Planar Systems Research Articles

A Kinematic Synthesis Methodology for P-Drivable Spatial Single Degree-of-Freedom Mechanisms to Avoid Singularities

Abstract For a single degree-of-freedom spatial mechanism, a reference frame attached to any of its links produces a continuous motion of this frame. Given the progression of this frame from the start through the end of the mechanism’s motion, this paper seeks to identify specific points relative to this moving reference frame. The points of interest are those that can be coupled with a second point determined in the fixed frame to act as the end joint locations for a spherical–prismatic–spherical (SPS) driving chain. If the selection of the point pair is made such that the change in distance between them as the mechanism moves is strictly monotonic, then the SPS chain they define is potentially capable of driving the mechanism over the desired range of motion. This motion is referred to as locally P-drivable because a global solution is not ensured by the process proposed herein. This synthesis process can avoid singularities encountered by actuating the mechanism at one of its original joints. The proposed approach enables the dimensional synthesis of a single degree-of-freedom mechanism to focus on creating circuit-defect-free solutions without concern for potential singular positions. The actuating chain can then be determined as a separate step in the synthesis process. This paper also considers motions that are not P-drivable and the specialization to planar systems with the synthesis of a P-drivable revolute–prismatic–revolute (RPR) chain.

Collective Spin-Wave Dynamics in Gyroid Ferromagnetic Nanostructures.

Expanding upon the burgeoning discipline of magnonics, this research elucidates the intricate dynamics of spin waves (SWs) within three-dimensional nanoenvironments. It marks a shift from traditionally used planar systems to exploration of magnetization configurations and the resulting dynamics within 3D nanostructures. This study deploys micromagnetic simulations alongside ferromagnetic resonance measurements to scrutinize magnetic gyroids, periodic chiral configurations composed of chiral triple junctions with a period in nanoscale. Our findings uncover distinctive attributes intrinsic to the gyroid network, most notably the localization of collective SW excitations and the sensitivity of the gyroid's ferromagnetic response to the orientation of the static magnetic field, a correlation closely tied to the crystallographic alignment of the structure. Furthermore, we show that for the ferromagnetic resonance, multidomain gyroid films can be treated as a magnonic material with effective magnetization scaled by its filling factor. The implications of our research carry the potential for practical uses such as an effective, metamaterial-like substitute for ferromagnetic parts and lay the groundwork for radio frequency filters. The growing areas of 3D magnonics and spintronics present exciting opportunities to investigate and utilize gyroid nanostructures for signal processing purposes.

The stability of smooth solitary waves for the [formula omitted]-family of Camassa–Holm equations

The b-family of Camassa–Holm (b-CH) equation is a one-parameter family of PDEs, which includes the completely integrable Camassa–Holm and Degasperis–Procesi equations but possesses different Hamiltonian structures. Motivated by this, we study the existence and the orbital stability of the smooth solitary wave solutions with nonzero constant background to the b-CH equation for the special case b=1, whose the Hamiltonian structure is different from that of b≠1. We establish a connection between the stability criterion for the solitary waves and the monotonicity of a singular integral along the corresponding homoclinic orbits of the spatial ODEs. We verify the latter analytically using the framework for the monotonicity of period function of planar Hamiltonian systems, which shows that the smooth solitary waves are orbitally stable. In addition, we find that the existence and orbital stability results for 0<b<1 are similar to that of b>1, particularly the stability criteria are the same. Finally, combining with the results for the case b>1, we conclude that the solitary waves to the b-CH equation is structurally stable under the variation of the parameter b>0.

Conformational Analysis of Conjugated Organic Materials: What Are My Heteroatoms Really Doing?

Organic semiconductor small molecules and polymers often incorporate heteroatoms into their chemical structures to affect the electronic properties of the material. A particular design philosophy has been to use these heteroatoms to influence torsional potentials, since the overlap of adjacent π-orbitals is most efficient in planar systems and is critical for charge delocalization in these systems. Since these design rules became popular, the messages from the earlier works have become lost in a sea of reports of "conformational locks", where the non-covalent interactions have relatively small contributions to planarizing torsional potentials. Greater influences can be found in the stabilization by extended conjugation, consideration of steric repulsion, and the interactions involving solubilizing chains and neighboring molecules or polymer chains in condensed phases.

Qualitative analysis and explicit solutions of perturbed Chen–Lee–Liu equation with refractive index

The main purpose of this article is to study the qualitative analysis and explicit solutions of perturbed Chen–Lee–Liu equation with refractive index. By utilizing wave transformation, two-dimensional dynamics in a plane are obtained. Using the qualitative theory of planar dynamical systems, a series of planar phase portraits are presented. By analyzing the bifurcations of these phase portraits and the characteristics of equilibrium points, some explicit solutions such as Jacobian function solutions and hyperbolic function solutions are constructed. These can further understand the dynamic behavior of perturbed Chen–Lee–Liu equation and their wave propagation.

Crossing Limit Cycles in Planar Piecewise Linear Systems Separated by a Nonregular Line with Node–Node Type Critical Points

In this paper, we investigate the existence and number of crossing limit cycles in a class of planar piecewise linear systems with node–node type critical points defined in two zones separated by a nonregular line formed by two rays emanated from the origin [Formula: see text], which are the positive [Formula: see text]- and [Formula: see text]-axes. We focus our attention on the existence of two-point crossing limit cycles, which intersect the switching line at two points. We obtain sufficient conditions under which the system has two two-point crossing limit cycles which intersect only one of the two rays. Moreover, we construct examples to show that this class of systems can have two, three or four two-point crossing limit cycles.

Impulsive Effects and Complexity Dynamics in the Anti-Predator Model with IPM Strategies

When confronted with the imminent threat of predation, the prey instinctively employ strategies to avoid being consumed. These anti-predator tactics involve individuals acting collectively to intimidate predators and reduce potential harm during an attack. In the present work, we propose a state-dependent feedback control predator-prey model that incorporates a nonmonotonic functional response, taking into account the anti-predator behavior observed in pest-natural enemy ecosystems within the agricultural context. The qualitative analysis of this model is presented utilizing the principles of impulsive semi-dynamical systems. Firstly, the stability conditions of the equilibria are derived by employing pertinent properties of planar systems. The precise domain of the impulsive set and phase set is determined by considering the phase portrait of the system. Secondly, a Poincaré map is constructed by utilizing the sequence of impulsive points within the phase set. The stability of the order-1 periodic solution at the boundary is subsequently analyzed by an analog of the Poincaré criterion. Additionally, this article presents various threshold conditions that determine both the existence and stability of an order-1 periodic solution. Furthermore, it investigates the existence of order-k (k≥2) periodic solutions. Finally, the article explores the complex dynamics of the model, encompassing multiple bifurcation phenomena and chaos, through computational simulations.

Bifurcations for Homoclinic Networks in Two-Dimensional Polynomial Systems

The bifurcation theory for homoclinic networks with singular and nonsingular equilibriums is a key to understand the global dynamics of nonlinear dynamical systems, which will help one determine the dynamical behaviors of physical and engineering nonlinear systems. In this paper, the appearing and switching bifurcations for homoclinic networks through equilibriums in planar polynomial dynamical systems are studied. The appearing and switching bifurcations are discussed for the homoclinic networks of nonsingular and singular sources, sinks, saddles with singular saddle-sources, saddle-sinks, and double-saddles in self-univariate polynomial systems. The first integral manifolds for nonsingular and singular equilibrium networks are determined. The illustrations of singular equilibriums to networks of nonsingular sources, sinks and saddles are given. The appearing and switching bifurcations are studied for homoclinic networks of singular and nonsingular saddles and centers with singular parabola-saddles and double-inflection saddles in crossing-univariate polynomial systems, and the first integral manifolds of such homoclinic networks are determined through polynomial functions. The illustrations of singular equilibriums to networks of nonsingular saddles and centers are given. This paper may help one understand higher-order bifurcation theory in nonlinear dynamical systems, which is completely different from the classic bifurcation theories.

Four Limit Cycles of Three-Dimensional Discontinuous Piecewise Differential Systems Having a Sphere as Switching Manifold

Because of their applications, the study of piecewise-linear differential systems has become increasingly important in recent years. This type of system already exists to model many different natural phenomena in physics, biology, economics, etc. As is well known, the study of the qualitative theory of piecewise differential systems focuses mainly on limit cycles. Most papers studying the problem of existence and the maximum number of limit cycles of piecewise differential systems have precisely considered planar systems. However, few papers have examined this problem in [Formula: see text]. In this paper, our main goal is to examine a class of discontinuous piecewise differential systems in [Formula: see text], where we consider the unit sphere as the separation surface that divides the entire space into two regions, each one has a linear vector field analogous to planar center. In general, it is hard to determine an exact upper bound for the number of limit cycles that a class of differential systems can exhibit. We prove that this class of differential systems can have at most four limit cycles. We show that there are examples of such differential systems with exactly 1, 2, 3 and 4 limit cycles.

Global Phase Portraits of Piecewise Quadratic Differential Systems with a Pseudo-Center

This paper deals with the global dynamics of planar piecewise smooth differential systems constituted by two different vector fields separated by one straight line that passes through the origin. From a quasi-canonical family of piecewise quadratic differential systems with a pseudo-focus point at the origin, which has six parameters, we investigate the subfamilies where the origin is indeed a pseudo-center. For such subfamilies, we classify its global phase portraits in the Poincaré disk and the associated bifurcation sets.

Nonintegrability of dissipative planar systems

We consider dissipative autonomous perturbations of planar Hamiltonian systems and give a sufficient condition for them not to be complex-meromorphically Bogoyavlenskij-integrable such that the first integral or commutative vector field also depends complex-meromorphically on the small parameter. We illustrate the theoretical result for three examples including systems with the Morse potential and effective potential of the Kepler problem.

Saddle–node canard cycles in slow–fast planar piecewise linear differential systems

By applying a singular perturbation approach, canard explosions exhibited by a general family of singularly perturbed planar Piecewise Linear (PWL) differential systems are analyzed. The performed study involves both hyperbolic and non-hyperbolic canard limit cycles appearing after both, a supercritical and a subcritical Hopf bifurcation. The obtained results are comparable with those obtained for smooth vector fields. In some sense, the manuscript can be understood as an extension towards the PWL framework of the results obtained for smooth systems by Dumortier and Roussarie in Mem. Am. Math. Soc. 1996, and Krupa and Szmolyan in J. Differ. Equ. 2001. In addition, some novel slow–fast behaviors are obtained. In particular, in the supercritical case, and under suitable conditions, it is proved that the limit cycles are organized along a curve exhibiting two folds. Each of these folds corresponds to a saddle–node bifurcation of canard limit cycles, one involving headless canard cycles, and the other involving canard cycles with head. This configuration also occurs in smooth systems with N-shaped fast nullcline. However, it has not been previously reported in the Van der Pol system. Our results provide justification for this observation.

Bifurcation, chaotic pattern and traveling wave solutions for the fractional Bogoyavlenskii equation with multiplicative noise

This paper presents a new study that incorporates the Stratonovich integral and conformal fractional derivative into the fractional stochastic Bogoyavlenskii equation with multiplicative noise. The study exposes the behavior of the system, including sensitivity, chaos and traveling wave solutions, by using the planar dynamical systems approach. Time series, periodic perturbation, phase portraits, and the Poincaré section are used to comprehensively study the dynamic properties. Notably, the research uses the planar dynamic systems method to build multiple traveling wave solutions, including kink wave, dark soliton, and double periodic solutions. Furthermore, a comparative study approach is applied to investigate the effects of fractional derivative and multiplicative noise on the traveling wave solutions, which demonstrate a significant influence of both variables. This work demonstrates the creative application of the planar dynamic system method to the analysis of nonlinear wave equations, offering insightful information that may be generalized to more complex wave phenomena.

Rational Involutions and an Application to Planar Systems of ODE

An involution refers to a function that acts as its own inverse. In this paper, our focus lies on exploring two-dimensional involutive maps defined by rational functions. These functions have denominators represented by polynomials of degree one and numerators by polynomials of a degree of, at most, two, depending on parameters. We identify the sets in the parameter space of the maps that correspond to involutions. The investigation relies on leveraging algorithms from computational commutative algebra based on the Groebner basis theory. To expedite the computations, we employ modular arithmetic. Furthermore, we showcase how involution can serve as a valuable tool for identifying reversible and integrable systems within families of planar polynomial ordinary differential equations.

Limit cycles and homoclinic networks in two-dimensional polynomial systems.

In this paper, the properties of equilibriums in planar polynomial dynamical systems are studied. The homoclinic networks of sources, sinks, and saddles in self-univariate polynomial systems are discussed, and the numbers of sources, sinks, and saddles are determined through a theorem, and the first integral manifolds are determined. The corresponding proof of the theorem is completed, and a few illustrations of networks for source, sinks, and saddles are presented for a better understanding of the homoclinic networks. Such homoclinic networks are without any centers even if the networks are separated by the homoclinic orbits. The homoclinic networks of positive and negative saddles with clockwise and counterclockwise limit cycles in crossing-univariate polynomial systems are studied secondly, and the numbers of saddles and centers are determined through a theorem, and the first integral manifolds are determined through polynomial functions. The corresponding proof of the theorem is given, and a few illustrations of networks of saddles and centers are given to show the corresponding geometric structures. Such homoclinic networks of saddles and centers are without any sources and sinks. Since the maximum equilibriums for such two types of planar polynomial systems with the same degrees are discussed, the maximum centers and saddles should be obtained, and maximum sinks, sources, and saddles should be achieved. This paper may provide a different way to determine limit cycles in the Hilbert 16th problem.

The number of limit cycles of a kind of piecewise quadratic systems with switching curve y = xm

We first study the expressions for the second-order Melnikov functions for planar piecewise smooth near-Hamiltonian systems with multiple zones separated by multiple switching curves through the origin. Next, up to the second-order Melnikov functions, we consider the number of limit cycles Z(m,2) of the system x˙=y,y˙=−x with two zones separated by the curve y=xm(2≤m∈N) under the piecewise perturbations of polynomials in x and y with degree 2. It is proved that Z(2,2)=9, 13≤Z(4,2)≤19, and Z(2k,2)≥14 for k≥3 and that Z(2k+1,2)=15 for k≥2. The results are new and improve some results in the literature.

A Novel Adaptive Fuzzy Control Scheme for a Class of Nonlinear Planar Systems Under State Constraints

A Novel Adaptive Fuzzy Control Scheme for a Class of Nonlinear Planar Systems Under State Constraints

The Solution of the Extended 16th Hilbert Problem for Some Classes of Piecewise Differential Systems

The limit cycles have a main role in understanding the dynamics of planar differential systems, but their study is generally challenging. In the last few years, there has been a growing interest in researching the limit cycles of certain classes of piecewise differential systems due to their wide uses in modeling many natural phenomena. In this paper, we provide the upper bounds for the maximum number of crossing limit cycles of certain classes of discontinuous piecewise differential systems (simply PDS) separated by a straight line and consequently formed by two differential systems. A linear plus cubic polynomial forms six families of Hamiltonian nilpotent centers. First, we study the crossing limit cycles of the PDS formed by a linear center and one arbitrary of the six Hamiltonian nilpotent centers. These six classes of PDS have at most one crossing limit cycle, and there are systems in each class with precisely one limit cycle. Second, we study the crossing limit cycles of the PDS formed by two of the six Hamiltonian nilpotent centers. There are systems in each of these 21 classes of PDS that have exactly four crossing limit cycles.

The Integrability and Modification to an Auxiliary Function Method for Solving the Strain Wave Equation of a Flexible Rod with a Finite Deformation

Our study focuses on the governing equation of a finitely deformed flexible rod with strain waves. By utilizing the well-known Ablowita–Ramani–Segur (ARS) algorithm, we prove that the equation is non-integrable in the Painlevé sense. Based on the bifurcation theory for planar dynamical systems, we modify an auxiliary equation method to obtain a new systematic and effective method that can be used for a wide class of non-linear evolution equations. This method is summed up in an algorithm that explains and clarifies the ease of its applicability. The proposed method is successfully applied to construct wave solutions. The developed solutions are grouped as periodic, solitary, super periodic, kink, and unbounded solutions. A graphic representation of these solutions is presented using a 3D representation and a 2D representation, as well as a 2D contour plot.

Bifurcations, chaotic behavior, sensitivity analysis, and various soliton solutions for the extended nonlinear Schrödinger equation

In this manuscript, our primary objective is to delve into the intricacies of an extended nonlinear Schrödinger equation. To achieve this, we commence by deriving a dynamical system tightly linked to the equation through the Galilean transformation. We then employ principles from planar dynamical systems theory to explore the bifurcation phenomena exhibited within this derived system. To investigate the potential presence of chaotic behaviors, we introduce a perturbed term into the dynamical system and systematically analyze the extended nonlinear Schrödinger equation. This investigation is further enriched by the presentation of comprehensive two- and 3D phase portraits. Moreover, we conduct a meticulous sensitivity analysis of the dynamical system using the Runge–Kutta method. Through this analytical process, we confirm that minor fluctuations in initial conditions have only minimal effects on solution stability. Additionally, we utilize the complete discrimination system of the polynomial method to systematically construct single traveling wave solutions for the governing model.