Stability is among the most important concepts in dynamical systems. Local stability is well-studied, whereas determining the ‘global stability’ of a nonlinear system is very challenging. Over the last few decades, many different ideas have been developed to address this issue, primarily driven by concrete applications. In particular, several disciplines suggested a web of concepts under the headline ‘resilience’. Unfortunately, there are many different variants and explanations of resilience, and often, the definitions are left relatively vague, sometimes even deliberately. Yet, to allow for a structural development of a mathematical theory of resilience that can be used across different areas, one has to ensure precise starting definitions and provide a mathematical comparison of different resilience measures. In this work, we provide a systematic review of the most relevant indicators of resilience in the context of continuous dynamical systems, grouped according to their mathematical features. The indicators are also generalised to be applicable to any attractor. These steps are important to ensure a more reliable, quantitatively comparable and reproducible study of resilience in dynamical systems. Furthermore, we also develop a new concept of resilience against certain nonautonomous perturbations to demonstrate how one can naturally extend our framework. All the indicators are finally compared via the analysis of a classic scalar model from population dynamics to show that direct quantitative application-based comparisons are an immediate consequence of a detailed mathematical analysis.

We investigate the existence of families of symmetric periodic solutions of second kind as continuation of the elliptical orbits of the two-dimensional Kepler problem for certain symmetric differentiable perturbations using Delaunay coordinates. More precisely, we characterize the sufficient conditions for its existence and its type of stability is studied. The estimate on the characteristic multipliers of the symmetric periodic solutions is the new contribution to the field of symmetric periodic solutions. In addition, we present some results about the relationship between our symmetric periodic solutions and those obtained by the averaging method for Hamiltonian systems. As applications of our main results, we get new families of periodic solutions for: the perturbed hydrogen atom with stark and quadratic Zeeman effect, for the anisotropic Seeligers two-body problem and to the planar generalized Størmer problem.

Flexibility is one of the most significant advantages of legged robots in unstructured environments. However, quadruped robots cannot interact with environments to complete some manipulation tasks. One effective way is to load a manipulation arm. In this paper, we exhibit a quadruped locomotion manipulation system (LMS) named HITPhanT. This system comprises a quadruped locomotion platform and a six-degree-of-freedom manipulation arm. Besides, when the LMS moves to a designated position for operation, it is necessary to constrain the foot contact points to avoid sliding. Therefore, the foot contact point is regarded as a spherical hinge. So the locomotion platform can be considered as a parallel mechanism. A hybrid kinematics model is established by considering the serial robotic arms connecting this parallel mechanism. Besides, the trajectory planning method, which improves the system’s manipulability in evaluating the system balance, is also proposed. Finally, corresponding experiments verify the overall system’s stabilization and algorithm’s effectiveness.

This paper presents a gait optimization method to generate the locomotion pattern for biped and discuss its stability. The main contribution of this paper is a newly proposed energy-based stability criterion, which permits the dynamic stable walking and could be straight-forwardly generalized to different locomotion scenarios and biped robots. The gait optimization problem is formulated subject to the constraints of the whole-body dynamics and kinematics. The constraints are established based on the modelling of bipedal hybrid dynamical systems. Following the whole-body modelling, the system energy is acquired and then applied to create the stability criterion. The optimization objective is also established on the system energy. The gait optimization is solved by being converted to a large-scale programming problem, where the transcription accuracy is improved via the spectral method. To further reduce the dimensionality of the large-scale problem, the whole-body dynamics is re-constructed. The generalization of the optimized gait is improved by the design of feedback control. The optimization examples demonstrate that the stability criterion naturally leads to a cyclic biped locomotion, though the periodicity was not previously imposed. Two simulation cases, level ground walking and slope walking, verify the generalization of the stability criterion and feedback control. The stability analyses are carried out by investigating the motions of centre of gravity and centre of pressure. It is revealed that if the tracked speed is above 0.3 m/s or the biped accelerates/climbs the slope, the stability criterion accomplishes the dynamic stable walking, where zero moment point criterion is not strictly complied.

We prove a stability theorem for spaces of smooth concordance embeddings. From it we derive various applications to spaces of concordance diffeomorphisms and homeomorphisms.

In this paper, we are interested in investigating notions of stability for generalized linear differential equations (GLDEs). Initially, we propose and revisit several definitions of stability and provide a complete characterization of them in terms of upper bounds and asymptotic behaviour of the transition matrix. In addition, we illustrate our stability results for GLDEs to linear periodic systems and linear impulsive differential equations. Finally, we prove that the well-known definitions of uniform asymptotic stability and variational asymptotic stability are equivalent to the global uniform exponential stability introduced in this article.

Interest in organic agriculture worldwide is growing and is mainly supported by a strong consumer interest. In the literature, a lot of attention has been paid to comparing organic and conventional systems, on studying the yield gap between the two systems and, how to reduce it. In the present work, based on the results from Polish organic and conventional series of field trials carried out in 2019–2021, organic and conventional systems were compared in terms of potato tuber yield. Moreover, we propose a Bayesian approach to the variety × environment × system data set and describe Bayesian counterparts of two stability measures. Using this methodology, we identify the most stable and highest tuber yielding varieties in the Polish potato organic and conventional series of field trials. It is shown that the tuber yield in the organic system was approx. 44% lower than the tuber yield in the conventional system. Moreover, varieties Tajfun and Otolia were the most stable and highest yielding varieties in the organic system, whereas in the conventional system, the variety Jurek was the most stable and highest yielding variety among the tested varieties. In the present work, the use of the Bayesian approach allowed us to calculate the probability that the mean of a given variety in given system exceeds the mean of control varieties in that system.

The stability and uniformity of the following gas mixtures: 90% argon; 85% argon and 15% carbon dioxide (CO2); 70% argon and 30% CO2; 98% nitrogen (N2); 92% N2 and 8% CO2; 90% N2 and 10% CO2; 85% N2 and 15% CO2; 80% N2 and 20% CO2; 70% N2 and 30% CO2; and 90% CO2 by volume in atmospheric air were assessed in a commercial dip-lift stunning system when the cradle was either stationary or in motion. The gas mixtures of 90% argon, 85% argon and 15% CO2, 70% argon and 30% CO2, 85% N2 and 15% CO2, 80% N2 and 20% CO2, 70% N2 and 30% CO2 and 90% CO2 by volume in atmospheric air could be sustained in a commercial dip-lift stunning system. The stability of the gas mixtures 92% N2 and 8% CO2, and 90% N2 and 10% CO2 by volume in atmospheric air were lower than in the previous cases. On the other hand, an N2 concentration higher than 94% by volume in atmospheric air could not be sustained in the stunning system. In addition, gas mixtures of argon and CO2 showed a higher stability than gas mixtures of N2 and CO2. The uniformity at different levels inside the pit (defined as the capacity of the gas to maintain its concentration constant at different levels inside the pit) was higher in 90% argon, or argon and CO2 mixtures and N2 and CO2 mixtures than in 90% CO2. This fact ensures that for the whole time the animals are inside the pit, the same conditions are applied, which is not the case for 90% CO2.

A widely used measure of individual propensity to utilize analytic processing is the Cognitive Reflection Test (CRT), a set of math problems with intuitively compelling but incorrect answers. Here, we ask whether scores on this measure are temporally stable. We aggregate data from 11 studies run on Amazon Mechanical Turk in which the Cognitive Reflection Test (CRT) was administered and identify N = 3,302 unique individuals who completed the CRT two or more times. We find a strong correlation between an individual’s first and last CRT performance, r = .806. This remains true even when constraining to data points separated by over 2 years, r = .755. Furthermore, we find that CRT scores from one timepoint correlated negatively with belief in God and social conservatism from the other timepoint (and to a similar extent as scores gathered at the same timepoint). These results show that CRT scores are stable over time, and – given the stable relationship between CRT and religious belief and ideology – provide some evidence for the stability of analytic cognitive style more generally.

The physics behind the transitions of natural Samaras, or the bio-inspired counterparts, to steady autorotation has been unclear. Theoretical and experimental investigations explore the inertial and aerodynamic characteristics required to guarantee stable transitions of an artificial Samara-like decelerator from chaotic tumbling motions to azimuthal autorotation. A non-dimensional inertial criterion is proposed, which is in accord with experiments.

The three-body problem is famously chaotic, with no closed-form analytical solutions. However, hierarchical systems of three or more bodies can be stable over indefinite timescales. A system is considered hierarchical if the bodies can be divided into separate two-body orbits with distinct time and length scales, such that one orbit is only mildly affected by the gravitation of the other bodies. Previous work has mapped the stability of such systems at varying resolutions over a limited range of parameters, and attempts have been made to derive analytic and semi-analytic stability boundary fits to explain the observed phenomena. Certain regimes are understood relatively well. However, there are large regions of the parameter space which remain unmapped, and for which the stability boundary is poorly understood. We present a comprehensive numerical study of the stability boundary of hierarchical triples over a range of initial parameters. Specifically, we consider the mass ratio of the inner binary to the outer third body ( $q_\mathrm{out}$ ), mutual inclination (i), initial mean anomaly and eccentricity of both the inner and outer binaries ( $e_\mathrm{in}$ and $e_\mathrm{out}$ respectively). We fit the dependence of the stability boundary on $q_\mathrm{ out}$ as a threshold on the ratio of the inner binary’s semi-major axis to the outer binary’s pericentre separation $a_\mathrm{in}/R_\mathrm{p, out} \leq 10^{-0.6 + 0.04q_\mathrm{out}}q_\mathrm{out}^{0.32+0.1q_\mathrm{out}}$ for coplanar prograde systems. We develop an additional factor to account for mutual inclination. The resulting fit predicts the stability of $10^4$ orbits randomly initialised close to the stability boundary with $87.7\%$ accuracy.

In this paper, we consider the existence and stability of singular patterns in a fractional Ginzburg–Landau equation with a mean field. We prove the existence of three types of singular steady-state patterns (double fronts, single spikes, and double spikes) by solving their respective consistency conditions. In the case of single spikes, we prove the stability of single small spike solution for sufficiently large spatial period by studying an explicit non-local eigenvalue problem which is equivalent to the original eigenvalue problem. For the other solutions, we prove the instability by using the variational characterisation of eigenvalues. Finally, we present the results of some numerical computations of spike solutions based on the finite difference methods of Crank–Nicolson and Adams–Bashforth.