Unstable Saddle Point Research Articles

Stability and control of tethered satellite with chemical propulsion in orbital plane

The tethered satellite with chemical propulsion has broad application prospects in the disposal of abandoned satellites, the orbital rescue of spacecrafts, and the transportation of space supplies, which is completely different from the traditional applications of tethered satellites. Therefore, new research on its dynamics, stability, and control becomes useful and interesting. In this article, based on a dumbbell model of tethered satellite, the dynamics equations of tethered system in orbital maneuvering are established. Furthermore, according to the definitions of transversal and radial propulsive coefficients, analytical solutions of the equilibrium position for librational angle are derived during maneuvering in orbital plane; meanwhile, the effects of propulsive coefficients on librational stability are analyzed, which provides a basis for a selection of expected attitude trajectory. Then, a method of hierarchical sliding-mode tension control is presented to track the expected in-plane angle. This method can address the underactuated problem of tethered systems without either complex coordinate transformation for the system state model or constraint equation restrictions. During orbital flight, in-plane and out-of-plane angles are decoupled, so the tether tension control cannot be conducted to inhibit the out-of-plane angle. To solve this problem, the binormal component of thrust acceleration normal to the orbital plane is adopted as a control variable, and a feedback linearization-based thrust controller is designed to damp out the out-of-plane angle. Afterwards, orbital transfer cases between two circular orbits are studied to demonstrate the effectiveness of the tethered satellite with chemical propulsion. Numerical simulation results indicate that the stability of librational angles has a close relation to propulsive coefficients, and distributions of stable centers and unstable saddle points are totally different on both sides of bifurcation point. In addition, tracking control requirements for tethered satellite are guaranteed by designed controllers, which ensure flight safety in orbital maneuvering.

CHAOS ENTANGLEMENT: A NEW APPROACH TO GENERATE CHAOS

A new approach to generate chaotic phenomenon, called chaos entanglement, is proposed in this paper. The basic principle is to entangle two or multiple stable linear subsystems by entanglement functions to form an artificial chaotic system such that each of them evolves in a chaotic manner. Firstly, a new attractor, entangling a two-dimensional linear subsystem and a one-dimensional one by sine function, is presented as an example. Dynamical analysis shows that both entangled subsystems are bounded and all equilibra are unstable saddle points when chaos entanglement is achieved. Also, numerical computation shows that this system has one positive Lyapunov exponent, which implies chaos. Furthermore, two conditions are given to achieve chaos entanglement. Along this way, by different linear subsystems and different entanglement functions, a variety of novel chaotic attractors have been created and abundant complex dynamics are exhibited. Our discovery indicates that it is not difficult any more to construct new artificial chaotic systems/networks for engineering applications such as chaos-based secure communication. Finally, a possible circuit is given to realize these new chaotic attractors.

Period doubling and Blazhko modulation in BL Herculis hydrodynamic models

We present the hydrodynamic BL Herculis-type models which display a long-term modulation of pulsation amplitudes and phases. The modulation is either strictly periodic or it is quasi-periodic, with the modulation period and modulation pattern varying from one cycle to the other. Such behaviour has not been observed in any BL Her variable so far, however, it is a common property of their lower luminosity siblings - RR Lyrae variables showing the Blazhko effect. These models provide a support for the recent mechanism proposed by Buchler & Kollath to explain this still mysterious phenomenon. In their model, a half-integer resonance that causes the period doubling effect, discovered recently in the Blazhko RR Lyrae stars, is responsible for the modulation of the pulsation as well. Although our models are more luminous than is appropriate for RR Lyrae stars, they clearly demonstrate, through direct hydrodynamic computation, that the mechanism can indeed be operational. Of great importance are models which show quasi-periodic modulation - a phenomenon observed in Blazhko RR Lyrae stars. Our models coupled with the analysis of the amplitude equations show that such behaviour may be caused by the dynamical evolution occurring in the close proximity of the unstable single periodic saddle point.

Frequency Switching in a 12-Cavity Relativistic Magnetron With Axial Extraction of Radiation

The possibility of mode switching from one pulse to another in a 6-cavity gigawatt magnetron with axial extraction of radiation through a horn antenna (such a magnetron is known as the MDO, i.e., magnetron with diffraction output) using a weak (200-300 kW), short (15-ns), and single-frequency RF signal was demonstrated using particle-in-cell simulations in our earlier work. This mode switching exploits the symmetric nature of the MDO that facilitates the use of any eigenmode as the operating mode. All scenarios of mode switching were considered using common properties of dynamical systems with two stable states separated by an unstable saddle point. In this paper, we continue to study the problem of mode switching, but this time for a 12-cavity MDO, for which we found splitting of the radiation frequency for each eigenmode owing to its different longitudinal distributions. Since splitting manifests as a bifurcation of frequency for definite values of the applied axial magnetic field, scenarios of frequency switching for this 12-cavity magnetron are considered.

Predicting the probability of persistence of HIV infection with the standard model

We consider the standard three-component differential equation model for the growth of an HIV virion population in an infected host in the absence of drug therapy. The dynamical properties of the model are determined by the set of values of six parameters which vary across host populations. There may be one or two critical points whose natures play a key role in determining the outcome of infection and in particular whether the HIV population will persist or become extinct. There are two cases which may arise. In the first case, there is only one critical point P_1 at biological values and this is an asymptotically stable node. The system ends up with zero virions and so the host becomes HIV-free. In the second case, there are two critical points P_1 and P_2 at biological values. Here P_1 is an unstable saddle point and P_2 is an asymptotically stable spiral point with a non-zero virion level. In this case the HIV population persists unless parameters change. We let the parameter values take random values from distributions based on empirical data, but suitably truncated, and determine the probabilities of occurrence of the various combinations of critical points. From these simulations the probability that an HIV infection will persist, across a population, is estimated. It is found that with conservatively estimated distributions of parameters, within the framework of the standard 3-component model, the chances that a within host HIV population will become extinct is between 0.6% and 6.9%. With less conservative parameter estimates, the probability is estimated to be as high as 24%. The many factors related to the transmission and possible spontaneous elimination of the virus are discussed.

Stability and sustained oscillations in a ventricular cardiomyocyte model

The Luo-Rudy I model, describing the electrophysiology of a ventricular cardiomyocyte, is associated with an 8-dimensional discontinuous dynamical system with logarithmic and exponential non-linearities depending on 15 parameters. The associated stationary problem was reduced to a nonlinear system in only two unknowns, the transmembrane potential V and the intracellular calcium concentration [Ca]( i ). By numerical approaches appropriate to bifurcation problems, sections in the static bifurcation diagram were determined. For a variable steady depolarizing or hyperpolarizing current (I (st)), the corresponding projection of the static bifurcation diagram in the (I (st), V) plane is complex, featuring three branches of stationary solutions joined by two limit points. On the upper branch oscillations can occur, being either damped at a stable focus or diverted to the lower branch of stable stationary solutions when reaching the unstable manifold of a homoclinic saddle, thus resulting in early after-depolarizations (EADs). The middle branch of solutions is a series of unstable saddle points, while the lower one a series of stable nodes. For variable slow inward and K(+) current maximal conductances (g (si) and g (K)), in a range between 0 and 4-fold normal values, the dynamics is even more complex, and in certain instances sustained oscillations tending to a limit cycle appear. All these types of behavior were correctly predicted by linear stability analysis and bifurcation theory methods, leading to identification of Hopf bifurcation points, limit points of cycles and period doubling bifurcations. In particular settings, e.g. one-fifth-of-normal g (si), EADs and sustained high amplitude oscillations due to an unstable resting state may occur simultaneously.

Shrinking Dimer Dynamics and Its Applications to Saddle Point Search

Saddle point search on an energy surface has broad applications in fields like materials science, physics, chemistry, and biology. In this paper, we present the shrinking dimer dynamics (SDD), a dynamic system which can be applied to locate a transition state on an energy surface corresponding to an index-1 saddle point where the Hessian has a negative eigenvalue. By searching for the saddle point and the associated unstable direction simultaneously in a single dynamic system defined in an extended space, we show that unstable index-1 saddle points of the energy become linearly stable steady equilibria of the SDD which makes the SDD a robust approach for the computation of saddle points. The time discretization of the SDD is connected to various iterative algorithms, including the popular dimer method used in many practical applications. Our study of these discretization schemes lays a rigorous mathematical foundation for the corresponding iterative saddle point search algorithms. Both linear local asymptotic stability analysis and optimal error reduction (convergence) rates are presented and further confirmed by numerical experiments. Global convergence and nonlinear asymptotic stability are also illustrated for some simpler systems. Applications of the SDD in both finite- and infinite-dimensional energy spaces are discussed.

A Dual Purpose Principal and Minor Subspace Gradient Flow

The dual purpose principal and minor subspace gradient flow can be used to track principal subspace (PS) and if altered simply by the sign, it can also serve as a minor subspace (MS) trackor. This is of practical significance in the implementations of algorithms. In this paper, a unified information criterion is proposed and a dual purpose principal and minor subspace gradient flow is derived based on the information criterion. In this dual purpose gradient flow, the weight matrix length is self-stabilizing, i.e., moving towards unit length in each learning step. The energy function associated with the dual purpose gradient flow for tracking PS and MS is given, and it exhibits a unique global minimum attained if and only if its state matrices span the PS or MS of the autocorrelation matrix of a vector data stream. The other stationary points of its energy function are (unstable) saddle points. The proposed dual purpose gradient flow can efficiently track an orthonormal basis of the PS or MS, which is illustrated through simulation experiments.

PREDICTING THE PROBABILITY OF PERSISTENCE OF HIV INFECTION WITH THE STANDARD MODEL

It is not well understood why the transmission of HIV may have a small probability of occurrence despite frequent high risk exposures or ongoing contact between members of a discordant couple. We explore the possible contributions made by distributions of system parameters beginning with the standard three-component differential equation model for the growth of a HIV virion population in an infected host in the absence of drug therapy. The overall dynamical behavior of the model is determined by the set of values of six parameters, some of which describe host immune system properties and others which describe virus properties. There may be one or two critical points whose natures play a key role in determining the outcome of infection and in particular whether the HIV population will persist or become extinct. There are two cases which may arise. In the first case, there is only one critical point P1at biological values and this is an asymptotically stable node. The system ends up with zero virions and so the host becomes HIV-free. In the second case, there are two critical points P1and P2at biological values. Here P1is an unstable saddle point and P2is an asymptotically stable spiral point with a non-zero virion level. In this case the HIV population persists unless parameters change. We let the parameter values take random values from distributions based on empirical data, but suitably truncated, and determine the probabilities of occurrence of the various combinations of critical points. From these simulations the probability that an HIV infection will persist, across a population, is estimated. It is found that with conservatively estimated distributions of parameters, within the framework of the standard 3-component model, the chances that a within-host HIV population will become extinct is between 0.6% and 6.9%. With less conservative parameter estimates, the probability is estimated to be as high as 24%. The many complicating factors related to the transmission and possible spontaneous elimination of the virus and the need for experimental data to clarify whether transient infections may occur are discussed. More realistic yet complicated higher dimensional models are likely to yield smaller probabilities of extinction.

Stability and Self-Sustained Oscillations in a Ventricular Cardiomyocyte Model

The Luo-Rudy I model, describing the electrophysiology of a ventricular cardiomyocyte, is associated with an 8-dimensional discontinuous dynamical system with logarithmic and exponential non-linearities depending on 15 parameters. By numerical approaches appropriate to bifurcation problems, sections in the static bifurcation diagram were determined. For different values of a steady depolarizing/hyperpolarizing current (Ist), the corresponding projection of the static bifurcation diagram in the (Ist, V) plane is complex, featuring three branches of stationary solutions delimited by two saddle-node bifurcation points. In addition, on the upper branch oscillations can occur within an Ist range [−4.45, −0.51 μA/cm2] where the Jacobian of the linearized system features two complex conjugated eigenvalues. Oscillations are either damped at a stable focus or amplified until the system trajectory is diverted to the lower branch of stable stationary solutions when reaching the unstable manifold of a homoclinic saddle. The middle branch of solutions is a series of unstable saddle points, while the lower one a series of stable nodes. For variable slow inward and K+ current maximal conductances (gsi and gK), in a range between 0 and 4-fold normal values, the dynamics is even more complex, and in certain instances self-sustained oscillations tending to a stable limit cycle appear. All these types of behavior were correctly predicted by linear stability analysis and bifurcation theory methods based on numerical continuation algorithms. Both unsustained oscillations, resulting in early after-depolarizations, and sustained oscillations may trigger dangerous ventricular arrhythmias by multiple mechanisms. In particular settings, e.g. for a normal gK and one-fifth-of-normal gsi, these two arrhythmia-threatening conditions may occur simultaneously.

Calculation of the magnetic energy barrier in nanostructured cells of synthetic ferrimagnets

A method is proposed to calculate the magnetic energy barrier of nanostructured cells of synthetic ferrimagnets. An important feature of the method is the use of an equation for the total energy that contains the magnetostatic fields at the saddle point as parameters of the energy equation. With no suitable methods of accessing the magnetic configuration at the unstable saddle point, it is difficult to obtain the saddle point magnetostatic fields. This difficulty is overcome with the use of equations that link the magnetostatic fields at the saddle point and critical fields, which are readily obtained by micromagnetic simulation. The present method is essentially based on the micromagnetic simulation, and, therefore, it should provide accurate results for the magnetic energy barrier. A contour diagram showing the thermal stability parameter is constructed as a function of the cell geometry and the thickness asymmetry, and the result should be of great value in designing magnetic cells for high density magnetic random access memory.

Coloured noise controlled dynamics of nonlinear polaritons in semiconductor microcavity

The coloured noise induced escape rate from the lower energy stable state of a driven nonlinear microcavity oscillator has been investigated by means of quasi-classical kinetic equations. We show that for coloured, i.e. narrow-band, relatively intense noise, the escape time is controlled by the interplay of two mechanisms: the noise induced drift and adiabatic regular shift of the oscillator state towards unstable saddle point. The cross-over between these mechanisms takes place in a particular range of the driving field intensity values, depending on the ratio between the oscillator damping and the coloured noise spectrum width. The dependence of the transition rate on the noise correlation time is analyzed for wide range of correlation time values.

Cation−π-Anion Interaction in Alkali Ion−Benzene−Halogen Ion Clusters

The present investigation on some ternary M(+)-benzene-X(-) aggregates follows previous studies concerning the binary, M(+)-benzene (M = Li, Na, K, Rb, Cs) and X(-)-benzene (X = F, Cl, Br, I), systems. A semimepirical model describing the intermolecular potential energy, formulated as a combination of few leading effective components, is here extended and applied to investigate more complex systems. The balancing of size repulsion with induction and dispersion attraction is described as a combination of ion-bond interactions and an electrostatic component, rationalized on the basis of the benzene quadrupole moment, is also included. For M(+)-benzene-X(-) aggregates, the simultaneous presence of a cation and an anion close to the pi system originates a strong Coulombic attraction and nonadditivity effects of the induction energy, which are carefully considered and explicitly included to obtain a proper analytical functional representation of the ternary compound. The proposed semiempirical methodology, providing the whole potential energy surface in a convenient analytical form, is useful to predict the main features of stable and unstable configurations, saddle points, and energy barriers, allowing for investigation of their influence on molecular dynamics simulations.

MULTISTABILITY AND MULTI 2π-KINKS IN THE FRENKEL–KONTOROVA MODEL: AN APPLICATION TO ARRAYS OF JOSEPHSON JUNCTIONS

A regular ring of Josephson junctions, connected in parallel, is studied analytically and numerically. We show that, depending on the strength of the r-well cosine potential the energy landscape of the Hamiltonian can have of the order of rN/N locally stable minima separated by large barriers specified by unstable saddle points. The counting problem for the degeneracy of the total energy is equivalent to a wellknown necklace problem in combinatorial mathematics. We also demonstrate that the distribution of the phase differences as well as the energy spectrum is fractal provided that the strength of the cosine potential is sufficiently strong.

Stability and bifurcation behaviour of electrostatic torsional NEMS varactor influenced by dispersion forces

The influences of Casimir and van der Waals forces on the nano-electromechanical systems (NEMS) electrostatic torsional varactor are studied. A one degree of freedom, the torsional angle, is adopted, and the bifurcation behaviour of the NEMS torsional varactor is investigated. There are two bifurcation points, one of which is a Hopf bifurcation point and the other is an unstable saddle point. The phase portraits are also drawn, in which periodic orbits are around the Hopf bifurcation point, but the periodic orbit will break into a homoclinic orbit when meeting the unstable saddle point.

Synchronization in a network of model neurons

We study the spatiotemporal dynamics of a network of coupled chaotic maps modelling neuronal activity, under variation of coupling strength epsilon and degree of randomness in coupling p. We find that at high coupling strengths (epsilon>epsilonfixed) the unstable saddle point solution of the local chaotic maps gets stabilized. The range of coupling where this spatiotemporal fixed point gains stability is unchanged in the presence of randomness in the connections, namely epsilonfixed is invariant under changes in p. As coupling gets weaker (epsilon<epsilonfixed), the spatiotemporal fixed point loses stability, and one obtains chaos. In this regime, when the coupling connections are completely regular (p=0), the network becomes spatiotemporally chaotic. Interestingly however, in the presence of random links (p>0) one obtains spatial synchronization in the network. We find that this range of synchronized chaos increases exponentially with the fraction of random links in the network. Further, in the space of fixed coupling strengths, the synchronization transition occurs at a finite value of p, a scenario quite distinct from the many examples of synchronization transitions at p-->0. Further we show that the synchronization here is robust in the presence of parametric noise, namely in a network of nonidentical neuronal maps. Finally we check the generality of our observations in networks of neurons displaying both spiking and bursting dynamics.

Geometrical effect on dislocation nucleation at crystal surface nanostructures

Crystal surface nanostructures such as steps and trenches in microelectronic layer structures have been considered as stress concentrators that may facilitate dislocation nucleation. Quantitative characterization of the critical condition of this atomic scale process is of considerable interest for the development of high performance electronic devices. This paper addresses this issue using a multiscale approach based on the variational boundary integral formulation of the Peierls–Nabarro dislocation model. By representing the profiles of embryonic dislocations as the relative displacements between the two adjacent atomic layers along the slip planes, the critical conditions for dislocation nucleation are obtained by solving the stress dependent activation energies required to activate embryonic dislocations from their stable to unstable saddle point configurations. The geometrical effect of surface nanostructures such as steps and trenches on dislocation nucleation is ascertained quantitatively. Our results show that the atomic scale surface nanostructures can reduce the critical stress for dislocation nucleation by nearly an order of magnitude and the trench configurations are more prone to dislocation nucleation than the step configurations. Nucleation of versatile dislocations in multiple slip systems at crystal surfaces may be attributed surface nanostructures of a variety of geometries.

Stability of Lithium in α‐Rhombohedral Boron

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Lynden-Bell and Tsallis distributions for the HMF model

Systems with long-range interactions can reach a Quasi Stationary State (QSS) as a result of a violent collisionless relaxation. If the system mixes well (ergodicity), the QSS can be predicted by the statistical theory of Lynden-Bell (1967) based on the Vlasov equation. When the initial distribution takes only two values, the Lynden-Bell distribution is similar to the Fermi-Dirac statistics. Such distributions have recently been observed in direct numerical simulations of the HMF model (Antoniazzi et al. 2006). In this paper, we determine the caloric curve corresponding to the Lynden-Bell statistics in relation with the HMF model and analyze the dynamical and thermodynamical stability of spatially homogeneous solutions by using two general criteria previously introduced in the literature. We express the critical energy and the critical temperature as a function of a degeneracy parameter fixed by the initial condition. Below these critical values, the homogeneous Lynden-Bell distribution is not a maximum entropy state but an unstable saddle point. We apply these results to the situation considered by Antoniazzi et al. For a given energy, we find a critical initial magnetization above which the homogeneous Lynden-Bell distribution ceases to be a maximum entropy state, contrary to the claim of these authors. For an energy U=0.69, this transition occurs above an initial magnetization M_{x}=0.897. In that case, the system should reach an inhomogeneous Lynden-Bell distribution (most mixed) or an incompletely mixed state (possibly fitted by a Tsallis distribution). Thus, our theoretical study proves that the dynamics is different for small and large initial magnetizations, in agreement with numerical results of Pluchino et al. (2004). This new dynamical phase transition may reconcile the two communities.

Stability of lithium in α-rhombohedral boron

The stability of lithium atoms in α-rhombohedral boron was studied by the density functional theory and Car–Parrinello molecular dynamics (MD) simulations. At a low Li concentration (1.03 at%), a Li atom at the center of the icosahedral B 12 site (the I-site) was found to be metastable, and the potential barrier was estimated at 775±25 K (=67±25 meV). Over 800 K, Li atoms began to escape from the B 12 cage and settled at the tetrahedral site (the T-site) or at the octahedral site (the O-site). Li at the T-site was also metastable below 1400 K, and Li at the O-site was energetically the most favorable. At a high Li concentration (7.69 at%), the I-site changed to an unstable saddle point. The T-site was still metastable, and the O-site was the most stable. Regardless of concentration, MD simulations showed that Li atoms at the O-site never jumped to other sites below 1400 K. The migration of Li would be very slow below this temperature.