is also estimated. Finally, two numerical examples with simulations are presented to show the effectiveness of the obtained results.

Stabilizing constrained chaotic system using a symplectic psuedospectral method

The problem of controlling chaotic systems has drawn much attention in the last two decades. However, the controlled system may be subjected to complicated constraints and few researches on controlling chaos take constraints into consideration. Therefore, the stabilization of constrained chaotic system is solved under the frame of nonlinear optimal control in this paper. A symplectic pseudospectral method based on qusilinearizaiton techniques and the parametric variational principle is developed to solve constrained nonlinear optimal control problems with arbitrary Lagrange-type cost functional. At the beginning of the proposed method, the original nonlinear optimal control problem is converted into a series of linear-quadratic constrained optimal control problems. Then each of the converted linear quadratic problems is transformed into a standard linear complementarity problem. The proposed method is successfully applied to stabilizing constrained chaotic systems around an unstable equilibrium point or an unstable periodic orbit. Numerical simulations demonstrate that the developed method is effective and efficient, and constraints are strictly satisfied.

The phase response and state space of slow wave contractions in the small intestine.

What is the central question of this study? What are the dynamical rules governing interstitial cell of Cajal (ICC)-generated slow wave contractions in the small intestine, as reflected in their phase response curve and state space? What is the main finding and its importance? The phase response curve has a region of phase advance surrounding a phase delay peak. This pattern is important in generating a stable synchrony within the ICC network and is related to the state space of the ICC; in particular, the phase delay peak corresponds to the unstable equilibrium point that threads the ICC's limit cycle. Interstitial cells of Cajal (ICCs) generate electrical oscillations in the gut. Synchronization of the ICC population is required for generation of coherent electrical waves ('slow waves') that cause muscular contraction and thereby move gut content. The phase response curve (PRC) is an experimental measure of the dynamical rules governing a population of oscillators that determine their synchrony and gives an experimental window onto the state space of the oscillator, its dynamical landscape. We measured the PRC of slow wave contractions in the mouse small intestine by the novel combination of diameter mapping and single pulse electrical field stimulation. Phase change (τ) was measured as a function of old phase (ϕ) and distance from the stimulation electrode (d). Plots of τ(ϕ, d) showed an arrowhead-shaped region of phase advance enclosing at its base a phase delay peak. The phase change mirrored the perturbed pattern of contraction waves in response to a pulse. The (ϕ, d) plane is the surface of a displacement tube extending from the limit cycle through state space. To visualize the state space vector field on this tube, latent phase (ϕlat ) was calculated from τ. At the transition from advance to delay, isochrons made boomerang turns before tightening and winding around the phase delay peak corresponding to the unstable equilibrium point that threads the limit cycle. This isochron foliation had previously been observed in oscillator models such as the Fitzhugh-Nagumo but has not been demonstrated experimentally. The spatial extension of the PRC afforded by diameter mapping allows a better understanding of the dynamical properties of ICCs and how they synchronize as a population.

A Visualization Tool for Real-Time Dynamic Contingency Screening and Remedial Actions

This paper proposes a real-time visual interactive transient stability screening and remedial action tool that uses an approach based on Lyapunov functions to enable the selection of appropriate remedial actions that stabilize power systems due to large disturbances and cascading failures. At present, there is no effective tool that enables making a well-informed choice from amongst a profusion of remedial action alternatives. Conventionally, transient stability analysis of power system is performed offline to assess the capability of the power system to withstand specific disturbances and to investigate the dynamic response of the power system as the network is restored to normal operation. In this paper, postfault stable and controlling unstable equilibrium points are determined using a homotopy-based approach. Subsequently, stability assessment of the system and the corresponding potential remedial actions are determined from the equilibrium points of the system dynamical model. The real-time transient stability analysis and remedial action algorithm are incorporated with the visualization tool to facilitate interactive decision making in real time. The transient stability analysis and remedial action algorithm, and the visualization tool are demonstrated on several test systems including the equivalent Western Electricity Coordinating Council system and the simplified New England 39 bus test system.

Variable Structure Tracking Control-Observer for a Perturbed Inertia Wheel Pendulum via Position Measurements

In this paper we address the robust tracking control problem, against unknown but bounded matched disturbances, for an inertia wheel pendulum. The interested periodic motion of the pendulum will be at the upright position which corresponds to an unstable equilibrium point of the unforced system. A two-relay controller based reference model was developed for generating the reference trajectories where a second order sliding modes controller enforces the unactuated link to follow such prescribed trajectory. The desired amplitude and frequency were tuned by choosing the two-relay control gains properly. A variable structure observer was developed to estimate the velocity of the pendulum and the wheel. Lyapunov stability analysis was made to demonstrate robustness of the closed-loop system. Performance issues of the constructed controller-observer were illustrated in a numerical study.

Multiple equilibrium points in global attractors for some p-Laplacian equations

In this paper, we continue to study the properties of the global attractor for some p-Laplacian equations with a Lyapunov function F in a Banach space when the origin is no longer a local minimum point but a saddle point of F. By using the abstract result established in our previous work, we prove the existence of multiple equilibrium points in the global attractor for some p-Laplacian equations under some suitable assumptions in the case that the origin is an unstable equilibrium point.

Anti-windup Fractional Order $$\textit{PI}^\lambda -\textit{PD}^\mu $$ PI λ - PD μ Controller Design for Unstable Process: A Magnetic Levitation Study Case Under Actuator Saturation

Many studies have been conducted by researchers globally in the recent past on the control scheme to stabilize the levitated object of magnetic levitation system (MLS) at the unstable equilibrium point though very small amount of work has been reported regarding the design and tuning of fractional order controller for unstable process, especially in the presence of actuator saturation. In this paper, a novel methodology of fractional order \(PI^\lambda -PD^\mu \) controller for MLS is proposed for both stabilizing the system and removing the actuator saturation. Fractional order control facilitates an extra degree of freedom to change the order of integrator, arise a new concept to diminish the error windup during actuator saturation. The tuning of gain parameter of PID controller is done using the conventional pole placement technique, while a simple tuning method is proposed here to tune fractional parameter using stability margin and sensitivity function. Stability of the proposed controller scheme employed to MLS is analyzed on the basis of Riemann sheets concept in w-plane. To compensate for saturation due to integral term, back-calculation anti-windup scheme is used here. Experimental demonstration of proposed scheme on magnetic levitation confirms the ability of the controller to reject the external random perturbations as well as its robustness against the variation of system parameters.

Invariant manifolds around artificial equilibrium points for low-thrust propulsion spacecraft

Low-thrust propulsion is incorporated into circular restricted three-body problem to balance the gravitational and centrifugal forces, and then artificial equilibrium points can be generated. The linear dynamics indicates that there are stable and unstable artificial equilibrium points. Around the unstable artificial equilibrium points, there are center and hyperbolic invariant manifolds. In this work, invariant manifolds around artificial equilibrium points are expressed as formal series of amplitudes corresponding to hyperbolic and center dynamics, and high-order series solutions are constructed up to an arbitrary order. By taking advantage of the series expansions constructed, the motions around unstable artificial equilibrium points can be parameterized. In order to check the validity, the practical convergence of series solutions truncated at different orders is considered. Finally, series expansions of invariant manifolds are applied to designing transfer trajectories from the primary to periodic orbits around artificial equilibrium points which are located inside $L_{1}$ and beyond $L_{2}$ points.

A high-precision real-time approach to calculate closest unstable equilibrium points

In this paper, an innovative real-time approach is presented in order to compute the closest unstable equilibrium points of system. The proposed approach estimates unstable equilibrium points of system at the first time step of fault duration and there is no need to post the fault data. To such aim, a new concept of equal area criteria is proposed in this paper which estimates the initial value of critical points of the system. This value is used in order to calculate corrected kinetic energy and as a result the closest unstable equilibrium point by taking into account the fault trajectory. Moreover, the details of power system are considered to calculate unstable equilibrium point by utilizing network preserving model. Finally, several case studies have been conducted on IEEE 9 bus and the New-England 39 bus test systems to illustrate the benefits of the proposed approach. It is worth noting that considering structure preserving in modeling and at the same time simplicity in implementation and low computational burden are the main salient features of the proposed approach. As a result, the proposed method is suitable for real-time applications.