Invariants Of Motion Research Articles

Integrable symplectic maps with a polygon tessellation

Identifying integrable dynamics remains a formidable challenge, and despite centuries of research, only a handful of examples are known to date. In this article, we explore a distinct form of area-preserving (symplectic) mappings derived from the stroboscopic Poincaré cross section of a kicked rotator—an oscillator subjected to an external force periodically switched on in short pulses. The significance of this class of problems extends to various applications in physics and mathematics, including particle accelerators, crystallography, and studies of chaos. Notably, Suris's theorem constrains the integrability within this category of mappings, outlining potential scenarios with analytic invariants of motion. In this paper, we challenge the assumption of the analyticity of the invariant by exploring piecewise linear transformations on a torus (T2) and associated systems on the plane (R2), incorporating arithmetic quasiperiodicity and discontinuities. Introducing a new automated technique, we discovered previously unknown scenarios featuring polygonal invariants that form perfect tessellations and, moreover, fibrations of the plane or torus. This work reveals a novel category of planar tilings characterized by discrete symmetries that emerge from the invertibility of transformations and are intrinsically linked to the presence of integrability. Our algorithm relies on the analysis of the Poincaré rotation number and its piecewise monotonic nature for integrable cases, contrasting with the noisy behavior in the case of chaos, thereby allowing for clear separation. Some of the newly discovered systems exhibit the peculiar behavior of “integrable diffusion,” characterized by infinite and quasirandom hopping between tiles while being confined to a set of invariant segments. Finally, through the implementation of a smoothening procedure, all mappings can be generalized to quasi-integrable scenarios with suppressed volume occupied by chaotic trajectories, thereby opening doors to potential practical applications. Published by the American Physical Society 2024

Snap-back repellers and chaos in a class of discrete-time memristor circuits

In the last decade the flux-charge analysis method (FCAM) has been successfully used to show that continuous-time (CT) memristor circuits possess for structural reasons first integrals (invariants of motion) and their state space can be foliated in invariant manifolds. Consequently, they display an initial condition dependent dynamics, extreme multistability (coexistence of infinitely many attractors) and bifurcations without parameters. Recently, a new discretization scheme has been introduced for CT memristor circuits, guaranteeing that the first integrals are preserved exactly in the discretization. On this basis, FCAM has been extended to discrete-time (DT) memristor circuits showing that they also are characterized by invariant manifolds and they display extreme multistability and bifurcations without parameters. This manuscript considers the maps obtained via DT-FCAM for a circuit with a flux-controlled memristor and a capacitor and it provides a thorough and rigorous investigation of the presence of chaotic dynamics. In particular, parameter ranges are obtained where the maps have snap-back repellers at some fixed points, thus implying that they display chaos in the Marotto and also in the Li–Yorke sense. Bifurcation diagrams are provided where it is possible to analytically identify relevant points in correspondence with the appearance of snap-back repellers and the onset of chaos. The dependence of the bifurcation diagrams and snap-back repellers upon the circuit initial conditions and the related manifold is also studied.

Linear coupling of transverse betatron oscillations: Dynamic stability and invariants of motion

Based on the technique of the discrete one-turn transfer maps, the problem of linear coupling between horizontal and vertical betatron oscillations in an accelerator has been treated exactly and entirely in explicit form. The stability region in the fractional part of the horizontal and the vertical betatron tune space as a function of the linear coupling strength has been obtained, and the increment/decrement of the horizontal and the vertical betatron oscillations in the case of the linear sum resonance has been shown to be approximately equal to the half of the coupling strength. The normal form parameterization of the one-turn linear map with horizontal-to-vertical coupling has been developed in detail in the spirit of Edwards and Teng formalism. The motion in the normal mode in the new normal form coordinates is decoupled by implying that two independent Courant-Snyder invariants exist, which have been found explicitly. Published by the American Physical Society 2024

New Class of Discrete-Time Memristor Circuits: First Integrals, Coexisting Attractors and Bifurcations Without Parameters

The use of ideal memristors in a continuous-time (CT) nonlinear circuit is known to greatly enrich the dynamic behavior with respect to the memristorless counterpart, which is a crucial property for applications in future analog electronic circuits. This can be explained via the flux–charge analysis method (FCAM), according to which CT circuits with ideal memristors have for structural reasons first integrals (or invariants of motion, or conserved quantities) and their state space can be foliated in infinitely many invariant manifolds where they can display different dynamics. The paper introduces a new discretization scheme for the memristor which, differently from those adopted in the literature, guarantees that the first integrals of the CT memristor circuits are preserved exactly in the discretization, and that this is true for any step size. This new scheme makes it possible to extend FCAM to discrete-time (DT) memristor circuits and rigorously show the existence of invariant manifolds and infinitely many coexisting attractors (extreme multistability). Moreover, the paper addresses standard bifurcations varying the discretization step size and also bifurcations without parameters, i.e. bifurcations due to varying the initial conditions for fixed step size and circuit parameters. The method is illustrated by analyzing the dynamics and flip bifurcations with and without parameters in a DT memristor–capacitor circuit and the Poincaré–Andronov–Hopf bifurcation in a DT Murali–Lakshmanan–Chua circuit with a memristor. Simulations are also provided to illustrate bifurcations in a higher-order DT memristor Chua’s circuit. The results in the paper show that DT memristor circuits obtained with the proposed discretization scheme are able to display even richer dynamics and bifurcations than their CT counterparts, due to the coexistence of infinitely many attractors and the possibility to use the discretization step as a parameter without destroying the foliation in invariant manifolds.

Visual sensitivity to biological motion invariants in humans at birth: Comment on "Motor invariants in action execution and perception" by Francesco Torricelli et al.

Visual sensitivity to biological motion invariants in humans at birth: Comment on "Motor invariants in action execution and perception" by Francesco Torricelli et al.

RSO proper elements: Concept, methods, and application to maneuver detection

Asteroid families consist of swarms of fragments generated after energetic interasteroidal collisions in the distant past that resulted in the breakup of their progenitors. They are recognized by searching for clusters in the three-dimensional space of “proper elements”; parameters characterizing the asteroid orbits that are very close to fundamental invariants of motion, and thus keep a dynamical record of the initial proximity of the orbits generated by a catastrophic fragmentation event. In the circumterrestrial context, proper elements have recently been applied to the dynamical taxonomy of resident space objects (RSO) for the association of debris from breakup into its “parent” satellite. Here, we clarify, adapt, and extend this fundamental concept to space situational awareness for on-orbit maneuver/anomaly detection. RSO proper elements, being linked to the underlying dynamical structure of orbits, can provide a more robust metric within existing maneuver-detection algorithms, through assessment of the induced changes in these quasi invariants of the motion. We highlight the deeper connection of proper elements to classical frozen orbits in Earth-satellite dynamics and to secular elements in artificial satellite theories based on canonical perturbation theory and showcase several techniques for their numerical computation, including a newly proposed Ptolemy method; applicable to both the asteroid and space-debris domains. We apply a Bayesian online changepoint-detection (BOCD) algorithm in both mean- and proper-element space and compare their performances for orbit-maneuver detection.

Study of generalized regularized long wave equation via septic Hermite collocation method with Crank–Nicolson and SSP-RK43 schemes to capture the various solitons

The paper aims to develop a powerful numerical technique for the generalized regularized long wave equation. The proposed technique is based on the orthogonal collocation on finite element method and septic Hermite interpolation polynomials as basis functions called septic Hermite collocation method (SHCM). These basis functions are C3 continuous, ensuring the continuity of dependent variable and its first three derivatives throughout the solution range. Two schemes viz one is implicit Crank–Nicolson scheme and other is Runge–Kutta method of four stages and third-order (SSP-RK43) are used for time discretization and SHCM is used for space discretization. The stability analysis of SHCM with Crank–Nicolson is performed and the scheme is found to be unconditionally stable. The stability of SHCM with SSP-RK43 by CFL conditions is also studied and the method is found to be conditionally stable. The proposed methods are applied to three problems involving single solitary waves, the interaction of two solitary waves, and the evolution of solitons via the Maxwellian initial condition. The formation and behavior of soliton waves are studied numerically. To demonstrate the robustness of proposed technique, error norms and the invariants of motion, i.e., mass, momentum, and energy are calculated. The results are in good agreement with the exact ones and found to be better than the numerical solutions available in the literature. The order of convergence along spatial and temporal directions has been calculated numerically. It is observed that the order of convergence of the explicit SSP-RK43 is higher than the implicit Crank–Nicolson and SHCM with SSP-RK43 provides better accuracy than the SHCM with Crank–Nicolson. The SHCM is capable of displaying solitary wave collision. The proposed technique is simple, fast, efficient, and easy to implement. It does not require unnecessary integration or calculation of weight functions as the case of other numerical methods for instance Galerkin methods, spectral methods, B-spline finite element methods etc.

Solitary wave solutions of the MRLW equation using a spatial five-point stencil of finite difference approximation

This paper proposes a finite difference scheme with a three-level time and a five-point stencil in space to solve an initial boundary value problem for the MRLW equation. The scheme is shown to be marginally stable and convergent with a fourth-order convergence in the space direction and a second-order convergence in the time variable direction with regard to the maximum norm. The conservation properties of the proposed scheme are assessed using the three motion invariants for mass, momentum, and energy. To validate the theoretical results, numerical experiments are given for both single and interaction of two and three solitary waves.

Convergence of Neural Networks with a Class of Real Memristors with Rectifying Characteristics

The paper considers a neural network with a class of real extended memristors obtained via the parallel connection of an ideal memristor and a nonlinear resistor. The resistor has the same rectifying characteristic for the current as that used in relevant models in the literature to account for diode-like effects at the interface between the memristor metal and insulating material. The paper proves some fundamental results on the trajectory convergence of this class of real memristor neural networks under the assumption that the interconnection matrix satisfies some symmetry conditions. First of all, the paper shows that, while in the case of neural networks with ideal memristors, it is possible to explicitly find functions of the state variables that are invariants of motions, the same functions can be used as Lyapunov functions that decrease along the trajectories in the case of real memristors with rectifying characteristics. This fundamental property is then used to study convergence by means of a reduction-of-order technique in combination with a Lyapunov approach. The theoretical predictions are verified via numerical simulations, and the convergence results are illustrated via the applications of real memristor neural networks to the solution of some image processing tasks in real time.

One-Dimensional Quantum Systems with Ground State of Jastrow Form Are Integrable.

Exchange operator formalism describes many-body integrable systems using phase-space variables involving an exchange operator that acts on any pair of particles. We establish an equivalence between models described by exchange operator formalism and the complete infinite family of parent Hamiltonians describing quantum many-body models with ground states of Jastrow form. This makes it possible to identify the invariants of motion for any model in the family and establish its integrability, even in the presence of an external potential. Using this construction we establish the integrability of the long-range Lieb-Liniger model, describing bosons in a harmonic trap and subject to contact and Coulomb interactions in one dimension. We further identify a variety of models exemplifying the integrability of Hamiltonians in this family.

Automatic Registration for Panoramic Images and Mobile LiDAR Data Based on Phase Hybrid Geometry Index Features

The registration of panoramic images and mobile light detection and ranging (LiDAR) data is quite challenging because different imaging mechanisms and viewing angle differences generate significant geometric and radiation distortions between the two multimodal data sources. To address this problem, we propose a registration method for panoramic images and mobile LiDAR data based on the hybrid geometric structure index feature of phase. We use the initial GPS/IMU to transform the mobile LiDAR data into an intensity map and align the two images to complete registration. Firstly, a novel feature descriptor called a hybrid geometric structure index of phase (HGIFP) is built to capture the structural information of the images. Then, a set of corresponding feature points is obtained from the two images using the constructed feature descriptor combined with a robust false-match elimination algorithm. The average pixel distance of the corresponding feature points is used as the error function. Finally, in order to complete the accurate registration of the mobile LiDAR data and panoramic images and improve computational efficiency, we propose the assumption of local motion invariance of 3D–2D corresponding feature points and minimize the error function through multiple reprojections to achieve the best registration parameters. The experimental results show that the method in this paper can complete the registration of panoramic images and the mobile LiDAR data under a rotation error within 12° and a translation error within 2 m. After registration, the average error of rotation is about 0.15°, and the average error of translation is about 1.27 cm. Moreover, it achieves a registration accuracy of less than 3 pixels in all cases, which outperforms the current five state-of-the-art methods, demonstrating its superior registration performance.

Solution of the generalized regularized long-wave equation with optimal spline collocation technique and implicit Crank–Nicolson as well as explicit SSP-RK43 scheme

In this work, a generalized regularized long-wave equation is solved using the optimal cubic B-spline collocation method for space discretization. Two different approaches are followed for the time-domain discretization. One is the implicit Crank–Nicolson scheme and the other is strong explicit stability preserving Runge–Kutta method of four stages and third-order. Also, the stability analysis of the techniques is carried out and it is shown that the implicit technique is unconditional stable, whereas the explicit technique is conditionally stable. These methods are applied to three problems involving a single solitary wave, the interaction of two solitary waves, and the evolution of solitons via the Maxwellian initial condition. These equations possess three invariants of motion that are mass, momentum, and energy. The value of these invariants is calculated, which is found to remain preserved for a long time. To demonstrate the robustness of both the techniques, the and error norms are calculated.

Second-order approximation pseudo-rigid model of circular arc flexure hinge

The paper presents a novel pseudo-rigid model for describing the elasto-kinematic behaviour of circular arc flexure hinge subjected to a pure moment. The model is based on the kinematic study of flexure hinge large deflection using second-order motion invariants (polodes and inflection circle), which accurately describes the relative motion between the connected bodies. The proposed pseudo-rigid model has a single degree of freedom. It is an epicyclic arrangement with two rolling without slipping external circles with the same radius and a torsional spring with constant stiffness. Analytical formulas for computing the radius and the location of the two rolling circles and the rotational spring stiffness are deduced. The proposed model has been compared with analytical formulas and structural finite element models in different configurations. The results show very good accordance even for large deflections, confirming the model's effectiveness.

Sixth order Hermite collocation method for analysis of MRLW equation

The septic Hermite collocation method (SHCM) is proposed for solving the modified regularized long wave equations (MRLW), which may be utilized to simulate wide range of problems of applied sciences. SHCM is a hybrid of the orthogonal collocation and the finite element methods with septic Hermite interpolating polynomials as the basis function. The SHCM is used for the space discretization and Crank–Nicolson (CN) scheme for the time discretization. The stability analysis of the proposed method is carried out using Fourier series analysis and the SHCM is shown to be unconditionally stable. The proposed method is found to be six order convergent in space direction and second order convergence in time direction. To demonstrate the applicability of proposed algorithm, several important forms of the MRLW equation are solved. Three motion invariants: mass, energy, and momentum have been computed numerically and proved to match their exact values. The behavior of solitary waves, as well as wave undulations, have been graphically depicted. The SHCM is capable of displaying solitary waves collision. Also, error norms are computed at various time levels and compared with literature results. The outcome is found to be superior than the previous values.

New method to analyze the invariant manifolds of memristor circuits

The paper considers a wide class of circuits containing memristors, coupled and nonlinear capacitors and inductors, linear resistive multi-ports and independent voltage and current sources. A new method is proposed to analyze the invariants of motion and invariant manifolds of memristor circuits in this class. The method permits to show the existence of invariant manifolds, and analytically find their expressions, under very weak conditions, namely, when there exists the differential algebraic equations (DAE) describing the circuit and there is at least one hybrid representation of a linear resistive multi-port where memristors are connected. These conditions are satisfied by all memristor circuits in the class, except for pathological cases only. One salient feature of the method is that the dynamic problem of finding the invariant manifolds is brought back to a static problem involving the analysis of a linear resistive multi-port. In the one-memristor case, this boils down to the familiar problem of finding the Thévenin or Norton equivalent of a linear resistive one-port. The results in the paper improve previous work along several directions. The technique here developed is applicable to a broader class of memristor circuits including coupled nonlinear reactive elements and linear multi-port resistors. More importantly, it is applicable even if the state equation (SE) description of the circuit does not exist, as in the case where there are singular points as impasse points. Moreover, when the SE representation exists, additional assumptions introduced in previous work are not needed. The effectiveness of the method is illustrated via the application to selected examples.

Demonstration of eigen-to-projected emittance mapping for an ellipsoidal electron bunch

Phase-space partitioning offers an attractive path for the precise tailoring of complex dynamical systems. In beam physics, the proposed approach involves (i) producing beams with cross-plane correlations to control kinematic-moment invariants known as eigenemittances and (ii) mapping them to invariants of motion associated with given degrees of freedom via a decoupling transformation. Here we report on the direct experimental demonstration of the mapping of eigenemittances to transverse emittances for an electron beam. Measured phase space density confirms the generation of beams with asymmetric transverse emittance ratio $>200$ consistent with the initiated eigenemittance values. In addition, the bunches were produced via photoemission in the space-charge-driven expansion regime ultimately resulting in the generation of uniformly charged three-dimensional ellipsoidal bunches. The ellipsoidal character of the distributions is experimentally confirmed.

Pinocchio: A language for action representation

The development of a language of action representation is a central issue for cognitive robotics, motor neuroscience, ergonomics, sport, and arts with a double goal: analysis and synthesis of action sequences that preserve the spatiotemporal invariants of biological motion, including the associated goals of learning and training. However, the notation systems proposed so far only achieved inconclusive results. By reviewing the underlying rationale of such systems, it is argued that the common flaw is the choice of the ‘primitives’ to be combined to produce complex gestures: basic movements with a different degree of “granularity”. The problem is that in motor cybernetics movements do not add: whatever the degree of granularity of the chosen primitives their simple summation is unable to produce the spatiotemporal invariants that characterize biological motion. The proposed alternative is based on the Equilibrium Point Hypothesis and, in particular, on a computational formulation named Passive Motion Paradigm, where whole-body gestures are produced by applying a small set of force fields to specific key points of the internal body schema: its animation by carefully selected force fields is analogous to the animation of a marionette using wires or strings. The crucial point is that force fields do add, thus suggesting to use force fields as a consistent set of primitives instead of basic movements. This is the starting point for suggesting a force field-based language of action representation, named Pinocchio in analogy with the famous marionette. The proposed language for action description and generation includes three main modules: 1) Primitive force field generators, 2) a Body-Model to be animated by the primitive generators, and 3) a graphical staff system for expressing any specific notated gesture. We suggest that such language is a crucial building block for the development of a cognitive architecture of cooperative robots.

Design of double-bend and multibend achromat lattices with large dynamic aperture and approximate invariants

A numerical method to design nonlinear double- and multi-bend achromat (DBA and MBA) lattices with approximate invariants of motion is investigated. The search for such nonlinear lattices is motivated by Fermilab's Integrable Optics Test Accelerator (IOTA), whose design is based on an integrable Hamiltonian system with two invariants of motion. While it may not be possible to design an achromatic lattice for a dedicated synchrotron light source storage ring with one or more exact invariants of motion, it is possible to tune the sextupoles and octupoles in existing DBA and MBA lattices to produce approximate invariants. In our procedure, the lattice is tuned while minimizing the turn-by-turn fluctuations of the Courant-Snyder actions $J_x$ and $J_y$ at several distinct amplitudes, while simultaneously minimizing diffusion of the on-energy betatron tunes. The resulting lattices share some important features with integrable ones, such as a large dynamic aperture, trajectories confined to invariant tori, robustness to resonances and errors, and a large amplitude-dependent tune-spread. Compared to the nominal NSLS-II lattice, the single- and multi-bunch instability thresholds are increased and the bunch-by-bunch feedback gain can be reduced.

New trigonometric B-spline approximation for numerical investigation of the regularized long-wave equation

Abstract The objective of this work is to propose a collocation technique based on new cubic trigonometric B-spline (NCTB-spline) functions to approximate the regularized long-wave (RLW) equation. This equation is used for modelling numerous problems occurring in applied sciences. The NCTB-spline collocation method is used to integrate the spatial derivatives. We use the Rubin–Graves linearization technique to linearize the non-linear term. The accuracy and efficiency of the technique are examined by employing it on three important numerical examples which have three invariants of motion viz. mass, momentum, and energy. It is observed that the error norms of the present method are less than the error norms of the methods available in the literature. The numerical values of these invariants have also been approximated, which remain conserved during the program run which shows that the propagation of the solitary wave is represented perfectly. The propagation of one and two solitary waves and undulations of waves are depicted graphically. The stability analysis shows that the method is unconditionally stable.

Visual Features and Their Own Optical Flow.

Symmetries, invariances and conservation equations have always been an invaluable guide in Science to model natural phenomena through simple yet effective relations. For instance, in computer vision, translation equivariance is typically a built-in property of neural architectures that are used to solve visual tasks; networks with computational layers implementing such a property are known as Convolutional Neural Networks (CNNs). This kind of mathematical symmetry, as well as many others that have been recently studied, are typically generated by some underlying group of transformations (translations in the case of CNNs, rotations, etc.) and are particularly suitable to process highly structured data such as molecules or chemical compounds which are known to possess those specific symmetries. When dealing with video streams, common built-in equivariances are able to handle only a small fraction of the broad spectrum of transformations encoded in the visual stimulus and, therefore, the corresponding neural architectures have to resort to a huge amount of supervision in order to achieve good generalization capabilities. In the paper we formulate a theory on the development of visual features that is based on the idea that movement itself provides trajectories on which to impose consistency. We introduce the principle of Material Point Invariance which states that each visual feature is invariant with respect to the associated optical flow, so that features and corresponding velocities are an indissoluble pair. Then, we discuss the interaction of features and velocities and show that certain motion invariance traits could be regarded as a generalization of the classical concept of affordance. These analyses of feature-velocity interactions and their invariance properties leads to a visual field theory which expresses the dynamical constraints of motion coherence and might lead to discover the joint evolution of the visual features along with the associated optical flows.