Phase Plane Analysis Research Articles

Higher Order Non-Planar Electrostatic Solitary Potential in a Streaming Electron-Ion Magnetoplasma: Phase Plane Analysis

We investigate cylindrical and spherical solitons in electron-ion (EI) plasma that contains hot (cold) electrons with stationary ions. The magneto-hydrodynamic equations are solved with the aid of the reductive perturbation (RP) technique, leading to the modified Korteweg–De Vries (mKdV) equation for the non-linear behaviour of the solitary waves in EI plasma. By employing the reduced differential transform method (RDTM), an approximate solution of the mKdV is obtained for solitary waves. Phase plane analysis reveals that these excitations exhibit periodic oscillations. The phase plane and periodic behaviour of the obtained model are studied. It is observed that the amplitude and width of the electron acoustic waves (EAWs) are affected by a slight change in the cold to hot electron temperature ratio (σc) and the number density of the cold to hot electron ratio (α). The effect of the streaming speed u(0) and superthermality index κe are investigated. This study is important for understanding the symmetric properties of cylindrical and spherical plasma, relying on the bifurcation analysis, impacted by the streaming effect in the EI plasma.

Lotka–Volterra competition-diffusion system: the critical competition case

We consider the reaction-diffusion competition system in the so-called critical competition case. The associated ODE system then admits infinitely many equilibria, which makes the analysis intricate. We first prove the nonexistence of ultimately monotone traveling waves by applying the phase plane analysis. Next, we study the large time behavior of the solution of the Cauchy problem with a compactly supported initial datum. We not only reveal that the “faster” species excludes the “slower” one (with a known spreading speed), but also provide a sharp description of the profile of the solution, thus shedding light on a new bump phenomenon.

Acute Inflammatory Recovery-A Matter of Sophisticated Interaction of Two Cell Populations?

Acute Inflammatory Recovery-A Matter of Sophisticated Interaction of Two Cell Populations?

Analysis of tumor-immune functional responses in a mathematical model of neoantigen cancer vaccines

Cancer neoantigen vaccines have emerged as a promising approach to stimulating the immune system to fight cancer. We propose a simple model including key elements of cancer-immune interactions and conduct a phase plane analysis to understand the immunological mechanisms of cancer neoantigen vaccines. Analytical results are obtained for two widely used functional forms that represent the killing rate of tumor cells by immune cells: the law of mass action (LMA) and the dePillis-Radunskaya Law (LPR). Using the LMA, our results reveal that a slowly growing tumor can escape the immune surveillance and that there is a unique periodic solution. The LPR offers richer dynamics, in which tumor elimination and uncontrolled tumor growth are both present. We show that tumor elimination requires sufficient number of initial activated T cells in relationship to the malignant cells, which lends support to using the neoantigen cancer vaccine as an adjuvant therapy after the primary tumor is surgically removed or treated using radiotherapy. We also derive a sufficient condition for uncontrolled tumor growth under the assumption of the LPR. The juxtaposition of analyses with these two different choices for the killing rate function highlights their importance on model behavior and biological implications, by which we hope to spur further theoretical and experimental work to understand mechanisms underlying different functional forms for the killing rate.

Subharmonic solutions for degenerate periodic systems of Lotka-Volterra type with impulsive effects

<abstract><p>In this paper, we are concerned with the existence of subharmonic solutions for the degenerate periodic systems of Lotka-Volterra type with impulsive effects. In our degenerate model, the variation of the predator and prey populations may vanish on a time interval, which imitates the (real) possibility that the predation is seasonally absent. Our proof is based on the Poincaré-Birkhoff theorem. By using phase plane analysis, we can find the large gap in the rotation numbers between the "small" solutions and the "large" solutions, which guarantees a suitable twist property. By applying the Poincaré-Birkhoff theorem, we then obtain the existence of subharmonic solutions. Our main theorem extends the associated results by J. López-Gómez et al.</p></abstract>

Multiple periodic solutions of second order parameter-dependent equations via rotation numbers

<abstract><p>We investigate the existence of multiple periodic solutions for a class of second order parameter-dependent equations of the form $ x''+f(t, x) = sp(t) $. We compare the behavior of its solutions with suitable linear and piecewise linear equations near positive infinity and infinity. Furthermore, in this context, the nonlinearity $ f $ does not satisfy the usual sign condition, and the global existence of solutions for the Cauchy problem associated to the equation is not guaranteed. Our approach is based on the Poincaré-Birkhoff twist theorem, a rotation number approach and the phase-plane analysis. Our result generalizes the result in Fonda and Ghirardelli <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> for second order parameter-dependent equations.</p></abstract>

A two-strain model of infectious disease spread with asymmetric temporary immunity periods and partial cross-immunity.

We introduce a two-strain model with asymmetric temporary immunity periods and partial cross-immunity. We derive explicit conditions for competitive exclusion and coexistence of the strains depending on the strain-specific basic reproduction numbers, temporary immunity periods, and degree of cross-immunity. The results of our bifurcation analysis suggest that, even when two strains share similar basic reproduction numbers and other epidemiological parameters, a disparity in temporary immunity periods and partial or complete cross-immunity can provide a significant competitive advantage. To analyze the dynamics, we introduce a quasi-steady state reduced model which assumes the original strain remains at its endemic steady state. We completely analyze the resulting reduced planar hybrid switching system using linear stability analysis, planar phase-plane analysis, and the Bendixson-Dulac criterion. We validate both the full and reduced models with COVID-19 incidence data, focusing on the Delta (B.1.617.2), Omicron (B.1.1.529), and Kraken (XBB.1.5) variants. These numerical studies suggest that, while early novel strains of COVID-19 had a tendency toward dramatic takeovers and extinction of ancestral strains, more recent strains have the capacity for co-existence.

Multiplicity of periodic solutions for weakly coupled parametrized systems with singularities

<abstract><p>We prove the existence of multiple periodic solutions for weakly coupled parametrized systems with a singularity of repulsive type at the origin and linear growth at infinity. The proof is based on a higher dimensional Poincaré-Birkhoff theorem and the phase-plane analysis of the solutions.</p></abstract>

Mesoscopic description of hippocampal replay and metastability in spiking neural networks with short-term plasticity.

Bottom-up models of functionally relevant patterns of neural activity provide an explicit link between neuronal dynamics and computation. A prime example of functional activity patterns are propagating bursts of place-cell activities called hippocampal replay, which is critical for memory consolidation. The sudden and repeated occurrences of these burst states during ongoing neural activity suggest metastable neural circuit dynamics. As metastability has been attributed to noise and/or slow fatigue mechanisms, we propose a concise mesoscopic model which accounts for both. Crucially, our model is bottom-up: it is analytically derived from the dynamics of finite-size networks of Linear-Nonlinear Poisson neurons with short-term synaptic depression. As such, noise is explicitly linked to stochastic spiking and network size, and fatigue is explicitly linked to synaptic dynamics. To derive the mesoscopic model, we first consider a homogeneous spiking neural network and follow the temporal coarse-graining approach of Gillespie to obtain a "chemical Langevin equation", which can be naturally interpreted as a stochastic neural mass model. The Langevin equation is computationally inexpensive to simulate and enables a thorough study of metastable dynamics in classical setups (population spikes and Up-Down-states dynamics) by means of phase-plane analysis. An extension of the Langevin equation for small network sizes is also presented. The stochastic neural mass model constitutes the basic component of our mesoscopic model for replay. We show that the mesoscopic model faithfully captures the statistical structure of individual replayed trajectories in microscopic simulations and in previously reported experimental data. Moreover, compared to the deterministic Romani-Tsodyks model of place-cell dynamics, it exhibits a higher level of variability regarding order, direction and timing of replayed trajectories, which seems biologically more plausible and could be functionally desirable. This variability is the product of a new dynamical regime where metastability emerges from a complex interplay between finite-size fluctuations and local fatigue.

A layered roll stability control strategy for commercial vehicles based on adaptive model predictive control

Due to the high centre of mass, commercial vehicles are prone to ultimate rollover under extreme steering conditions. The rollover of commercial vehicles leads to serious traffic accidents and poses a threat to road traffic. In this paper, the mechanism of vehicle rollover is revealed by static equilibrium and phase plane analysis. Meanwhile, a robust roll angle observer is designed by combining Kalman filter and sliding mode observation. Considering the influence of wheel lifting on roll dynamics, a novel layered control strategy is proposed by employing engine torque limit, differential braking and active front-wheel steering. Moreover, the layered control of roll stability is realised by adaptive model predictive control with time-varying weights and constraints. To verify the validity and reliability of the proposed algorithm, the hardware-in-loop tests under typical driving conditions have been carried out. The results show that the proposed method can effectively restrain the divergence of roll state and prevent vehicle rollover.

Control of PMSM Based on Switched Systems and Field-Oriented Control Strategy

Starting from the problem of studying the parametric robustness in the case of the control of a permanent magnet-synchronous motor (PMSM), although robust control systems correspond entirely to this problem, due to the complexity of the algorithms of the robust type, in this article the use of switched systems theory is proposed as a study option, given the fact that these types of systems are suitable both for the study of systems with variable structure and for systems with significant parametric variation under conditions of lower complexity of the control algorithms. The study begins by linearizing a PMSM model at a static operating point and continues with a systematic presentation of the basic elements and concepts concerning the stability of switched systems by applying these concepts to the control system of a PMSM based on the field-oriented control (FOC) strategy, which usually changes the value of its parameters during operation (stator resistance Rs, stator inductances Ld and Lq, but also combined inertia of PMSM rotor and load J). The numerical simulations performed in Simulink validate the fact that, for parametric variations of the PMSM structure, the PMSM control switched systems preserve qualitative performance in terms of its control. A series of Matlab programs are presented based on the YALMIP toolbox to obtain Pi matrices, by solving Lyapunov–Metzler type inequalities, and using dwell time to demonstrate stability, as well as the qualitative study of the performance of PMSM control switched systems by presenting in phase plane and state space analysis of the evolution of state vectors: ω PMSM rotor speed, iq current, and id current.

Co-optimization strategies for connected and automated fuel cell hybrid vehicles in dynamic curving scenarios

The performance of speed planning and energy management for connected and automated fuel cell hybrid vehicles (CAFCHVs) in the curve directly affects the curve passage, operating safety and energy economy. However, the uncertainty of complex traffic conditions (such as the dynamic state of the preceding and ego vehicle, road adhesion coefficient, and curve radius) and the lateral stability of CAFCHV lead to the difficulty of online speed planning and energy management. To address this problem, a co-optimization strategy is implemented in this study. First, according to the stability condition of CAFCHV in the curve, the critical safe speed is obtained by phase plane analysis. In addition, combing the timeliness, future information of driving conditions, and the current state of preceding and ego-vehicles, the gradient-based model prediction control (GRAMPC) leveraging the fast projection gradient method is adopted to calculate the safe and optimal speed sequence. Meanwhile, the energy management strategy (EMS) based on the power ratio adaptive equivalent consumption minimization strategy (PR-AECMS) is utilized for energy distribution. A multi-objective performance function is introduced to evaluate the total cost of hydrogen consumption and battery life extension. The simulation results reveal that the proposed strategy can obtain a safe and optimal speed sequence when CAFCHV operates on the curve road. And compared with the mode of tracking the speed of the preceding vehicle, the hydrogen consumption, SOC, battery degradation, and total cost are reduced by 1.4%, 1.9%, 9.9%, and 1.8% regulated by the planning mode, respectively.

Real-Time Predictive Control of Path Following to Stabilize Autonomous Electric Vehicles Under Extreme Drive Conditions

A novel real-time predictive control strategy is proposed for path following (PF) and vehicle stability of autonomous electric vehicles under extreme drive conditions. The investigated vehicle configuration is a distributed drive electric vehicle, which allows to independently control the torques of each in-wheel motor (IWM) for superior stability, but bringing control complexities. The control-oriented model is established by the Magic Formula tire function and the single-track vehicle model. For PF and direct yaw moment control, the nonlinear model predictive control (NMPC) strategy is developed to minimize PF tracking error and stabilize vehicle, outputting front tires’ lateral force and external yaw moment. To mitigate the calculation burdens, the continuation/general minimal residual algorithm is proposed for real-time optimization in NMPC. The relaxation function method is adopted to handle the inequality constraints. To prevent vehicle instability and improve steering capacity, the lateral velocity differential of the vehicle is considered in phase plane analysis, and the novel stable bounds of lateral forces are developed and online applied in the proposed NMPC controller. Additionally, the Lyapunov-based constraint is proposed to guarantee the closed-loop stability for the PF issue, and sufficient conditions regarding recursive feasibility and closed-loop stability are provided analytically. The target lateral force is transformed as front steering angle command by the inversive tire model, and the external yaw moment and total traction torque are distributed as the torque commands of IWMs by optimization. The validations prove the effectiveness of the proposed strategy in improved steering capacity, desirable PF effects, vehicle stabilization, and real-time applicability.

Phase plane analysis applied to non-explicit multibody vehicle models

A new methodology for constructing stability maps (phase-plane analysis) is presented and validated for application to complex multibody vehicle models implemented in Multibody Dynamics simulation software (Adams®). Traditional methodologies are developed to be applied to explicit mathematical models. Given the complexity of some special multibody systems, particularly in vehicle dynamics, simplifications are needed to apply this stability analysis technique. The main limitation when using simplified models is the need to neglect components which could have a significant influence on the dynamic behavior of the system and therefore on its stability. In the proposed methodology it is not necessary to have access to explicit mathematical models of multibody systems. Thus, the stability map of a vehicle model can be constructed by considering highly nonlinear dynamic elements, such as tires and silent-blocks components, modeled using the nonlinear finite element technique.

On the Dirichlet problem associated with bounded perturbations of positively-(p, q)- homogeneous Hamiltonian systems

The existence of solutions for the Dirichlet problem associated with bounded perturbations of positively-(p, q)-homogeneous Hamiltonian systems is considered both in nonresonant and resonant situations. To deal with the resonant case, the existence of a couple of lower and upper solutions is assumed. Both the well-ordered and the non-well-ordered cases are analysed. The proof is based on phase-plane analysis and topological degree theory.

An Intelligent Predictive Algorithm for the Anti-Rollover Prevention of Heavy Vehicles for Off-Road Applications

Rollover detection and prevention are among the most critical aspects affecting the stability and safety assessment of heavy vehicles, especially for off-road driving applications. This topic has been studied in the past and analyzed in depth in terms of vehicle modelling and control algorithms design able to prevent the rollover risk. However, it still represents a serious problem for automotive carmakers due to the huge counts among the main causes for traffic accidents. The risk also becomes more challenging to predict for off-road heavy vehicles, for which the incipient rollover might be triggered by external factors, i.e., road irregularities, bank angles as well as by aggressive input from the driver. The recent advances in road profile measurement and estimation systems make road-preview-based algorithms a viable solution for the rollover detection. This paper describes a model-based formulation to analytically evaluate the load transfer dynamics and its variation due to the presence of road perturbations, i.e., road bank angle and irregularities. An algorithm to detect and predict the rollover risk for heavy vehicles is also presented, even in presence of irregular road profiles, with the calculation of the ISO-LTR Predictive Time through the Phase-Plane analysis. Furthermore, the artificial intelligence techniques, based on the recurrent neural network approach, is also presented as a preliminary solution for a realistic implementation of the methodology. The paper finally assess the efficacy of the proposed rollover predictive algorithm by providing numerical results from the simulation of the most severe maneuvers in realistic off-road driving scenarios, also demonstrating its promising predictive capabilities.

Design of Approximate Explicit Model Predictive Controller Using Parametric Optimization

AbstractThis paper introduces a new technique, called state-parameterized nonlinear programming control (sp-NLPC), for designing a feedback controller that can stabilize intrinsically unstable nonlinear dynamical systems using parametric optimization. Stability-preserving constraints are included in the optimization problem solved offline by the predictive parameterized Pareto genetic algorithm (P3GA), a constrained nonlinear parametric optimization algorithm. The optimal control policy is approximated from P3GA output using radial basis function (RBF) metamodeling. The sp-NLPC technique requires fewer assumptions and is more data-efficient than alternative methods. Two nonlinear systems (single and double inverted pendulums on a cart) are used as benchmarking problems. Performance and computational efficiency are compared to several competing control design techniques. Results show that sp-NLPC outperforms and is more efficient than competing methods. A qualitative investigation on phase plane analysis for the controlled systems is presented to ensure stability. The approximating state-dependent solution for the control input lends itself to applications of control design for control co-design (CCD). Such extensions are discussed as part of future work.

Oscillation and Nonoscillation for Nonlinear Delay Difference Equations by Phase Plane Analysis

Oscillation and Nonoscillation for Nonlinear Delay Difference Equations by Phase Plane Analysis

Decentralized integration of constrained active steering and torque vectoring systems to energy-efficient stability control of electric vehicles

In this paper, the optimal driving torques of four wheels in an electric vehicle (EV) are obtained by minimizing the losses of four in-wheel motors. In order to slightly change these optimal torques for vehicle stability recovery, a new constrained active front steering (AFS) system is analytically designed and integrated with the torque vectoring (TV) system in a novel decentralized structure. In the proposed structure, the required external yaw moment is applied by the TV controller only when the constrained AFS is filled to capacity determined by a stability index extracted from the nonlinear phase plane analysis. As a result of this integration strategy, the external yaw moment is used as low as possible. Consequently, the torques in electrical motors are used near the optimal values consistent with optimal energy consumption. Comparative simulation studies with the standalone TV are conducted in the CarSim software environment to show the efficiency of the proposed decentralized control structure in terms of energy consumption and stability. Moreover, the suitability of the constrained control method used in the integration structure is shown in comparison with the well-known nonlinear model predictive control method in terms of practical implementation.

Flow dynamics for coupled surface and internal deep-water waves

This article determines the fluid motion underlying coupled linear internal and surface waves in a deep-water two-fluid-layer model (with the lower layer being of infinite depth). A detailed Eulerian description of the wave-field kinematics for coupled linear travelling waves is achieved using phase-plane analysis. The qualitative motion of individual fluid particles is elucidated through analysis of the relevant nonlinear dynamical systems from the Lagrangian viewpoint.