Stability is understood as the stability of a one-parameter orbit of stationary rotation of a vortex system, that is, orbital stability. Instability of stationary rotation is interpreted as a spectral instability.
Based on the ideas of statistical topography, the stochastic boundary value problem of the occurrence of anomalously large structures on the sea surface is considered in [1]. The boundary condition on the sea surface is considered as a closed stochastic quasilinear equation in the kinematic approximation. Starting from the stochastic Liouville equation [2], under the assumption of the random nature of the hydrodynamic velocity field within the diffusion approximation, an equation is obtained for a single point in space and simultaneous in time joint probability density of the fields of sea surface elevation and its gradient, taking into account stochastic topographic inhomogeneities of the seabed. It is shown that, for the deep sea, clustering of the field of the gradient modulus of the sea surface occurs with a probability of one, which corresponds to the occurrence of such rare events as anomalously large structures and deep depressions on the sea surface in almost all realizations of the stochastic velocity field. In this paper, we test the assumption that clusters of the gradient modulus lead to the occurrence of anomalously large surface elevations. Numerical modeling of a stochastic quasilinear equation for surface elevation demonstrated that anomalous structures do indeed arise. The mechanism by which such structures arise was also analyzed.
In this paper, the stability problem of trivial equilibrium positions is addressed for two classes of nonlinear mechanical systems with unbounded delays. It should be noticed that the delay-dependent terms in the equations considered can be interpreted as integral parts of proportional-integral-differential (PID) controllers. It is known that such terms can improve the characteristics of transient processes and provide the damping of undesirable vibrations. First, assuming that strongly nonlinear dissipative and positional forces acting on the system are homogeneous of different homogeneity degrees, an original approach to the Lyapunov – Krasovskii functional construction is proposed. With the aid of this functional, it is proved that the asymptotic stability of the auxiliary delay-free system implies the asymptotic stability for the original time-delay system. Next, we study a mechanical system that is subject to linear gyroscopic forces in addition to nonlinear homogeneous dissipative and positional forces. To derive asymptotic stability conditions for such a system, a special technique to the application of the decomposition method is developed. The investigated system composed of the second-order equations is represented as a complex system describing interaction of two isolated subsystems consisting of the first-order equations. This form of the decomposition is an extension of the classical one for delay-free linear gyroscopic systems. However, in the linear case, the asymptotic stability was guaranteed only under an additional restriction on the system. It was assumed that there is a large positive parameter at the vector of the gyroscopic forces. In the present contribution, it is proved that, for systems with strongly nonlinear dissipative and positional forces, such a constraint is not
The paper handles the problem of automatic path tracking by an autonomous off-road eight-wheeled robotic vehicle subject to uncertain longitudinal and lateral wheel slips. A sliding mode guidance law is proposed that solves this problem, explicitly takes into account the constraints on the control inputs, and ensures tracking of any path whose contortion lies within given bounds. Necessary conditions for the mission feasibility are first established. Under slight and partly inevitable enhancement of them, nonlocal convergence and robust stability of the proposed guidance law are theoretically justified. In doing so, the slipping effects are treated as bounded uncertainties. Simulation results confirm the applicability and performance of the control law.
This study numerically investigates boundary layer separation criteria for flows over small surface irregularities on a flat plate at high Reynolds numbers using the double-deck model framework. By solving the Prandtl equations with self-induced pressure, critical amplitude values (i. e., the height of a hump or the depth of a pit) separating attached laminar flow from separated flow with a stationary vortex are determined for Gaussian-shaped irregularities. The results show that separation begins at points of zero curvature of the streamlined surface. Importantly, no geometric parameter (such as maximum curvature or tangent angle) remains invariant along the obtained critical amplitude values, refuting prior hypotheses of a universal critical curvature of the irregularity. Furthermore, the critical amplitude values differ for humps and pits of identical shape. Thus, a separation criterion based solely on the geometry of the irregularity is not attainable for arbitrary shapes.
The effectiveness of the proposed approach is illustrated through the stabilization of a periodic trajectory in the Pendubot system, where the second link oscillates around the horizontal position.
Motivated by problems in robotic interaction control, we present a model-based method for robust orbital stabilization. Our objective is to design a time-invariant feedback law for a model of a nonlinear system, or for its digital twin, that makes the distance between its solutions and a planned periodic trajectory decay exponentially. The method uses transverse coordinates, which are functions that vanish on the orbit and remain independent in the first-order approximation. We regulate the linearized dynamics of transverse coordinates to zero. The novelty of the method is that it replaces the projection-based modification of a stabilizing time-periodic controller with a combination of a time-invariant control law for a subsystem and a discontinuous sliding-mode term. The sliding-mode part forces the state to a switching
This study introduces an approach for open-loop geometric calibration of industrial manipulators that integrates three widely used kinematic formulations: Denavit – Hartenberg (DH), Product of Exponentials (POE), and Complete Parametric Continuous (CPC) models. The proposed method focuses on identifying optimal measurement configurations within a local, spatially narrow workspace, which is a common operational scenario in industrial robotic applications. To achieve high calibration efficiency, a linear approximation model was employed, and the measurement configurations were selected using the D-optimality criterion to maximize parameter identifiability. Experimental validation was performed on an ABB IRB 1600 (10/1.45) manipulator equipped with an API Radian Laser Tracker EMSD3 measurement system, providing a linear accuracy of $0.7$ $\mu$m per meter. The system was equipped with a Smart Track Sensor offering an orientation accuracy of $0.005$ degrees. Independent measurement sets were used for experiments for each model in several variations to identify the best parameter estimates that can be used in the future for this robot. The results demonstrate a substantial enhancement in calibration accuracy. Specifically, applying the POE-based identification procedure within the narrow workspace region reduced the average error in the Tool Center Point (TCP) position by a factor of $22$ when compared to the uncalibrated nominal parameters, with the mean error decreasing from $2.852$ mm to $0.13$ mm. Additionally, the repeatability analysis showed that the standard deviation of TCP position errors across repeated measurements did not exceed $0.007$ mm. These results confirm that the proposed approach ensures high calibration precision and robustness suitable for high-accuracy industrial robotic tasks.
This paper investigates the topological conjugacy of skew products defined on the total space of locally trivial fiber bundles. We prove that under certain restrictions on the chaotic dynamics of the homeomorphism on the base, namely, the inability to choose a closed connected subset of the supporting manifold which is also an invariant set of the base homeomorphism (indecomposable chaos), any homeomorphism conjugating the skew products must also be the skew product. Thus, the topological conjugacy of homeomorphisms on the base spaces is a necessary condition for the conjugacy of such skew products. As a consequence, we establish that the direct products of indecomposably chaotic homeomorphisms with homeomorphisms having finite $\omega$-limit sets are topologically conjugate if and only if they are conjugate componentwise.
Analytically the following results were obtained: existence of optimal trajectories was proved, absence of abnormal trajectories was shawn, a Hamiltonian system for normal extremals was derived, its symmetry and Maxwell points was described. Symbolically all polynomial integrals of the Hamiltonian system of degree not greater than $54$ were described. Numerically the optimal synthesis was constructed.
Sliding mode control (SMC) provides strong robustness against matched disturbances. Among SMC schemes, the super-twisting algorithm (STA) provides continuous control action and finite-time convergence for systems of relative degree one. However, in real applications, actuator imperfections and unmodeled dynamics prevent true finite-time convergence and cause high-frequency oscillations called chattering. The chattering effect can be mitigated by tuning control parameters, for instance, through frequency-domain analysis. Yet, most existing methods rely on simplified system models, limiting their applicability to complex systems. This work proposes a generalized frequency-domain framework for STA gain tuning based on a first-order plus dead time (FOPDT) model. The method identifies FOPDT parameters from the reaction curve and employs describing function and harmonic balance analysis to predict analytically the chattering amplitude, average energy, and
This paper deals with the problem of autonomously driving a mobile robot to the source of an unknown planar scalar environmental field based on pointwise measurement of its value at the current location of the robot. An underactuated nonholonomic Dubins car-like robot with a bounded control range is handled. A novel bioinspired algorithm based on hybrid control paradigm is proposed and justified via mathematically rigorous results on nonlocal convergence. This algorithm assumes the use of only two field sensors and does not require measuring or estimating the field gradient. It is also shown that, in the case of an extended source represented by a curve, the proposed algorithm endows the robot with the ability to track this curve, thereby exhibiting its layout. Theoretical results are confirmed with computer simulation experiments. The proposed control law is efficient both computationally and in terms of the resulting motion.
Physics-informed neural networks (PINNs) have demonstrated great promise in solving partial differential equations without labeled data, yet their performance often deteriorates for highly nonlinear systems — particularly steady non-Newtonian flows governed by power-law or yield-stress rheologies. In this work, we present a systematic comparative study of PINN training strategies for Couette flow of pseudoplastic fluids between coaxial cylinders, governed by the Ostwald – de Waele and Herschel – Bulkley constitutive laws. We evaluate three established differential-form PINN variants — baseline (fixed loss weights), curriculum learning, and adaptive loss weighting — and introduce a novel variational PINN (vPINN) derived from the dissipation potential of Herschel – Bulkley fluids. Crucially, the proposed vPINN embeds the principle of minimum dissipation directly into the loss functional via the analytically integrated shear-dependent potential $\Phi(\dot{\gamma})=q_{0}\dot{\gamma}^{2}+\frac{2q_{1}\dot{\gamma}^{z+1}}{z+1}$, thereby enforcing physics through a variational principle rather than residual minimization. Using an exact analytical solution as ground truth, we benchmark all models on velocity and pressure reconstruction across varying gap geometries $\Bigl{(}$dimensionless parameter $\gamma=\frac{R_{1}}{R_{2}-R_{1}}\Bigr{)}$. While adaptive weighting improves pressure recovery by $6$–$8\%$ (MAE, $L_{\infty}$, RSD), all differential PINNs exhibit nearly identical — and limited — accuracy for velocity prediction, with no benefit from curriculum scheduling. In contrast, the vPINN achieves a substantial and consistent gain in velocity accuracy: for $\gamma=4.0$, RSD drops from $1.09\%$ (all PINNs) to $0.85\%$ ($-22\%$); for the widest gap ($\gamma=0.67$), RSD falls
This article presents the results of a study exploring sensorimotor integration in upper-limb prostheses through the development of a prototype noninvasive adaptive control system for a bionic hand prosthesis. The study focuses on creating sensory feedback that replicates the properties of biofeedback with a focus on signals from the fingertips, unlike most studies that focus on recognizing patterns in electromyogramm (EMG) signals. The prototype integrates a two-component sensor system into a bionic hand prosthesis model with five independent servomotors. This system consists of a surface EMG sensor, which detects muscle activation intent, and thin-film resistive pressure sensors embedded in the fingertips. The algorithm processes normalized EMG and pressure data in real time using a programmable microcontroller, implementing closed-loop grip force adjustment. Key developments include dynamic calibration using the RMS signal envelope, multi-input PID controllers (tuned using the Ziegler – Nichols method)
This paper addresses the problem of orbital stabilization of a periodic walking gait for a model or a digital twin of a three-link planar biped robot with a single actuator. A Lyapunov equation-based approach is proposed for the synthesis of a stabilizing controller for the corresponding impulsive mechanical system. The method ensures exponential vanishing of transverse coordinates, defining deviations from the nominal periodic trajectory, by solving Lyapunov matrix inequalities, which provide sufficient conditions for orbital stability of the closed-loop dynamics in the nominal case of no disturbances. The proposed approach allows systematic
Every discrete dynamical system (cascade) generated by a homeomorphism induces a continuous dynamic system (flow) — a suspension. However, not every flow is equivalent to a suspension over a cascade, a necessary and sufficient condition for this is the existence of a global section for the flow. In the case of the existence, the flow is equivalent to a suspension over a Poincaré map on the global section. The basis of the topological dynamics is the topological classification of cascades (flows) up to a conjugacy (equivalence) realized by a homeomorphism that sends the trajectories of one system into the trajectories of another while preserving the direction of the motion. The paper explores the deep relationship between a homeomorphism and its suspension. The core question is: if two of such suspensions are topologically equivalent, does it mean the original homeomorphisms were topologically conjugate? Usually, the answer is “no”, and here a natural question arises about the relationship between the invariants of the topological conjugacy and the topological equivalence of suspensions for homologically reducible homeomorphisms. In this paper we identify the exact boundary where the answer becomes “yes”. We find conditions under which the topological conjugacy of the homeomorphisms on manifolds is tantamount to the equivalence of the suspensions over them. The found condition, called homological irreducibility, consists of the absence of the eigenvalue $1$ in the first homology group action. It allows us to distinguish some classes of homologically irreducible homeomorphisms. In particular, to give an exhaustive description of them in the class of homeomorphisms of surfaces with nonnegative Euler characteristic.
The spatiotemporal dynamics of coupled nonlinear oscillators provide a natural substrate for the high-dimensional feature transformations required for complex pattern classification. We investigate this principle using an electronically implemented chain of FitzHugh\,--\,Nagumo neurons operating in the excitable regime. Static two-dimensional inputs are encoded by boundary pulse-train frequencies applied at the terminal nodes, driving the chain into reproducible, input-dependent voltage patterns distributed across space and time. We formalize this device as a physical kernel: a deterministic mapping from a low-dimensional input space to an explicit feature space derived from measured dynamics. To characterize the resulting kernel beyond accuracy alone, we introduce kernel tomography diagnostics based on the spectrum of the centered Gram matrix, including effective dimension and centered kernel alignment, and we benchmark against standard software classifiers defined directly on the input coordinates. Using controlled synthetic tasks with increasingly complex decision boundaries, we show that the physical-kernel representation supports strong performance with simple convex readouts and benefits from a~hierarchical design that combines local spectral features with coupling-aware observables such as internodal coherence. Furthermore, a lightweight readout incorporating quadratic feature interactions significantly improves performance by capturing mode interactions while retaining convex training. Finally, we demonstrate cross-task reuse of a single precomputed response map by applying nearest-grid lookup to a~real vowel formant dataset, achieving nontrivial separability without any hardware retuning. Overall, the proposed framework provides mechanistic insight into how forced excitable chains induce task-relevant feature geometry and offers principled guidance for designing and evaluating neuromorphic hardware as physical kernels for static classification.
While a previously proposed method for estimating inertial manifold dimension, based on explicitly computing angles between pairs of covariant Lyapunov vectors (CLVs), employs efficient algorithms, it remains computationally demanding due to its substantial resource requirements. In this work, we introduce an improved method to determine this dimension by analyzing the angles between tangent subspaces spanned by the CLVs. This approach builds upon a fast numerical technique for assessing chaotic dynamics hyperbolicity. Crucially, the proposed method requires significantly less computational effort and minimizes memory usage by eliminating the need for explicit CLV computation. We test our method on two canonical systems: the complex Ginzburg – Landau equation and a diffusively coupled chain of Lorenz oscillators. For the former, the results confirm the accuracy of the new approach by matching prior dimension estimates. For the latter, the analysis demonstrates the absence of a low-dimensional inertial manifold, highlighting a complex regime that merits further investigation. The presented method offers a practical and efficient tool for characterizing attractors in infinite-dimensional dynamical systems.
On a closed orientable surface, we consider the set of axiom A diffeomorphisms whose nonwandering sets consist of connected one-dimensional expanding attractors and contracting repellers (any attractor/repeller is locally homeomorphic to the product of segment and Cantor set). This set consists of $\Omega$-stable diffeomorphisms and structurally unstable diffeomorphisms. We classify such diffeomorphisms up to the global conjugacy on its nonwandering sets.
The planar restricted problem of three bodies moving under gravitational attraction is considered. The motions close to the Lagrange libration points are studied. The orbital eccentricity of the smaller of the two main attracting bodies and their mass ratio are chosen as problem parameters. A linear canonical transformation that is $2\pi$-periodic in true anomaly is obtained analytically up to the 4th degree of eccentricity inclusive, which reduces the Hamiltonian function of the linearized equations of perturbed motion to a real normal form corresponding to two harmonic oscillators independent of each other. The oscillations frequencies and the coefficients in the transformation are obtained explicitly in terms of the problem parameters.
