Unstable Saddle Research Articles

Numerical study of saddle-node bifurcation for longitudinal flight with CFD/RBD technique

The square cross section missile was found instability in the longitudinal direction at a certain Mach number in a range of angle of attack both in CFD and wind tunnel test results. To deeply understand the flight behavior and the aerodynamics of the missile, the coupled Computational Fluid Dynamic/Rigid Body Dynamics (CFD/RBD) approach was employed to simulate the free flight process, and the predict technique of dynamic stability derivatives was founded to analyze the types of equilibrium points. The study shows that the type of trim angle of attack is turned from stable node to an unstable saddle with the increase of Mach number, known as the saddle-node bifurcation. There are two extra stable nodes are produced and located on the left and right sides of the saddle point when the Mach number is bigger than the transition Mach number. Further simulation results of free pitching motions reveal the decrease of the attraction region of the two extra stable modes, which makes the pitching motion more complicated and sensitive. A slight change of the initial angle of attack may make the missile move from one trim point to another.

Dependence of chaotic behavior on optical properties and electrostatic effects in double-beam torsional Casimir actuation.

We investigate the influence of Casimir and electrostatic torques on double-beam torsional microelectromechanical systems with materials covering a broad range of conductivities of more than three orders of magnitude. For the frictionless autonomous systems, bifurcation and phase space analysis shows a significant difference between stable and unstable operating regimes for equal and unequal applied voltages on both sides of the double torsional system giving rise to heteroclinic and homoclinic orbits, respectively. For equal applied voltages, only the position of a symmetric unstable saddle equilibrium point is dependent on the material optical properties and electrostatic effects, while in any other case stable and unstable equilibrium points are dependent on both factors. For the periodically driven system, a Melnikov function approach is used to show the presence of chaotic motion rendering predictions of whether stiction or stable actuation will take place over long times impossible. Chaotic behavior introduces significant risk for stiction, and it is more likely to occur for the more conductive systems that experience stronger Casimir forces and torques. Indeed, when unequal voltages are applied, the sensitive dependence of chaotic motion on electrostatics is more pronounced for the highest conductivity systems.

Pursue high economic development with less pollution: implications from a Resource-Economy-Pollution dynamic system

How to reduce pollution intensity and realize the coordinated development of the economy and environment has become a key issue for environment governance. This paper investigates the pollution intensity by applying the Resource-Economy-Pollution dynamical system (REP). The dynamic behavior of the system is analyzed by means of Lyapunov exponents and bifurcation diagrams. A new REP chaotic attractor is observed. In addition, the existence of two unstable saddles is the reason for the occurrence of oscillation phenomenon. The integrated pollution intensity is studied. We detect how integrated pollution intensity is influenced by the patterns of resource consumption, recycling intensity of waste and the speed of economic development over time. We find that increasing the intensity of waste recycling is effective in reducing pollution intensity, while the effect of slowing the resource consumption and economy is not obvious. From the point of view of energy conservation and economic development, a full analysis of factors that may relate to pollution intensity should be conducted before making energy policies.

Lateral motion stability of high-temperature superconducting maglev systems derived from a nonlinear guidance force hysteretic model

High-temperature superconducting (HTS) maglev, owning to the capability of passive stabilization, is potentially promising for high-speed transportation. The guidance force of bulk HTS materials above a permanent magnetic guideway has a nonlinear response due to the hysteresis effect. As a kind of rail transit, when the vehicle runs along the track, the curve and other disturbances will cause vibrations to the vehicle system. These physical factors will pose dynamic loads on the components, reducing structural reliability as well as affecting the ride comfort. The lateral motion, as an important part of the vehicle system dynamics, needs to be studied in the pursuit of HTS maglev realization. In this paper, we first measured the guidance forces of HTS bulks under different motion conditions, and analyzed the relationship between the lateral displacement, the movement velocity and the guidance force. Then, a mathematical model was built based on these experimental data. The key feature of this mathematical model is that it can describe the hysteresis characteristic of the guidance force. Based on this model, we investigated the lateral motion stability of the HTS levitation system, and found three singular points, one stable focus point, and two unstable saddle points. Lastly, a phase portrait was proposed to indicate the safe working region of the HTS maglev vehicle where the vehicle can automatically return to its equilibrium position. These experimental and simulation results are important to clarify the lateral motion stability under external disturbance or shock, and provide a reference basis for the design of levitation systems.

Oscillation phenomenon and its mechanism of an energy-saving and emission-reduction system

PurposeThis paper aims to study the oscillation phenomenon before chaos as well as its mechanism of occurrence in the energy-saving and emission-reduction system.Design/methodology/approachThe system dynamics analysis, phase portrait analyses, equilibrium point analysis and bifurcation curve were applied to this paper.FindingsFirst, the authors find an oscillation phenomenon previous to chaos. Second, on the one hand, the existence of two unstable saddles is the reason for the occurrence of oscillation phenomenon. On the other hand, the increasing of carbon emissions can arouse oscillation phenomenon.Originality/valueThis paper finds an oscillation phenomenon previous to chaos in the energy-saving and emission-reduction system. The mechanism of occurrence of oscillation phenomenon is studied. The existence of two unstable saddles is the reason for the occurrence of such oscillation phenomenon. The oscillation is related with fold bifurcation. The study also provides a theoretical basis for the further study of chaos control.

Flow behaviour of drop and jet modes of a laminar falling film on horizontal tubes

The flow behaviour of a falling film on horizontal tubes has significant impacts on the efficiency of heat and mass transfer for the horizontal tubular evaporators and absorbers. Thus it is crucial to investigate the effect of contact angle and specific flow rate on the hydrodynamics of films and droplets in the drop and jet modes, which are the most important modes for evaporators and absorbers. A three-dimensional CFD model, using the volume of fluid (VOF) method, has been built to capture the gas-liquid interface. By employing mesh refinement and independence tests, the film spreading and droplet formation, liquid bridge breakup, droplet detachment, impact and fluctuation waviness of the simulations at the completely wetting condition are found to be in good agreement with experimental data and theory. In the drop mode, the behaviours of the droplets alternately falling from neighboring sites, including unstable slender liquid bridge and complex saddle waves were observed. This indicates that the wave propagation speed at first is higher than the impaction speed. In the in-line jet mode, the flow pattern is almost stationary but it is perturbed weakly by the impaction wave. The circumferential ring film has its maximum thickness located at the middle of the two-column in-line jet due to the wave interaction. In addition, the wetting area, spreading speed and instability time of the falling film are reduced with an increase in the contact angle, while the first two parameters increase at higher specific flow rates. In particular, an overshoot occurred at the larger contact angle. Finally, the results showed that the falling film flow pattern is sensitive to the initial wetting condition. Further, it reveals that the final steady wetted region depends on the receding contact angle, thus the introduction of the dynamic contact angle is the best approach for accurately determining the behaviour of falling films.

Statistics and dynamics of blockings with a point vortex model

We investigate a reduced point vortex model for statistical and dynamical analyses of atmospheric blocking phenomena. Thereby, we consider High-over-low and Omega blocking as relative equilibria of two and three point vortices. Under certain conditions, such point vortex systems move westward opposing the mean westerly flow and hence can become stationary. Based on the kinematic vorticity number, two novel, independent methods, the contour and the trapezoid method, are introduced in order to identify the vortices that form the blocking pattern, their local positions and circulation magnitudes. While the contour method takes into account the observed stationarity of blocking, the trapezoid method minimizes the total circulation of the vortex system following point vortex theory. Using an instantaneous blocking index, a total number of 347 blocking periods were identified in NCEP-NCAR Reanalysis data for the Euro-Atlantic region during the time period 1990–2012. This procedure provides the basis to corroborate the applicability of the point vortex model to atmospheric blocking in a statistical framework. The calculated translation speed of point vortex systems associated with the atmospheric blocking appears to match the zonal mean velocity reasonably well. This model explains the stationary behaviour of blocking patterns. A comparison between the theoretical and a statistical model further reveals that the circulation of the blocking high follows the principles of the point vortex model to a large extent. However, the low-pressure systems behave more variable. Moreover, the stability of point vortex equilibria is analysed regarding the relative distances by considering linear stability analysis and simulations. This reveals that the point vortex blocking model corresponds to an unstable saddle point. Furthermore, we take viscosity and a Brownian motion into account to simulate the influence of the smaller, subgrid-scale disturbances. As a result, a clustering near the equilibrium state emerges indicating the persistence of the atmospheric blocking pattern.

New families of vortex patch equilibria for the two-dimensional Euler equations

Various modified forms of contour dynamics are used to compute multipolar vortex equilibria, i.e., configurations of constant vorticity patches which are invariant in a steady rotating frame. There are two distinct solution families for “N + 1” point vortex-vortex patch equilibria in which a finite-area central patch is surrounded by N identical point vortices: one with the central patch having opposite-signed vorticity and the other having same-signed vorticity to the satellite vortices. Each solution family exhibits limiting states beyond which no equilibria can be found. At the limiting state, the central patch of a same-signed equilibrium acquires N corners on its boundary. The limiting states of the opposite-signed equilibria have cusp-like behaviour on the boundary of the central patch. Linear stability analysis reveals that the central patch is most linearly unstable as it approaches the limiting states. For equilibria comprising a central patch surrounded by N identical finite-area satellite patches, again two distinct families of solutions exist: one with the central patch and satellite patches having the same-signed vorticity and the other in which they are opposite-signed. In each family, there are two limiting behaviours in which either the central patch or the satellite patches develop corners or cusps. Streamline plots and time-dependent simulations indicate that opposite-signed multipolar equilibria are robust structures and same-signed equilibria are generally less stable. Streamlines also reveal stable and unstable (saddle point) stagnation points, indicating the existence of new equilibria in which additional patches of vorticity are “grown” at the stagnation points. Examples of such equilibria are computed, and a general numerical routine is briefly described for finding even more complex finite-area equilibria. Finally, new nested polygonal vortex equilibria consisting of two sets of polygonally arranged vortex patches (named “N + N” equilibria here) are computed for two distinct cases: one with the corners of the polygons aligned with each other and the other when they are staggered. Various limiting states are computed for these equilibria. Time-dependent simulations reveal that the aligned equilibria are susceptible to instability, while the staggered equilibria survive a relatively long time. In some parameter regimes, following instability, these structures evolve into known structures such as “N + 1” multipolar vortex equilibria and N-polygon co-rotating equilibria.

Motion stability of the magnetic levitation and suspension with YBa2Cu3O7-x high-Tc superconducting bulks and NdFeB magnets

The flux pinning effect of YBa2Cu3O7-x high temperature superconducting (HTS) bulk can achieve self-stable levitation over a permanent magnet or magnet array. Devices based on this phenomenon have been widely developed. However, the self-stable flux pinning effect is not unconditional, under disturbances, for example. To disclose the roots of this amazing self-stable levitation phenomenon in theory, mathematical and mechanical calculations using Lyapunov's stability theorem and the Hurwitz criterion were performed under the conditions of magnetic levitation and suspension of HTS bulk near permanent magnets in Halbach array. It is found that the whole dynamical system, in the case of levitation, has only one equilibrium solution, and the singular point is a stable focus. In the general case of suspension, the system has two singular points: one is a stable focus, and the other is an unstable saddle. With the variation of suspension force, the two first-order singular points mentioned earlier will get closer and closer, and finally degenerate to a high-order singular point, which means the stable region gets smaller and smaller, and finally vanishes. According to the center manifold theorem, the high-order singular point is unstable. With the interaction force varying, the HTS suspension dynamical system undergoes a saddle-node bifurcation. Moreover, a deficient damping can also decrease the stable region. These findings, together with existing experiments, could enlighten the improvement of HTS devices with strong anti-interference ability.

Numerical Simulation of the Supersonic Disk-Gap-Band Parachute by Using Implicit Coupling Method

Abstract The implicit coupling method is applied to model the 0.8 m disk-band-gap parachute at Mach 2.0. The fluid and structure governing equations are solved by the Lower-Upper Symmetric Gauss-Seidel (LU-SGS) algorithm and Newmark scheme, respectively. By exchanging the numerical results of the coupling surface with Gauss-Seidel algorithm, high accuracy solutions at every physical time step are obtained. The numerical results of the canopy drag coefficient and projected area fit well with the wind tunnel test results. The simulation reproduces the shock oscillation and breathing phenomenon of the canopy that are usually observed in these systems at Mach 2.0. Furthermore, it is found that the unstable saddle point is the main reason for the shock oscillation of the canopy. And the unsynchronized phases of the canopy area and shock oscillation curves lead to the drag of the canopy oscillate in irregular state.

A fast and effective principal singular subspace tracking algorithm

In this paper, we propose a fast and effective neural network algorithm to perform singular value decomposition (SVD) of a cross-covariance matrix between two high-dimensional data streams. Firstly, we derive a dynamical system from a newly proposed information criterion. This system exhibits a single stable stationary point if and only if the weight matrices of the left and right neural networks span the left and right principal singular subspace of a cross-covariance matrix, respectively, and the other stationary points are (unstable) saddle points. Then, a principal singular subspace (PSS) tracking algorithm is obtained from the dynamical system. Moreover, convergence analysis shows that the proposed algorithm converges to a stationary point that relates to the principal singular values. Thus, compared with traditional algorithms who can only track the PSS, the proposed algorithm can not only track the PSS but also estimate all of the corresponding principal singular values based on the extracted subspace. Finally, numerical simulations and practical application are carried to further demonstrate the efficiency of the proposed algorithm.

Sifat-Sifat Sistem Pendulum Terbalik dengan Lintasan Berbentuk Lingkaran


 The inverted pendulum is a high-order non-linear, miltivariable and highly unstable dynamic system. High-order non-linear systems in the inverted pendulum must be dilutarized to be solved easily. From the calculations that have been done can be deduced that the system from the inverted pendulum is unstable saddle system, can be controlled and can be observed. In addition the system can also be formed into a system of controlled companions and observable forms of kompanion.

On the topology of wall turbulence in physical space

The topological features dominating the different sublayers of the near wall turbulence are revisited through direct numerical simulations performed in large computational domains up to the Karman number of 1100. The averages of the invariants of the velocity gradient tensor, conditioned by fixed magnitudes of the velocity components, are adequately performed. In the physical space, the dominating topology in the viscous sublayer is the stable focus stretching cohabiting with the unstable node/saddle/saddle points. The stable and unstable focuses mostly populate the inner layer with equal importance, while the former become predominant in the meso and outer layers. Eddies within the low speed streaks with intense streamwise negative velocity of very high shear stress are located near the zero-discriminant facets. The enstrophy and straining prevails the stable focus stretching and the unstable compressing, respectively. The Eulerian velocity field conditioned by a given topology seen by an observer moving with the local velocity is also considered. It is found that the wall normal velocity marks the stretching line of the unstable saddles and the compressing line of the stable nodes/saddles/saddles. The conditional velocity fields linked to the unstable focuses compressing and stable focuses stretching are closely similar.

An Improved Memristive Diode Bridge-Based Band Pass Filter Chaotic Circuit

By replacing a series resistor in active band pass filter (BPF) with an improved memristive diode bridge emulator, a third-order memristive BPF chaotic circuit is presented. The improved memristive diode bridge emulator without grounded limitation is equivalently achieved by a diode bridge cascaded with only one inductor, whose fingerprints of pinched hysteresis loop are examined by numerical simulations and hardware experiments. The memristive BPF chaotic circuit has only one zero unstable saddle point but causes complex dynamical behaviors including period, chaos, period doubling bifurcation, and coexisting bifurcation modes. Specially, it should be highly significant that two kinds of bifurcation routes are displayed under different initial conditions and the coexistence of three different topological attractors is found in a narrow parameter range. Moreover, hardware circuit using discrete components is fabricated and experimental measurements are performed, upon which the numerical simulations are validated. Notably, the proposed memristive BPF chaotic circuit is only third-order and has simple topological structure.

Coupled cross-correlation neural network algorithm for principal singular triplet extraction of a cross-covariance matrix

This paper proposes a novel coupled neural network learning algorithm to extract the principal singular triplet (PST) of a cross-correlation matrix between two high-dimensional data streams. We firstly introduce a novel information criterion (NIC), in which the stationary points are singular triplet of the crosscorrelation matrix. Then, based on Newton's method, we obtain a coupled system of ordinary differential equations (ODEs) from the NIC. The ODEs have the same equilibria as the gradient of NIC, however, only the first PST of the system is stable (which is also the desired solution), and all others are (unstable) saddle points. Based on the system, we finally obtain a fast and stable algorithm for PST extraction. The proposed algorithm can solve the speed-stability problem that plagues most noncoupled learning rules. Moreover, the proposed algorithm can also be used to extract multiple PSTs effectively by using sequential method.

Entanglement storage by classical fixed points in the two-axis countertwisting model

We analyze a scheme for storage of entanglement quantified by the quantum Fisher information in the two-axis countertwisting model. A characteristic feature of the two-axis countertwisting Hamiltonian is the existence of the four stable center and two unstable saddle fixed points in the mean-field phase portrait. The entangled state is generated dynamically from an initial spin coherent state located around an unstable saddle fixed point. At an optimal moment of time the state is shifted to a position around stable center fixed points by a single rotation, where its dynamics and properties are approximately frozen. We also discuss evolution with noise. In some cases the effect of noise turns out to be relatively weak, which is explained by parity conservation.

Analytical modeling of dynamic pull-in instability behavior of torsional nano/micromirrors under the effect of Casimir force

In this study, the dynamic stability of torsional electrostatic micro/nano-electromechanical systems (MEMS/NEMS) using single degree of freedom model under Casimir force and size effect is investigated. First, the equation of motion of system is developed. Afterwards, dimensionless equation of motion by using normalized parameters is presented and by considering state space method, dynamic equations are solved and intended results are achieved. Afterwards, pull-in angle and pull-in voltage of system which include the parameters affecting its performance, are investigated and its dependence on Casimir force and size effect are shown. Results show that equilibrium points include center points, stable focus points and unstable saddle points. The phase portraits related to these points exhibit periodic orbits, heteroclinic orbits and homoclinic orbits which with regard to considered particular circumstances for system are illustrated. Furthermore, in present study by using modified couple stress model, the influence of size effect on amplitude and frequency of system is considered. Obtained results show that proposed model is in better agreement with experimental data and is able to shorten the distance between the classic theory and the experimental results.

Dynamic instability of vibrating carbon nanotubes near small layers of graphite sheets based on nonlocal continuum elasticity

This article presents a new asymptotic method to predict dynamic pull-in instability of nonlocal clamped–clamped carbon nanotubes (CNTs) near graphite sheets. Nonlinear governing equations of carbon nanotubes actuated by an electric field are derived. With due allowance for the van der Waals effects, the pull-in instability and the natural frequency–amplitude relationship are investigated by a powerful analytical method, namely, the parameter expansion method. It is demonstrated that retaining two terms in series expansions is sufficient to produce an acceptable solution. The obtained results from numerical methods verify the strength of the analytical procedure. The qualitative analysis of system dynamics shows that the equilibrium points of the autonomous system include center points and unstable saddle points. The phase portraits of the carbon nanotube actuator exhibit periodic and homoclinic orbits.

Models of dark matter halos based on statistical mechanics: The fermionic King model

We discuss the nature of phase transitions in the fermionic King model which describes tidally truncated quantum self-gravitating systems. This distribution function takes into account the escape of high energy particles and has a finite mass. On the other hand, the Pauli exclusion principle puts an upper bound on the phase space density of the system and stabilizes it against gravitational collapse. As a result, there exists a statistical equilibrium state for any accessible values of energy and temperature. We plot the caloric curves and investigate the nature of phase transitions as a function of the degeneracy parameter in both microcanonical and canonical ensembles. We consider stable and metastable states and emphasize the importance of the latter for systems with long-range interactions. Phase transitions can take place between a "gaseous" phase unaffected by quantum mechanics and a "condensed" phase dominated by quantum mechanics. The phase diagram exhibits two critical points, one in each ensemble, beyond which the phase transitions disappear. There also exist a region of negative specific heats and a situation of ensemble inequivalence for sufficiently large systems. We apply the fermionic King model to the case of dark matter halos made of massive neutrinos. The gaseous phase describes large halos and the condensed phase describes dwarf halos. Partially degenerate configurations describe intermediate size halos. We argue that large dark matter halos cannot harbor a fermion ball because these nucleus-halo configurations are thermodynamically unstable (saddle points of entropy). Large dark matter halos may rather contain a central black hole resulting from a dynamical instability of relativistic origin occurring during the gravothermal catastrophe.

Hidden dynamics and multi‐stability in an improved third‐order Chua's circuit

By utilising an active diode pair to substitute the classical Chua's diode, an improved third-order Chua's circuit is proposed in this study. This circuit has an unstable zero saddle point and two stable non-zero saddle-foci in a local parameter region, from which hidden dynamics and multi-stability are discovered. The parameter region of hidden dynamics is figured out and hidden attractors are captured numerically and experimentally. Furthermore, multi-stability in the proposed circuit are also exhibited by numerical and circuit simulations.