Isolated Equilibrium Point Research Articles

Generalized model for steady-state bifurcations without parameters in memristor-based oscillators with lines of equilibria

We demonstrate how the pitchfork, transcritical and saddle-node bifurcations of steady states observed in dynamical systems with a finite number of isolated equilibrium points occur in systems with lines of equilibria. The exploration is carried out by using the numerical simulation and linear stability analysis applied to a model of a memristor-based circuit. All the discussed bifurcation scenarios are considered in the context of models with the piecewise-smooth memristor current-voltage characteristic (Chua’s memristor), as well as on examples of oscillators with the memristor nonlinearity that is smooth everywhere. Finally, we compare the dynamics of ideal-memristor-based oscillators with the behavior of models taking into account the memristor forgetting effect. The presented results are obtained for electronic circuit models, but the studied bifurcation phenomena can be exhibited by systems with lines of equilibria of any nature.

Orbital Hypernormal Forms

In this paper, we analyze the problem of determining orbital hypernormal forms—that is, the simplest analytical expression that can be obtained for a given autonomous system around an isolated equilibrium point through time-reparametrizations and transformations in the state variables. We show that the computation of orbital hypernormal forms can be carried out degree by degree using quasi-homogeneous expansions of the vector field of the system by means of reduced time-reparametrizations and near-identity transformations, achieving an important reduction in the computational effort. Moreover, although the orbital hypernormal form procedure is essentially nonlinear in nature, our results show that orbital hypernormal forms are characterized by means of linear operators. Some applications are considered: the case of planar vector fields, with emphasis on a case of the Takens–Bogdanov singularity.

Subcritical Andronov-Hopf scenario for systems with a line of equilibria.

Using numerical simulation methods and analytical approaches, we demonstrate hard self-oscillation excitation in systems with infinitely many equilibrium points forming a line of equilibria in the phase space. The studied bifurcation phenomena are equivalent to the excitation scenario via the subcritical Andronov-Hopf bifurcation observed in classical self-oscillators with isolated equilibrium points. The hysteresis and bistability accompanying the discussed processes are shown and explained. The research is carried out on an example of a nonlinear memristor-based self-oscillator model. First, a simpler model including Chua's memristor with a piecewise-smooth characteristic is explored. Then, the memristor characteristic is changed to a function being smooth everywhere. Finally, the action of the memristor forgetting effect is taken into consideration.

Zero-Hopf Bifurcation in the Generalized Stretch-Twist-Fold Flow

A zero-Hopf equilibrium in a three-dimensional system is an isolated equilibrium point which has a zero eigenvalue and a simple pair of purely imaginary eigenvalues. In general, for such an equilibrium, there is no theory for finding when some periodic solutions are bifurcated by perturbing the parameters of the system. In this work, we describe the values of the parameters for which a zero-Hopf equilibrium occurs at the equilibrium points in the generalized stretch-twist-fold flow. Thus, only one condition for parameters of the generalized stretch-twist-fold flow introduced in a system (Eq. 1) is found for which the equilibrium point is a zero-Hopf equilibrium. For this condition, we use the averaging method to provide the existence of a periodic solution, which bifurcates from the zero-Hopf equilibrium point. The main result in this paper is Theorem 1, which gives a periodic solution of the generalized stretch-twist-fold flow.

Structural Analysis and Control of a Model of Two-Site Electricity and Heat Supply

This paper introduces a control problem of regulation of energy flows in a two-site electricity and heat supply system, where two combined heat and power (CHP) plants are interconnected via electricity and heat flows. The control problem is motivated by recent development of fast operation of CHP plants to provide ancillary services of power system on the order of tens of seconds to minutes. Due to the physical constraint that the responses of the heat subsystem are not necessary as fast as those of the electric subsystem, the target controlled state is not represented by any isolated equilibrium point, implying that stability of the system is lost in the long-term sense on the order of hours. In this paper, we first prove in the context of nonlinear control theory that the state-space model of the two-site system is nonminimum phase due to nonexistence of isolated equilibrium points of the associated zero dynamics. Instead, we locate a one-dimensional (1D) invariant manifold that represents the target controlled flows completely. Then, by utilizing a virtual output under which the state-space model becomes minimum phase, we synthesize a controller that achieves not only the regulation of energy flows in the short-term regime but also stabilization of an equilibrium point in the long-term regime. Effectiveness of the synthesized controller is established with numerical simulations with a practical set of model parameters.

Robust control of a class of under-actuated mechanical systems with model uncertainty

ABSTRACTRobust control of under-actuated mechanical systems (UMSs) with model uncertainty is still a challenging problem. For UMSs, the model parametric uncertainties make it difficult to precisely calculate the isolated equilibrium point corresponding to a fixed input. Without an accurate destination state, many set-point control methods cannot eliminate the positioning errors. An improved sliding mode control (ISMC) method is proposed to solve the robust control problem for a class of UMSs with model uncertainty and input disturbance. A balance variable is introduced in the sliding surface design to compensate for the disturbance caused by the inaccurate destination state, and the ISMC method is proposed to make the system state reach the sliding surface in finite time. Linear matrix inequality approach and particle swarm optimisation algorithm are applied to design the sliding mode surface parameters. The simulation results on an UMS are presented to show the effectiveness of the proposed scheme.

Real-time classifier based on adaptive competitive self-organizing algorithm

This article introduces a novel adaptive competitive self-organizing (ACS) model, with applicability for real-time clustering and vector quantization. An important feature of this model is its dynamic structure and self-adjusting parameters that also offers a solution to the problem of parasitic limit points and consequently in more accurate label assignments. This unsupervised classifier is free of any external control mechanism. Its self-organizing (SO) dynamic is governed by the gradient descent (GD) theory in cooperation with a competition mechanism based on Lotka–Volterra competitive exclusion. The core algorithm of this classifier is based on developing an energy function, where its minima or equilibrium points correspond to the centroid of similar input patterns. Since this energy function is a form of Lyapunov function, it guarantees stabilization of the dynamical trajectories of labels in finite numbers of isolated equilibrium points. This energy function along with other control parameter functions, then, will be the base for the set of ordinary differential equations (ODEs) describing the overall dynamic of our system. Finally, the effectiveness of the proposed ACS model is demonstrated by implementing it on both real and artificial data sets as well as comparing with other well-known clustering methods. ACS method showed a better clustering performance in some categories and an overall comparable rendition.

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.

Zero-Hopf bifurcation in a Chua system

A zero-Hopf equilibrium is an isolated equilibrium point whose eigenvalues are ±ωi≠0 and 0. In general for a such equilibrium there is no theory for knowing when it bifurcates some small-amplitude limit cycles moving the parameters of the system. Here we study the zero-Hopf bifurcation using the averaging theory. We apply this theory to a Chua system depending on 6 parameters, but the way followed for studying the zero-Hopf bifurcation can be applied to any other differential system in dimension 3 or higher.In this paper first we show that there are three 4-parameter families of Chua systems exhibiting a zero-Hopf equilibrium. After, by using the averaging theory, we provide sufficient conditions for the bifurcation of limit cycles from these families of zero-Hopf equilibria. From one family we can prove that 1 limit cycle bifurcates, and from the other two families we can prove that 1, 2 or 3 limit cycles bifurcate simultaneously.

On Formal Stability of Stratified Shear Flows

A novel linear stability criterion is established for the equilibria of general three-dimensional (3D) rotating flows of an ideal gas satisfying Boyle-Charles\rq\ law by a newly refined energy-Casimir convexity (ECC) method that can exploit a larger class of Casimir invariants. As a conventional ECC method cannot be applied directly to stratified shear flows, in our new approach, rathern than checking the local convexity of a Lyapunov functional L\equiv E+C_{E} defined as a sum of the total energy and a ceratin Casimir, we seek the condition for non-existence of unstable manifolds: orbits (physically realisable flow in phase space) on the leaves of invariants including L as well as other Casimirs connecting a given equilibrium point O and other points in the neighbourhood of it. We argue that the separatrices of the second variation of L ( \delta^{2}L=0 ) generally consist of such unstable manifolds as well as pseudo unstable ones for which either the total energy or Casimirs actually serves as a barrier for escaping orbits. The significance of new method lies in the fact that it eliminates the latter so as to derive a condition that O being an isolated equilibrium point in terms of orbital connections.

Constructing Chaotic Systems with Total Amplitude Control

A general method is introduced for controlling the amplitude of the variables in chaotic systems by modifying the degree of one or more of the terms in the governing equations. The method is applied to the Sprott B system as an example to show its flexibility and generality. The method may introduce infinite lines of equilibrium points, which influence the dynamics in the neighborhood of the equilibria and reorganize the basins of attraction, altering the multistability. However, the isolated equilibrium points of the original system and their stability are retained with their basic properties. Electrical circuit implementation shows the convenience of amplitude control, and the resulting oscillations agree well with results from simulation.

Maxwell’s conjecture on three point charges with equal magnitudes

Maxwell’s conjecture on three point charges states that the number of non-degenerate equilibrium points of the electrostatic field generated by them in R3 is at most four. We prove the conjecture in the cases when three point charges have equal magnitudes and show the number of isolated equilibrium points can only be zero, two, three, or four.Specifically, fixing positions of two positive charges in R3, we know exactly where to place the third positive charge to have two, three, or four equilibrium points. All equilibrium points are isolated and there are no other possibilities for the number of isolated equilibrium points.On the other hand, if both two of the fixed charges have negative charge values, there are always two equilibrium points except when the third positive charge lies in the line segment connecting the two negative charges. The exception cases are when the field contains only a curve of equilibrium points.In this paper, computations assisted by computer involve symbolic and exact integer computations. Therefore, all the results are proved rigorously.

Parameter identification of unknown biased sinusoid

An algorithm composed of a normalised frequency estimator and a first-order low-pass filter is introduced to identify online the unknown values of bias, amplitude, frequency and phase angles of a sinusoid signal. Slow integral manifold technique and averaging method are used to educe a non-linear autonomous system for frequency estimation. The global exponential stability of isolated equilibrium point and the ultimate boundedness under unknown bounded disturbance are investigated respectively. The identification and the estimation errors of bias, amplitude and phase angle under bounded disturbance are also formulated and exhibited by simulation.

An upper bound of the index of an equilibrium point in the plane

We give an upper bound of the index of an isolated equilibrium point of a C1 vector field in the plane. The vector field is decomposed in gradient and Hamiltonian components. This decomposition is related with the Loewner vector field. Associated to this decomposition we consider the set Π where the gradient and Hamiltonian components are linearly dependent. The number of branches of Π starting at the equilibrium point determines the upper bound of the index.

Exponential Stability, Semistability, and Boundedness of a Multi-ANF System

A nonlinear system being composed of multiadaptive notch filters in parallel is introduced to track the sinusoidal components of an almost periodic signal and estimate their frequencies and amplitudes. An almost periodic nonlinear dynamic system for estimation of frequencies is achieved after the existence of a slow integral manifold is proved and results in a nonlinear autonomous system with averaging method employed. Three types of local stability of the autonomous system, the exponential stability in isolated equilibrium point and the semistability on center manifold and the ultimate boundedness under unknown periodic disturbance, are respectively investigated. The transient performance is affected by frequency adaptive gain independent of bandwidth parameter. The validity of the proposed algorithm is verified by simulation results.

Stability Analysis of Normalized Frequency Adaptive Comb Filter

A normalized frequency adaptive comb filter composed of a number of normalized frequency estimators in parallel is proposed to track unknown frequency and unknown amplitude of each component of an almost periodic signal. The filter consists of two coupled nonlinear differential equations respectively for updating frequencies and estimating state variables. A nonlinear autonomous equation for frequency estimation is deduced after average method is applied to the almost periodic dynamic system resulting from system decoupling by slow integral manifold. Three kinds of local stability of the autonomous system:the exponential stability of isolated equilibrium point,the semistability on center manifold,and the robustness under unknown periodic disturbance are proved. The convergence and the boundedness of amplitude estimation and signal tracking as well as the effect of parameters on the transient and steady-state performance of frequency and amplitude are investigated. Each sinusoidal component and its amplitude are accurately tracked on the condition of given intervals instead of the values of frequencies. The response speed is independent of the component amplitude. Simulation results reveal the validity of the proposed algorithm.

On stability and convergence of the population-dynamics in differential evolution

Theoretical analysis of the dynamics of evolutionary algorithms is believed to be very important to understand the search behavior of evolutionary algorithms and to develop more efficient algorithms. In this paper we investigate the dynamics of a canonical Differential Evolution (DE) algorithm with DE/rand/1 type mutation and binomial crossover. Differential Evolution (DE) is well known as a simple and efficient algorithm for global optimization over continuous spaces. Since its inception in 1995, DE has been finding many important applications in real-world optimization problems from diverse domains of science and engineering. The paper proposes a simple mathematical model of the underlying evolutionary dynamics of a one-dimensional DE-population. The model shows that the fundamental dynamics of each search-agent (parameter vector) in DE employs the gradient-descent type search strategy (although it uses no analytical expression for the gradient itself), with a learning rate parameter that depends on control parameters like scale factor F and crossover rate CR of DE. The stability and convergence-behavior of the proposed dynamics is analyzed in the light of Lyapunov's stability theorems very near to the isolated equilibrium points during the final stages of the search. Empirical studies over simple objective functions are conducted in order to validate the theoretical analysis.

Sliding mode control of underactuated multibody systems and its application to shape change control

In this article, we introduce an approach based on sliding mode control to design full state feedback controllers for stabilisation of underactuated non-linear multibody systems. We define first order sliding surfaces as a linear combination of actuated and unactuated coordinate tracking errors. Lyapunov stability analysis guarantees that all system trajectories reach and remain on the sliding surfaces. However, stability of the sliding surfaces depends on the equilibrium manifold. If the system has isolated equilibrium points, it is linearly controllable and asymptotic stability can be guaranteed under certain conditions. Otherwise, the control system fails Brockett's necessary condition for existence of a smooth stabilising feedback. In the latter case, if the total momentum is conserved, the closed-loop control system will be marginally stable. Consequently, a procedure is proposed to achieve an asymptotically stable discontinuous control law through sliding surface redefinition and shape changes. It is proposed that repetitive application of shape changes will lead to asymptotic convergence of the system to the desired configuration. Simulation results are presented for an inverted pendulum as an example of a system with isolated equilibrium points and an existing communication satellite as an example of shape change control. In both cases, the control is shown to be effective and robust with respect to uncertainties and disturbances.

Robust stability analysis of static neural network with S-type distributed delays

In this paper, we investigate local robust stability of static neural network (SNN) with S-type distributed delays. We derive some new sufficient conditions for local robust stability of equilibrium points and estimate attracting domains of equilibrium points except isolated equilibrium points. Our results not only show local robust stability of equilibrium points but also allow much broader application for static neural network with or without delays. It is shown that our results are new and improve corresponding results existing in the previous literature.

Multistability of neural networks with discontinuous activation function

In this paper, the multistability is studied for two-dimensional neural networks with multilevel activation functions. And it is showed that the system has n2 isolated equilibrium points which are locally exponentially stable, where the activation function has n segments. Furthermore, evoked by periodic external input, n2 periodic orbits which are locally exponentially attractive, can be found. And these results are extended to k-neuron networks, which is really enlarge the capacity of the associative memories. Examples and simulation results are used to illustrate the theory.