Phase Plane Research Articles

The Discrete Ueda System and Its Fractional Order Version: Chaos, Stabilization and Synchronization

The Ueda oscillator is one of the most popular and studied nonlinear oscillators. Whenever subjected to external periodic excitation, it exhibits a fascinating array of nonlinear behaviors, including chaos. This research introduces a novel fractional discrete Ueda system based on Y-th Caputo fractional difference and thoroughly investigates its chaotic dynamics via commensurate and incommensurate orders. Applying numerical methods like maximum Lyapunov exponent spectra, bifurcation plots, and phase plane. We demonstrate the emergence of chaotic attractors influenced by fractional orders and system parameters. Advanced complexity measures, including approximation entropy (ApEn) and C0 complexity, are applied to validate and measure the nonlinear and chaotic nature of the system; the results indicate a high level of complexity. Furthermore, we propose a control scheme to stabilize and synchronize the introduced Ueda map, ensuring the convergence of trajectories to desired states. MATLAB R2024a simulations are employed to confirm the theoretical findings, highlighting the robustness of our results and paving the way for future works.

Control optimization of air traffic emissions in a two-variable dynamic model

Since 1999, every report released by the International Panel on Climate Change has advocated a decrease in the greenhouse gas emissions associated with aviation in order to preserve the current climate. This study used a two variable differential equations model with a non-linear control term to address several aspects of the emissions stabilization issue. By optimizing the control term parameter, several management alternatives can be obtained based on the properties of the phase plane of the model solutions, as identified by a stability analysis. The system can be stabilised around an equilibrium point that maintains the present number of passengers, or maintains the emissions level or is nearest to its present state. Each of these options entails different issues of growth or reduction in the number of passengers and/or the emissions rate, directly obtained from the model results. The last option seems especially novel and promising, since only short-distance flight passengers are severely reduced, while long-distance and international passengers are allowed to growth, and their associated emissions are reduced to below 50 percent of their actual value. Moreover, in a scenario of slow growth in air traffic, these rates could improve, with fewer reductions in the number of short-distance passengers.

Dynamics of the temporal evolution in radiating stars

We study the dynamics of a charged radiating star in general relativity. The junction conditions at the surface of the star lead to a restriction that connects the radial pressure to the heat flux. The master equation reduces to a nonlinear second order differential equation which determines the temporal evolution. The dynamical behaviour is studied via a phase plane analysis which reveals interesting behaviour. The presence of both the electromagnetic field and the cosmological constant are included in our treatment. They affect the temporal evolution of the gravitating star. We identify the restrictions on the parameters that lead to a stable asymptotic end state of the star.

A fuzzy systems approach to modeling the US-China arms race

This paper presents a fuzzy approach to modeling the arms race between the United States and China. In the proposed model, we integrate fuzzy logic with differential equations, enabling the modeling of strategic interactions between the two nations in a non-linear manner, differing from the original Richardson model. This methodology allows for the incorporation of quantitative data and expert knowledge on conflicts into the differential equations-based mathematical model. By studying the phase plane of the fuzzy model, it is possible to analyze scenarios of stability and tension escalation in the dynamics of the arms race.

Molecular Mechanistic Analysis of Liquid-Crystalline Polymers Composed of p-Hydroxybenzoic Acid I: Thermal Properties.

All-atom molecular dynamics (MD) calculations of the crystalline polymeric p-hydroxybenzoic acid (pHBA) were conducted at various temperatures to investigate its thermal response. The calculated structure factor equivalent to the X-ray diffraction pattern of pHBA clearly showed two phase transitions occurring at 600 and 650 K. The first transition at 600 K occurred from the orthorhombic phase to the pseudohexagonal phase, identified by the presence of the 211-peak. This peak disappeared during the second transition at 650 K, indicating that the phase at 650 K was hexagonal. The structure of the pseudohexagonal phase was anisotropic with respect to the ab plane but isotropic in the hexagonal phase. Discontinuous changes in the calculated unit cell volume and unit cell length were observed at 600 K, associated with the first phase transition. We also calculated the linear expansion coefficients in three directions. An anisotropic expansion was observed in three directions for the orthorhombic crystal. In particular, the linear expansion coefficient in the c-direction was negative. In contrast to this, an isotropic expansion was found in the a- and b-directions for the hexagonal crystal, while the expansion in the c-direction is still negative. This study provides valuable insights into the thermal behavior of polymeric pHBA, which is essential for understanding its structural transformations and designing crystalline polymers with tailored thermal properties.

Regulating Crystalline Phase/Plane of Polymer Electrolyte for Rapid Lithium Ion Transfer

Electronic‐rich functional groups and flexible segments have long been perceived to be the decisive factors influencing lithium‐ion transfer in polymer electrolytes, while crystallinity is regarded as the great scourge. Actually, the research on the influence of crystalline phase and crystalline plane is still in scarcity. Herein, taking poly(vinylidene fluoride)‐hexafluoropropylene (PVDF‐HFP) as an example, new (111/201) crystal planes (belonged to β‐phase) are regulated by dissolving process and clarified by Synchrotron radiation X‐ray diffraction and X‐ray diffraction. Density functional theory calculation indicates that the newly exposed (111/201) crystal planes provide higher binding energy with lithium ions and are conducive to provide more ion transport channels.7Li nuclear magnetic resonance of new crystalline planes contained PVDF‐HFP based electrolyte shows lower field and sharper peak intensity, further proves the rapid lithium ion transfer. Therefore, a high ionic conductivity of 6.42×10−4 S cm−1 and a large lithium‐ion transfer number of 0.7 are achieved. This study offers a new insight into the influence of crystalline phase and crystalline plane on the transfer of lithium ion for polymer electrolytes.

Regulating Crystalline Phase/Plane of Polymer Electrolyte for Rapid Lithium Ion Transfer.

Electronic-rich functional groups and flexible segments have long been perceived to be the decisive factors influencing lithium-ion transfer in polymer electrolytes, while crystallinity is regarded as the great scourge. Actually, the research on the influence of crystalline phase and crystalline plane is still in scarcity. Herein, taking poly(vinylidene fluoride)-hexafluoropropylene (PVDF-HFP) as an example, new (111/201) crystal planes (belonged to β-phase) are regulated by dissolving process and clarified by Synchrotron radiation X-ray diffraction and X-ray diffraction. Density functional theory calculation indicates that the newly exposed (111/201) crystal planes provide higher binding energy with lithium ions and are conducive to provide more ion transport channels. 7Li nuclear magnetic resonance of new crystalline planes contained PVDF-HFP based electrolyte shows lower field and sharper peak intensity, further proves the rapid lithium ion transfer. Therefore, a high ionic conductivity of 6.42×10-4 S cm-1 and a large lithium-ion transfer number of 0.7 are achieved. This study offers a new insight into the influence of crystalline phase and crystalline plane on the transfer of lithium ion for polymer electrolytes.

Generalization of Interarticulator Timing Control: Evidence From Tongue-Jaw and Lip-Jaw Kinematics Using Electromagnetic Articulography.

In skilled speech production, the motor system coordinates the movements of distinct sets of articulators to form precise and consistent constrictions in the vocal tract at distinct locations, across contextual variations in movement rate and amplitude. Research efforts have sought to uncover the critical control parameters governing interarticulator coordination during constriction formation, with a focus on two parameters: (a) latency of movement onset of one articulator relative to another (temporal parameters) and (b) phase angle of movement onset for one articulator relative to another (spatiotemporal parameters). Consistent interarticulator timing between jaw and tongue tip movements, during the formation of constrictions at the alveolar ridge, was previously found to scale more reliably than phase angles across variation in production rate and syllable stress. In the present study, we test whether these temporal regularities generalize to another set of articulators, namely, the jaw and lower lip, during the formation of constrictions at the lips. Eight talkers produced vowel-consonant-vowel (VCV) sequences, recorded using electromagnetic articulography, with variation in production rate and syllable stress. V was /ɑ/-/ɛ/ and C was alveolar /t/-/d/ or bilabial /p/-/b/. Two measures were obtained: (a) the timing of tongue tip/lower lip raising onset for intervocalic C, relative to jaw opening-closing cycles for the flanking vowels, and (b) the angle of tongue tip/lower lip raising onset for intervocalic C, relative to the jaw phase plane. Across both sets of articulators, consonant-related movement onset latencies scaled more consistently with variation in the jaw opening-closing cycle than phase angles. Furthermore, movement onset latencies were more strongly affiliated with utterance type than phase angles. Findings demonstrate that precise temporal coordination of articulator movements regulates the formation of precise constrictions, independent of the specific set of articulators involved or where in the vocal tract the constriction is produced.

CONTROL DESIGN FOR A MULTIDIMENSIONAL SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS WITH RELAY HYSTERESIS AND PERTURBATION

A multidimensional controllable system with a constant matrix, a significant nonlinearity of the twoposition relay type with hysteresis as a control and a continuous periodic perturbation function is considered. The system matrix has simple, real, non-zero eigenvalues, among which one can be positive. Conditions for the system parameters, including the nonlinearity ones, are established under which there is a single two-point oscillatory periodic solution with a period comparable to the period of the perturbation function in the case of a special type of the feedback vector. The asymptotic stability of the solution has been proven using the phase plane method. The results obtained are illustrated by examples for three-dimensional systems.

Efficient Locomotion for Space Robots Inspired by the Flying Snake

Robots are becoming an integral part of space facilities construction and maintenance, and may need to move to and from different work locations. Robotic arms that are widely employed, which are mounted on fixed bases, have difficulty coping with increasingly complex missions. The challenge discussed in this paper is the problem of the efficient locomotion of robotic systems. Inspired by the gliding motion of a flying snake launched from a tree and combined with the weightlessness of the space environment, we design similar motions for the robot, including the following three steps. First, the robot folds its body like a snake and initiates flight by accelerating the global center of mass (CM), focusing on the movement direction and generating suitable momentum. Then, during the flight (free-floating) phase, the joint motions are planned using a nonlinear optimization technique, considering the nonholonomic constraints introduced by the momentum conservation and the system states at the initial and final states of the floating. Meanwhile, the difficulties caused by long-distance flights are addressed to reduce the motion computational cost and energy consumption by introducing the phase plane analysis method. Finally, the landing motion is designed to avoid rigid collisions and rollover on the radial plane. The numerical simulations illustrate that the three phases of maneuvers are smooth and continuous, which can help the space robots efficiently traverse the environment.

How plasticity shapes the formation of neuronal assemblies driven by oscillatory and stochastic inputs.

Synaptic connections in neuronal circuits are modulated by pre- and post-synaptic spiking activity. Previous theoretical work has studied how such Hebbian plasticity rules shape network connectivity when firing rates are constant, or slowly varying in time. However, oscillations and fluctuations, which can arise through sensory inputs or intrinsic brain mechanisms, are ubiquitous in neuronal circuits. Here we study how oscillatory and fluctuating inputs shape recurrent network connectivity given a temporally asymmetric plasticity rule. We do this analytically using a separation of time scales approach for pairs of neurons, and then show that the analysis can be extended to understand the structure in large networks. In the case of oscillatory inputs, the resulting network structure is strongly affected by the phase relationship between drive to different neurons. In large networks, distributed phases tend to lead to hierarchical clustering. The analysis for stochastic inputs reveals a rich phase plane in which there is multistability between different possible connectivity motifs. Our results may be of relevance for understanding the effect of sensory-driven inputs, which are by nature time-varying, on synaptic plasticity, and hence on learning and memory.

The use of modelling in the learning of differential equations in an economics course

Abstract This study examines the experience of using an open-ended modelling problem to introduce economics students to the mathematical subject of differential equations. We show the potential of such an approach in motivating students whilst at the same time helping them construct a new differential equations Schema that is coordinated with their developing understanding of economics. In a classroom context of collaborative group work, general class discussions and carefully chosen homework exercises, students showed that they could develop an interest in the relation between mathematics, specifically differential equations, and economics, and ‘discover’ uses of mathematics that are new to them whilst modelling the economics problem of diffusion of an innovation. Students were able to anticipate important mathematical ideas, like the phase plane qualitative analysis of dynamical systems, on their own. This was accompanied by activities that formalized the mathematical aspects of the course as students developed them.

Model predictive control parameter optimization method for autonomous vehicles

Aiming at the problem of insufficient tracking accuracy and long computational time of model predictive control algorithm in trajectory tracking of autonomous vehicles, a model predictive control parameter optimization method based on a nondominated sorting whale optimization algorithm was proposed. In this method, the parameter optimization problem of the model predictive control algorithm was transformed into a multiobjective optimization problem, with the predictive horizon, control horizon, and sample time as optimized objectives and the sum of squared lateral trajectory errors and computational time as optimized variables. The multiobjective optimization problem was solved using the nondominated sort whale optimization algorithm, and then the Pareto optimal solution set was obtained. The best solution was determined using the expert scoring method, continuous order weighted average operator method, game theory combination weighting method, and technique for order preference by similarity to the ideal solution method. Finally, the phase plane method is used to analyze the vehicle handling stability. The experimental results show that compared with the initial method, the lateral trajectory errors of the proposed method are reduced by 66.09% on average, and the computational time is reduced by 54.68% on average. Therefore, the proposed method is considered to be an effective method for parameter optimization of model predictive control.

Advanced tumor growth modeling: A numerical study integrating phase plane analysis with finite volume method

This study investigates tumor density dynamics through computational modeling, focusing on key parameters: proliferation rate m, diffusion coefficient τ, and additional factors c and κ. Systematic exploration reveals nuanced interactions shaping tumor growth, regression, and dispersal. Higher m values accelerate proliferation, while increased τ facilitates dispersal. Parameters c and κ further refine our understanding of tumor behavior. These insights inform the development of computational models for oncological research, with implications for targeted therapeutic interventions. The aim can be realized by utilizing the propose Finite Volume Method to develop a representation of cancer models and improve the efficiency of their simulations. Furthermore, the numerical results will be validated by comparing them with the Tanh-Coth method formulation to assess the model's accuracy and efficiency. The findings suggest that accurate modeling of tumor dynamics can aid in predicting tumor progression and response to treatments, ultimately contributing to personalized medicine and more effective therapeutic strategies. Phase plane analysis shows the schemes are conditionally stable and accurate to the fourth order in time. The method demonstrates exceptional agreement and error-free findings compared to 3D plots, underscoring its potential utility in clinical and research settings.

Bifurcation analysis and control of the full velocity difference model with delayed velocity difference.

With the increase in the number of urban vehicles, various traffic problems have gradually emerged. Studying the causes of traffic congestion and proposing effective mitigation strategies have important practical significance. This paper proposes a macroscopic traffic flow model that considers the delayed speed difference. This paper applies nonlinear bifurcation to describe and predict nonlinear traffic phenomena on highways from the perspective of global stability of the traffic system. By using the traveling wave transformation, the proposed car-following model is converted into a macroscopic traffic flow model. Next, this paper employs the linear stability analysis to find the bifurcation points of the stability transition in the traffic system, exploring the qualitative characteristics of the inhomogeneous continuous traffic flow model. Theoretical derivations demonstrate the existence of bifurcation points within the model. Additionally, this paper plots the density-time space diagrams and phase plane diagrams of the system to visually present the sudden changes in traffic flow as variable parameters pass through these bifurcation points. Finally, this paper designs a feedback controller to regulate the Hopf bifurcation, aiming to delay or eliminate the occurrence of Hopf bifurcations in the stochastic system.

Traveling wave solutions for a Keller-Segel system with nonlinear chemical gradient

In this paper we establish a complete existence result on traveling wave solutions for a generalized one-dimensional Keller-Segel system for the density u(x,t) of cells and the concentration v(x,t) of a chemical signal. The system distinctively incorporates the nonlinear gradient response of the cells to the chemical signal and nonlinear and non-monotone degradation rate of the chemical signal. Using the geometric singular perturbation theory and novel phase plane analysis, we establish that for every pair of equilibria (u⁎,v−⁎) and (u⁎,v+⁎) with v−⁎>v+⁎ there exists a maximal speed s⁎ with 0<s⁎≤∞ such that for each 0<s<s⁎ and sufficiently small parameter ε>0, the system has a traveling wave solution (u(x−st),v(x−st)) connecting the equilibria (u⁎,v−⁎) and (u⁎,v+⁎) at x=−∞ and x=∞, respectively. Moreover, we obtain the explicit necessary and sufficient condition for s⁎=∞ and upper and lower bounds for s⁎ in the case s⁎<∞. We also prove that the component v of the traveling wave solutions are strictly decreasing for a range of wave speeds s with 0<s<s⁎⁎<s⁎ and is oscillatory near ξ=∞ for all large speeds s.

Proportional-Navigation-Based All-Aspect Approach Against Nonmaneuvering Target Using Phase-Plane Trajectory Shaping

Approaching a target from a desired heading is crucial for various modern applications. Existing literature mostly addresses terminal angle-constrained guidance for stationary targets, which also includes strategies for achieving the entire angular spectrum (all-aspect approach). However, existing strategies against moving targets can achieve only a restricted set of terminal angles. To this end, this paper presents a proportional-navigation-based multiphase guidance strategy against a nonmaneuvering target, moving at a lower speed than the pursuer, with any desired terminal angle in 2-D, by shaping the trajectory on an angular phase plane. First, a generalized version of the two-phase guidance scheme is presented for achieving terminal angles in one half-space. This is subsequently extended to attaining any desired terminal angle in the other half-space by using an additional phase for line-of-sight rotation reversal. By shaping the angular phase-plane trajectory, the pursuer’s field-of-view constraint is also seamlessly incorporated in the guidance formulation. Further, an all-aspect approach in a 3-D environment is achieved by splitting the entire 3-D engagement into two planar engagement phases and employing the developed planar guidance on each engagement plane. The performance of the developed guidance strategies is validated through extensive numerical simulations considering first-order lag and lateral acceleration limits.

Nonlinear pitch oscillation and control of a gravitational stabilized sailcraft perturbed by solar pressure torque

The sailcraft based space missions is deeply influenced by its attitude dynamics since the solar radiation pressure force for propellantless orbital maneuver is dependent on its orientation with respect to the sun line. Therefore, it is essential to explore the attitude motion evolution of a sailcraft and understand its attitude dynamic characteristics. To address this, nonlinear pitch dynamics of a sailcraft in an Earth orbit with a sliding mass as an attitude control actuator is focused on. Firstly, the Lagrange equation method is adopted to establish the pitch dynamics of a sailcraft subjected to the gravitational gradient torque, solar radiation pressure torque considering the Earth's occlusion of sunlight, damping torque, and the attitude control torque (this torque is generated by positioning the sliding mass at proper location). Secondly, the possible occurrence of chaotic pitch motion free of control torque is analytically predicted for the sailcraft in the circular and elliptical orbits using the Melnikov method respectively. The effectiveness of the Melnikov method is verified using various numerical methods such as phase plane, Poincaré sections, and power spectrum density. Furthermore, the influence of various parameters on the chaotic evolution is analyzed in detail. Lastly, to control the chaotic pitch motion onto the selected unstable periodic orbit obtained using the close return pairs, a sliding mode controller based on control input anti-saturation considering the position restriction of the sliding mass is developed. The effectiveness of the developed pitch chaos stabilization controller is verified by two numerical simulation cases.

Study of the resonant motion of a test particle inside Kepler 69 using circular restricted three body problem

Summary: Scientific researches to discover extraterrestrial life are very intense and important. For a long time, humanity has speculated about the existence of the planetary systems different from our solar system, where extraterrestrial life can be existing elsewhere in the universe. The understanding of the origin of planets and planetary systems has indeed become a major focus of research in modern Astrophysics. Exoplanet research is one of the most field developing subjects in Astrophysics and Astronomy. Today, the number of discovered exoplanets, where life is thought to exist, is large. One of the main goals of scientists is to find Earth 2, where there is possibility of extraterrestrial life. Kepler's mission was, and is very important in the field of exoplanet research. This mission has revolutionized our understanding of exoplanets. The number of discovered exoplanets has exceeded 5780 in 4314 planetary systems, with 969 systems having more than one planet [1]. Among them, one of the most interesting is Kepler 69 c, a super-Earth around Kepler 69. This exoplanet is located at the boundary between the habitable zone and the outer one. In this study let’s focus on the classical gravitational problem called Circular Restricted Three-Body Problem (CRTBP) [2, 3] and use an iterative method to perform numerical calculations through MATLAB software. Therefore, in this paper, let’s try to study the movement of a “test particle”, under the action of the gravitational field of the system Kepler 69–Kepler 69 c. The aim of this paper is to study the motion of a “test particle” (for example an extrasolar asteroid test) inside the exoplanetary system Kepler 69. Results are presented in the (x, y) plane in the rotating and inertia system. Also, some of the results are presented in the phase planes (x, vx) and (y, vy)

On the Lateral Stability System of Four-Wheel Driven Electric Vehicles Based on Phase Plane Method

To improve the handling and stability of four-wheel independent drive electric vehicles (FWID EVs), this paper introduces a hierarchical architecture lateral stability control system. The upper-level controller is responsible for generating the additional yaw moment required by the vehicle. This includes a control strategy based on feedforward control and a Linear Quadratic Regulator (LQR) for handling assistance control, an LQR-based stability control, a PID controller-based speed-following control, and a stability assessment method. The lower-level controller uses Quadratic Programming (QP) to optimally distribute the additional yaw moment to the four wheels. A “normalized” method was proposed to determine vehicle stability. After comparing it with the existing double-line method, diamond method, and curved boundary method through the open-loop Sine with Dwell test and the closed-loop Double Lane Change (DLC)test simulation, the results demonstrate that this method is more sensitive and accurate in determining vehicle stability, significantly enhancing vehicle handling and stability.