Limit Cycle Research Articles

Oscillating latent dynamics in robot systems during walking and reaching.

Sensorimotor control of complex, dynamic systems such as humanoids or quadrupedal robots is notoriously difficult. While artificial systems traditionally employ hierarchical optimisation approaches or black-box policies, recent results in systems neuroscience suggest that complex behaviours such as locomotion and reaching are correlated with limit cycles in the primate motor cortex. A recent result suggests that, when applied to a learned latent space, oscillating patterns of activation can be used to control locomotion in a physical robot. While reminiscent of limit cycles observed in primate motor cortex, these dynamics are unsurprising given the cyclic nature of the robot's behaviour (walking). In this preliminary investigation, we consider how a similar approach extends to a less obviously cyclic behaviour (reaching). This has been explored in prior work using computational simulations. But simulations necessarily make simplifying assumptions that do not necessarily correspond to reality, so do not trivially transfer to real robot platforms. Our primary contribution is to demonstrate that we can infer and control real robot states in a learnt representation using oscillatory dynamics during reaching tasks. We further show that the learned latent representation encodes interpretable movements in the robot's workspace. Compared to robot locomotion, the dynamics that we observe for reaching are not fully cyclic, as they do not begin and end at the same position of latent space. However, they do begin to trace out the shape of a cycle, and, by construction, they are driven by the same underlying oscillatory mechanics.

Branch-Chain-Rich Diisopropyl Ether with Steric Hindrance Facilitates Stable Cycling of Lithium Batteries at - 20°C.

Li metal batteries (LMBs) offer significant potential as high energy density alternatives; nevertheless, their performance is hindered by the slow desolvation process of electrolytes, particularly at low temperatures (LT), leading to low coulombic efficiency and limited cycle stability. Thus, it is essential to optimize the solvation structure thereby achieving a rapid desolvation process in LMBs at LT. Herein, we introduce branch chain-rich diisopropyl ether (DIPE) into a 2.5M Li bis(fluorosulfonyl)imide dipropyl ether (DPE) electrolyte as a co-solvent for high-performance LMBs at - 20°C. The incorporation of DIPE not only enhances the disorder within the electrolyte, but also induces a steric hindrance effect form DIPE's branch chain, excluding other solvent molecules from Li+ solvation sheath. Both of these factors contribute to the weak interactions between Li+ and solvent molecules, effectively reducing the desolvation energy of the electrolyte. Consequently, Li (50μm)||LFP (mass loading ~ 10mgcm-2) cells in DPE/DIPE based electrolyte demonstrate stable performance over 650 cycles at - 20°C, delivering 87.2 mAh g-1, and over 255 cycles at 25°C with 124.8 mAh g-1. DIPE broadens the electrolyte design from molecular structure considerations, offering a promising avenue for highly stable LMBs at LT.

A Nanocrystallite CuS/Nitrogen‐Doped Carbon Host Improves Redox Kinetics in All‐Solid‐State Li2S Batteries

AbstractAll‐solid‐state Li–S batteries are a promising energy storage system that can solve the shuttle effects of polysulfides in liquid Li–S batteries. However, sluggish solid‐state reaction kinetics and the low conductivity of cathode materials have impeded their development. Here, a N‐doped carbon embedded with CuS nanoparticles (CuSNC) is reported as a host for Li2S in all‐solid‐state batteries that addresses some of these issues. Electrochemical studies, supported by a combination of X‐ray diffraction (XRD), X‐ray photoelectron spectroscopy (XPS), X‐ray absorption spectroscopy (XAS), electron microscopy, and density functional theory (DFT) calculations reveal that CuSNC provides good affinity to Li2S. This lowers the activation barrier for the conversion of Li2S to sulfur on the charge, suggesting an electrocatalytic effect on the CuS surface. Li+ diffusion in the cathode and the reaction kinetics are enhanced compared to N‐doped graphene. The CuSNC/Li2S cathode reaches an areal capacity of 1.8 mAh cm−2 and a retention rate of 94% after 100 cycles. At a 1.0 mA cm−2 current density, CuSNC/Li2S maintains stable performance over 500 cycles with a low decay rate (0.05% per cycle); at a higher Li2S loading, delivers a capacity of 9.6 mAh cm−2, albeit with more limited cycling. This study provides a promising way to design Li2S cathodes to achieve improved reaction kinetics and better electrochemical performance.

Theoretico-experimental analysis of bistability in the oscillatory response of a TaOx ReRAM to pulse train stimuli

Fading memory is the capability of a physical system to approach a unique asymptotic behaviour, irrespective of the initial conditions, when stimulated by an input from a certain class. Standard stimuli from the AC periodic class typically induce fading memory effects in non-volatile memristors, as uncovered for the first time back in 2016. Very recently, a deep investigation of resistance switching phenomena in a TaOx resistive random access memory cell revealed the capability of the nano-device to exhibit one of two possible oscillatory behaviours, depending upon the initial condition, when subject to a particular periodic excitation. This interesting finding was, however, left unexplained. Bistability is the simplest form of local fading memory. In a system, endowed with local fading memory under a given stimulus, the initial condition does not affect the long-term behaviour of the state as long as it is drawn from the basin of attraction of either of the distinct coexisting state-space attractors (two limit cycles for the periodically forced memristor acting as a bistable oscillator). Here, the history of the system, encoded in the initial condition, is, thus, erasable only locally through ad hoc stimulation. Motivated by the discovery of local history erase effects in our resistive random access memory cell, this study applies a powerful system-theoretic tool, enabling the analysis of the response of first-order systems to square pulse train-based periodic stimuli, known as the time-average state dynamic route, to an accurate physics-based mathematical model, earlier fitted to the nano-device, to determine a strategy for specifying the parameters of an excitation signal, consisting of the sequence of two square pulses of opposite polarity per period so as to induce various forms of monostability or multistability in the non-volatile memristor. In particular, as an absolute novelty in the literature, experimental measurements validate the theoretical prediction on the capability of the device to operate as one of two distinct oscillators, depending upon the initial condition, under a specific pulse train excitation signal. The coexistence of multiple oscillatory operating modes in the periodically forced resistive random access memory cell, an example par excellence of their unique non-linear dynamics, may inspire the development and circuit implementation of novel sensing and mem-computing paradigms.

Learning an Autonomous Dynamic System to Encode Periodic Human Motion Skills.

Learning an autonomous dynamic system (ADS) encoding human motion rules has been shown as an effective way for human motion skills transfer. However, most existing approaches focus on goal-directed motion skills transfer, and the study on periodic motion skills transfer is rare. One popular approach for periodic motion skills transfer is learning periodic dynamic movement primitive (DMP); however, periodic DMP is sensitive to spatial disturbances due to the introduction of the phase parameters. To solve this issue, this brief presents a novel approach to learn an ADS with a stable limit cycle without introducing phase parameters. First, a data-driven Lyapunov function (energy function) is learned, such that one of its level surfaces is consistent with periodic human demonstration trajectories. Then, an ADS is learned by sequentially solving energy function-related constrained optimization problems. With a proper design of constraint functions, we can ensure that the trajectory generated by the ADS will converge to an energy function-level surface, of which the shape is similar to periodic human demonstration trajectories. Experiments are conducted to show the effectiveness of the proposed approach (PA).

Alleviating limit cycling in training GANs with an optimization technique

Alleviating limit cycling in training GANs with an optimization technique

Bifurcation Analysis of a Holling–Tanner Model with Generalist Predator and Constant-Yield Harvesting

In this paper, we introduce constant-yield prey harvesting into the Holling–Tanner model with generalist predator. We prove that the unique positive equilibrium is a cusp of codimension 4. As the parameter values change, the system exhibits degenerate Bogdanov–Takens bifurcation of codimension 4. Using the resultant elimination method, we show that the positive equilibrium is a weak focus of order 2, and the system undergoes degenerate Hopf bifurcation of codimension 2 and has two limit cycles. By numerical simulations, we demonstrate that the system exhibits homoclinic bifurcation and saddle–node bifurcation of limit cycles as the parameters are varied. The main results show that constant-yield prey harvesting and generalist predator can lead to complex dynamic behavior of the model.

Control of large amplitude limit cycle of a multi-dimensional nonlinear dynamic system of a composite cantilever beam

For the first time, a control strategy based on Fuzzy Sliding Mode Control is implemented in the control of a large amplitude limit cycle of a composite cantilever beam in a multi-dimensional nonlinear form. In the dynamic model establishment of the investigated structure, the higher-order shearing effect is applied, as well as the second-order discretization. Numerical simulation demonstrates that a multi-dimensional nonlinear dynamic system of the investigated structure is demanded for accurate estimation of large amplitude limit cycle responses. Therefore, a control strategy is employed to effectively suppress such responses of the beam in multi-dimensional nonlinear form.

Pulsing in the Ahu‘ail‘au pond–spillway system during the 2018 Klauea eruption: a dynamical systems perspective

During the 2018 K $\unicode{x012B}$ lauea lower East Rift Zone eruption, lava from 24 fissures inundated more than 8000 acres of land, destroying more than 700 structures over three months. Eruptive activity eventually focused at a single vent characterized by a continuously fed lava pond that was drained by a narrow spillway into a much wider, slower channelized flow. The spillway exhibited intervals of ‘pulsing’ behaviour in which the lava depth and velocity were observed to oscillate on time scales of several minutes. At the time, this was attributed to variations in vesiculation originating at depth. Here, we construct a toy fluid dynamical model of the pond–spillway system, and present an alternative hypothesis in which pulsing is generated at the surface, within this system. We posit that the appearance of pulsing is due to a supercritical Hopf bifurcation driven by an increase in the Reynolds number. Asymptotics for the limit cycle near the bifurcation point are derived with averaging methods and compare favourably with the cycle periodicity. Because oscillations in the pond were not observable directly due to the elevation of the cone rim and an obscuring volcanic plume, we model the observations using a spatially averaged Saint-Venant model of the spillway forced by the pond oscillator. The predicted spillway cycle periodicity and waveforms compare favourably with observations made during the eruption. The unusually well-documented nature of this eruption enables estimation of the viscosity of the erupting lava.

Nonsmooth dynamics of a Filippov predator–prey ecological model with antipredator behavior

This article proposes a class of nonsmooth Filippov pest–predator ecosystems with intermittent control strategies based on the pest’s antipredator behavior. aiming to investigate the influence of control strategies and switching thresholds on pest control. First, a comprehensive theoretical analysis of various equilibria within the Filippov system is undertaken, emphasizing the presence and stability of sliding mode dynamics and pseudoequilibrium. Secondly, through numerical simulations, the article discusses boundary-focus, boundary-node, and boundary-saddle bifurcation. Finally, the nonexistence of limit cycles in the Filippov system is theoretically studied. The research indicates that the solution trajectories of the model ultimately stabilize either at the real equilibria or at pseudoequilibrium on the model’s switching surface. Moreover, when the model has multiple coexisting real equilibrium and pseudoequilibrium, the pest-control strategy is correlated with the initial density of both the pest and the predator population.

Hunting stability control of high-speed bogie based on active yaw damper

Hunting stability is an inherent property of railway vehicles that determines the operational speed. This paper establishes a half-vehicle model with nonlinear wheel/rail equivalent conicity and interactive forces. Additionally, the dynamic performances of vehicle under various levels of wheel wear with passive suspension are compared and analyzed. Importantly, the investigation delves into the hunting stability of the vehicle system, employing both linear and nonlinear control approaches. The results demonstrate a notable reduction in critical speed during the end-worn period with a passive suspension. However, this reduction can be substantially countered through the application of active control, resulting in a significant speed increase. The implementation of stiffness control raises the frequency of limit cycles, whereas damping control serves to diminish it. Notably, an appropriate linear cubic stiffness control effectively mitigates the amplitudes of limit cycles during instances of instability. Moreover, the control strategy derived from the simplified model is extended to enhance the stability of the entire vehicle system. The research findings hold the potential to offer a promising strategy for the active control of high-speed vehicles, particularly during periods of wheel wear.

Four crossing limit cycles of a family of discontinuous piecewise linear systems with three zones separated by two parallel straight lines

Four crossing limit cycles of a family of discontinuous piecewise linear systems with three zones separated by two parallel straight lines

IMPACT OF HUNTING COOPERATION AND FEAR EFFECT IN A GENERALIST PREDATOR–PREY MODEL

Predator foraging facilitation or hunting cooperation and the antipredator behavior of prey are essential mechanisms in evolutionary biology and ecology and may strongly influence the predator–prey dynamics. In a real-world scenario, this behavioral tendency is well documented, but less is known about how it could affect the dynamics between predator and prey. Here, we investigate the impact of the fear of predator on prey and the hunting cooperation in predator on the predator–prey dynamics, where the predator is assumed to be of generalist type. We observe that without fear, even with the high level of hunting cooperation, both populations may coexist, though the increasing level of hunting cooperation reduces the prey density at coexistence equilibrium. Moreover, increasing level of fear also destabilizes the system with and without hunting cooperation. Further, in the presence of hunting cooperation and fear effect, the model shows three different types of bistability phenomena: bistability between two coexisting equilibria, bistability between coexisting equilibria and prey-free equilibrium, and bistability between stable limit cycle and coexisting equilibria. In addition, saddle-node, Hopf, transcritical bifurcation of codimension one, Bautin (generalized Hopf), Bogdanov–Takens, and cusp bifurcation of codimension two are observed.

Influence of the magnetic flux on the dynamics of a self-sustaining System: Analytical, numerical and analogical investigations

Abstract This paper investigates the nonlinear dynamics of a ferroelectric enzyme-substrate
reaction modeled by \textcolor{red}{the} bi-rhythmic \emph{van der Pol} oscillator coupled to \textcolor{red}{the} magnetic
flux. We derive the equilibrium points and study their stability.
We analyze some bifurcation structures and the variation of the Lyapunov exponents.
The phenomena of symmetric attractors and the
 anti-monotonicity are observed. \textcolor{red}{By} increasing the magnetic flux,
 \textcolor{red}{we find that the equilibrium points are stable},
 tends to control chaotic regimes, and affects regular and quasi-regular ones.
 As the magnetic flux increases, the amplitude of the oscillations around the equilibrium
 points decreases and the \textcolor{red}{amplitude of the}
 limit cycles at the Hopf bifurcation \textcolor{red}{tends} to disappear.
 Further \textcolor{red}{increasing the magnetic flux gives} rise to chaotic dynamics.
 The electrical circuit and analogical simulations are \textcolor{red}{derived} using the PSpice software.
\textcolor{red}{The agreement between analogical and numerical results is acceptable}.

The Limit Cycles for a Class of Non-autonomous Piecewise Differential Equations

The Limit Cycles for a Class of Non-autonomous Piecewise Differential Equations

Estimates for the Number of Limit Cycles in Discontinuous Generalized Liénard Equations

Estimates for the Number of Limit Cycles in Discontinuous Generalized Liénard Equations

Non-Linear Dynamics of Simple Elastic Systems Undergoing Friction-Ruled Stick–Slip Motions

The stick–slip phenomenon is a jerking motion that can occur while two objects slide over each other with friction. There are several situations in which this phenomenon can be observed: between the slabs of the friction dampers used to mitigate vibrations in buildings, as well as between the components of the base isolation systems used for seismic protection. The systems of this kind are usually designed to work in a smooth and flawless manner, but under particular conditions undesired jerking motions may develop, yielding complex dynamic behavior even when only a few degrees of freedom are involved. A simplified approach to the problems of this kind leads to the mechanical model of a rigid block connected elastically to a rigid support and at the same time with friction to a second rigid support, both the supports having a prescribed motion. Despite the apparent simplicity of this model, it is very useful for studying important features of the non-linear dynamics of many physical systems. In this work, after a suitable formulation of the problem, the equations of motion are solved analytically in the sticking and sliding phases, and the influence of the main parameters of the system on its dynamics and limit cycles is investigated and discussed.

Nonlinear vibrations, bifurcations and chaos of piezoelectric composite lattice sandwich plate with four simply supported edges

This paper is devoted to investigate the nonlinear dynamics, bifurcation and chaos of the piezoelectric composite lattice sandwich plate with 1:3 internal resonance. Galerkin method is used to discretize the nonlinear partial differential governing equations of motion into the ordinary differential equations. The multi-scale method is used to obtain the modulation equations of the piezoelectric composite lattice sandwich plate in the polar coordinates and Cartesian coordinates in the primary and parametric resonances, respectively. Using Newton-Raphson method, the bifurcation diagrams of the steady-state equilibrium solution are obtained for the modulation equation with the change of the system parameters. The stability of the steady-state equilibrium solution is analyzed for the modulation equation. The existence of the static bifurcations, such as the saddle-node bifurcation, pitchfork bifurcation and Hopf dynamic bifurcations are found. The complex nonlinear jump phenomena caused by the existence of the multiple nonlinear steady-state solution regions are analyzed in detail. Fourth-order Runge-Kutta method is used to continue tracing the nonlinear periodic solution of the modulation equation in Cartesian coordinates. The time-history diagram, phase portrait and Poincaré map are obtained for the piezoelectric composite lattice sandwich plate in the nonlinear resonance. The dynamic process of the nonlinear periodic solution is described for the modulation equation from the limit cycle oscillation to the chaotic attractor in detail.

Limit Cycles and Local Bifurcation of Critical Periods in a Class of Switching Z2 Equivariant Quartic System

In this paper, the limit cycles and local bifurcation of critical periods for a class of switching Z2 equivariant quartic system with two symmetric singularities are investigated. First, through the computation of Lyapunov constants, the conditions of the two singularities to become the centers are determined. Then, we prove that there are at most 18 limit cycles with a distribution pattern of 9-9 around the two symmetric singular points of the system. Numerical simulation is conducted to validate the obtained results. Furthermore, by calculating the period constants, we determine the conditions for the critical point to be a weak center of finite order. Finally, the number of local critical periods that bifurcate from the equilibrium point under the center conditions is discussed. This study presents the first example of a quartic switching smooth system with 18 limit cycles and 4 local critical periods bifurcating from two symmetric singular points.

Self-assembled nano-MnS@N,P dual-doped lignite based carbon as high-performance sodium-ion batteries anode

Manganese sulfide (MnS) is a suitable electrode material for use in sodium-ion batteries (SIBs) due to its high theoretical capacity and low cost. However, its practical application is still hampered by disadvantages such as large volume expansion during charging/discharging and limited cycle life. Therefore, reasonable structural design is of great importance in order to achieve an excellent sodium storage performance of the electrode materials for SIBs. Herein, nano-MnS@N,P dual-doped lignite based porous carbon (MnS@N,P-LPC) composite with a single MnS cube encapsulated in an N,P dual-doped lignite derived porous C is successfully prepared as superior anode material for SIBs. The multilayer pore structure and the combined effect of N and P dual-doping can enhance electrochemical performance by supplying more sodium storage sites. After 200 cycles at 0.1 A g−1, the composite complex produced a maximum capacity of 519 mA h g−1 and demonstrated a capacity of 392.5 mA h g−1 even at 1.6 A g−1. It exhibits outstanding cycling stability and superior rate performance. This excellent electrochemical performance is ascribed to the presence of the multilayer pore N,P dual-doped C skeleton, which effectively addresses the volume expansion of MnS particles during charging/discharging and facilitates efficient transport of ions and electrons through an effective conductive network.