Phase-plane Behavior Research Articles

Imaging phase plane models

In application-oriented mathematics, particularly in the context of nonlinear system analysis, phase plane analysis through SageMath offers a visual display of the qualitative behaviour of solutions to differential equations without inundating students with cumbersome calculations of the plane-phase. A variety of examples is usually given to illustrate phase-plane behaviour. We approach these problems by considering a problem containing a single real parameter that exemplifies the various situations clearly and simply. We developed two computer programs in SageMath: one program calculates aspects of the Jacobian matrix and displays the phase plane portraits, the other determines the centre manifold. Computations and images generated with computer codes are useful in understanding dynamic models in biology, physics and engineering that involve planar non-linear autonomous differential equations. This paper covers analytical and computational skills that are helpful for students and teachers.

Qualitative dynamics of interacting vacuum cosmologies

We present a phase-space analysis of the qualitative dynamics cosmologies where dark matter exchanges energy with the vacuum component. We find fixed points corresponding to power-law solutions where the different components remain a constant fraction of the total energy density and given an existence condition for any fixed points with nonvanishing energy transfer. For some interaction models we find novel fixed points in the presence of a third noninteracting fluid with constant equation of state, such as radiation, where the interacting vacuum+matter tracks the evolution of the third fluid, analogous to tracker solutions previously found for self-interacting scalar fields. We illustrate the phase-plane behavior, determining the equation of state and stability of the fixed points in the case of a simple linear interaction model, for interacting vacuum and dark matter, including the presence of noninteracting radiation. We give approximate solutions for the equation of state in matter- or vacuum-dominated solutions in the case of small interaction parameters.

Gross Moist Stability Analysis: Assessment of Satellite-Based Products in the GMS Plane

Abstract New diagnostic applications of the gross moist stability (GMS) are proposed with demonstrations using satellite-based data. The plane of the divergence of column moist static energy (MSE) against the divergence of column dry static energy (DSE), referred to as the GMS plane here, is utilized. In this plane, one can determine whether the convection is in the amplifying phase or in the decaying phase; if a data point lies below (above) a critical line in the GMS plane, the convection is in the amplifying (decaying) phase. The GMS plane behaves as a phase plane in which each convective life cycle can be viewed as an orbiting fluctuation around the critical line, and this property is robust even on the MJO time scale. This phase-plane behavior indicates that values of the GMS can qualitatively predict the subsequent convective evolution. This study demonstrates that GMS analyses possess two different aspects: time-dependent and quasi-time-independent aspects. Transitions of time-dependent GMS can be visualized in the GMS plane as an orbiting fluctuation around the quasi-time-independent GMS line. The time-dependent GMS must be interpreted differently from the quasi-time-independent one, and the latter is the GMS relevant to moisture-mode theories. The authors listed different calculations of the quasi-time-independent GMS: (i) as a regression slope from a scatterplot and (ii) as a climatological quantity, which is the ratio of climatological MSE divergence to climatological DSE divergence. It is revealed that the latter, climatological GMS, is less appropriate as a diagnostic tool. Geographic variations in the quasi-time-independent GMS are plotted.

Cycles of maximin and utilitarian policies under the veil of ignorance

A conceptual and mathematical model of a social community behavior in a choice situation under a veil of ignorance, where two alternative policies—Rawlsian maximin and Harsanyian utilitarianism—can be implemented through the aggregation of individual preferences over these two policies, is constructed and investigated. We first incorporate in our conceptual model psychological features such as risk-aversion and prosocial preferences that likely underlie choices of welfare policies. We secondly develop and select the mathematical model presented it by means of an autonomous system of ordinary differential equations. A qualitative analysis of this system global phase-plane behavior shows possible tendencies of community development under social choices over Rawlsian or utilitarian societies depending on psychological parameters such as risk aversion and prosocial preferences.

多入力非線形システムに対する位相面における幾何学的考察に基づく非線形制御とその応用

多入力非線形システムに対する位相面における幾何学的考察に基づく非線形制御とその応用

1入力非線形システムに対する位相面における幾何学的考察に基づく非線形コントローラの設計

1入力非線形システムに対する位相面における幾何学的考察に基づく非線形コントローラの設計

Analytical design of dual-tone signal for ADC phase-plane testing

An analytical description of the phase-plane behavior of a dual-tone signal for nonlinear analog-to-digital converters testing is presented. A quality index for the phase-plane coverage of the dual-tone test signal is proposed and evaluated. Conditions for optimal input frequency selection are analytically derived for practical calibration applications, and numerical results highlighting the working of the proposed approach are provided.

Modelling of congenital nystagmus waveforms produced by saccadic system abnormalities.

Models of the mechanisms of normal eye movements are typically described in terms of the block diagrams which are used in control theory. An alternative approach to understanding the mechanisms of normal eye movements involves describing the eye movement behaviour in terms of smooth changes in state variables. The latter approach captures the burst cell firing against motor error (difference between target gaze angle and current gaze angle) phase plane behaviour which is found experimentally and facilitates the modelling of variations in burst cell behaviour. A novel explanation of several types of congenital nystagmus waveforms is given in terms of a saccadic termination abnormality.

Process dynamics: modeling, analysis, and simulation

I. PROCESS MODELING. 1. Introduction. Motivation. Models. Systems. Background of the Reader. How To Use This Textbook. Courses Where This Textbook Can Be Used. 2. Process Modeling. Background. Balance Equations. Material Balances. Constitutive Relationships. Material and Energy Balances. Distributes Parameter Systems. Dimensionless Models. Explicit Solutions to Dynamic Models. General Form of Dynamic Models. II. NUMERICAL TECHNIQUES. 3. Algebraic Equations. Notations. General Form for a Linear System of Equations. Nonlinear Functions of a Single Variable. MATLAB Routines for Solving Functions of a Single Variable. Multivariable Systems. MATLAB Routines for Systems of Nonlinear Algebraic Equations. 4. Numerical Integration. Background. Euler Integration. Runge-Kutta Integration. MATLAB Integration Routines. III. LINEAR SYSTEMS ANALYSIS. 5. Linearization of Nonlinear Models: The State-Space Formulation. State Space Models. Linearization of Nonlinear Models. Interpretation of Linearization. Solution of the Zero-Input Form. Solution of the General State-Space Form. MATLAB Routines step and initial. 6. Solving Linear nth Order ODE Models. Background. Solving Homogeneous, Linear ODEs with Constant Coefficients. Solving Nonhomogeneous, Linear ODEs with Constant Coefficients. Equations with Time-Varying Parameters. Routh Stability Criterion-Determining Stability Without Calculating Eigenvalues. 7. An Introduction to Laplace Transforms. Motivation. Definition of the Laplace Transform. Examples of Laplace Transforms. Final and Initial Value Theorems. Application Examples. Table of Laplace Transforms. 8. Transfer Function Analysis of First-Order Systems. Perspective. Responses of First-Order Systems. Examples of Self-Regulating Processes. Integrating Processes. Lead-Lag Models. 9. Transfer Function Analysis of Higher-Order Systems. Responses of Second-Order Systems. Second-Order Systems with Numerator Dynamics. The Effect of Pole-Zero Locations on System Step Responses. Pad Approximation for Deadtime. Converting the Transfer Function Model to State-Space Form. MATLAB Routines for Step and Impulse Response. 10. Matrix Transfer Functions. A Second-Order Example. The General Method. MATLAB Routine ss2tf. 11. Block Diagrams. Introduction to Block Diagrams. Block Diagrams of Systems in Series. Pole-Zero Cancellation. Systems in Series. Blocks in Parallel. Feedback and Recycle Systems. Routh Stability Criterion Applied to Transfer Functions. SIMULINK. 12. Linear Systems Summary. Background. Linear Boundary Value Problems. Review of Methods for Linear Initial Value Problems. Introduction to Discrete-Time Models. Parameter Estimation of Discrete Linear Systems. IV. NONLINEAR SYSTEMS ANALYSIS. 13. Phase-Plane Analysis. Background. Linear System Examples. Generalization of Phase-Plane Behavior. Nonlinear Systems. 14. Introduction Nonlinear Dynamics: A Case Study of the Quadratic Map. Background. A Simple Population Growth Model. A More Realistic Population Model. Cobweb Diagrams. Bifurcation and Orbit Diagrams. Stability of Fixed-Point Solutions. Cascade of Period-Doublings. Further Comments on Chaotic Behavior. 15. Bifurcation Behavior of Single ODE Systems. Motivation. Illustration of Bifurcation Behavior. Types of Bifurcations. 16. Bifurcation Behavior of Two-State Systems. Background. Single-Dimensional Bifurcations in the Phase-Plane. Limit Cycle Behavior. The Hopf Bifurcation. 17. Introduction to Chaos: The Lorenz Equations. Introduction. Background. The Lorenz Equations. Stability Analysis of the Lorenz Equations. Numerical Study of the Lorenz Equations. Chaos in Chemical Systems. Other Issues in Chaos. IV. REVIEW AND LEARNING MODULES. Module 1 Introduction to MATLAB. Module 2 Review of Matrix Algebra. Module 3 Linear Regression. Module 4 Introduction to SIMULINK. Module 5 Stirred Tank Heaters. Module 6 Absorption. Module 7 Isothermal Continuous Stirred Tank Chemical Reactors. Module 8 Biochemical Reactors. Module 9 Diabatic Continuous Stirred Tank Reactors. Module 10 Ideal Binary Distillation. Index.

A Global Analysis of a Temperature-Dependent Model System for a Mite Predator-Prey Interaction

The nonlinear behavior of a particular Bazykin-type predator-prey differential equation system, with model parameters chosen appropriately for a temperature dependent mite interaction on fruit trees, is examined by means of the numerical code AUTO-86. Within this context, six of the seven types of nontrivial phase plane behavior qualitatively described for such systems by Bazykin (1976, Int. Inst. Appl. Syst. Analysis, Laxenburg) are numerically demonstrated. In addition, three types of qualitative behavior not previously described for such systems are demonstrated, including one in which two unstable fixed points and a saddle point are all surrounded in the phase plane by one stable limit cycle. The ecological implications of the metastability present for certain values of the parameters are discussed with respect to population outbreaks caused by temperature increases and population perturbations, and with respect to population crashes caused by degradation of the environmental carrying capacity and overexploitation.

On the structural stability of the cubic autocatalytic scheme in a closed vessel

This paper is concerned with the cubic autocatalator, A+2B→2B, B→C, in a closed system, in particular when the extra uncatalysed reaction A→B, the quadratic autocatalytic reaction A+B→2B and the additional decay step A→C are included. Previous work has shown that the oscillatory behaviour in the original scheme is structurally unstable with respect to the inclusion of the uncatalysed reaction. Here the equations exhibit a more complex pattern of behaviour than was present in previous studies and this has been fully investigated using constructional methods developed by Gray and Roberts. It is shown that the basic scheme is also structurally unstable with respect to the inclusion of the quadratic autocatalytic step. However, the inclusion of the extra decay step A→C has the effect of removing this structural instability by bringing into the finite phase plane behaviour that was previously seen at infinity.

Identification of observed dynamic bifurcations and development of qualitative models

A systematic study of dynamical systems requires finding a series of bifurcation diagrams instead of just cases with different phase plane behaviour. A scheme is developed for identifying experimentally observed bifurcations (transitions) in the dynamic features of a system characterized by two state variables. Maps of experimentally found bifurcation points in a two parameter (operating conditions) plane may be used to estimate the location of singular points and their nature (defining conditions). This information may be used to predict new types of dynamic behaviour and corresponding operating conditions. Moreover, it can be used to develop a simple (qualitative) mathematical model which predicts all the observed dynamic features and transitions.

Phase-plane analyses of the Michaelis-Menten reaction equations

The irreversible Michaelis-Menten scheme may be reduced to a pair of autonomous first-order differential equations. The phase-plane behaviour of these is investigated.

Graphical stability, enrichment, and pest control by a natural enemy

A two-dimensional mathematical model for predator-prey interactions is proposed. Stability of the equilibrium is examined in the light of a graphical method of Rosenzweig and MacArther. A second graphical method is also discussed. The phase-plane behavior of the model is analyzed. A perturbed form of the model is introduced, and its relation to pest control is discussed. In varying the carrying capacity, the relation to enrichment is analyzed. In addition the stabilizing effects due to the addition of a source term are noted.

Non-equilibrium thermodynamics of temporally oscillating chemical reactions

Two thermodynamic quantities are introduced: the entropy change due to a variation of chemical affinity from a steady, state. to some other state and the corresponding entropy production. The entropy change is always negative definite except at the steady state and is capable of being a Liapunov function. The phase-plane behaviour of the entropy production along the trajectory generated by kinetic equations is investigated in connection with the stability of steady state. The examples taken on that occasion are the Volterra-Lotka and Prigogine-Lefever models. The non-equilibrium thermodynamic properties common to the oscillating reactions in two-variable system are in general considered with an emphasis on the thermodynamic analysis for the direction of rotation of the trajectory generated by the two-variable kinetic equations.

Control of second-order systems at large amplitudes

Unconstrained optimal control of dynamical systems governed by second-order nonlinear differential equations is considered. Large amplitude behavior is investigated, and a control problem defined by the dominating nonlinearities is treated exactly. This asymptotic problem is reduced to a single differential equation by means of a similarity transformation, and control strategies are determined from the phase-plane behavior. The control of the Duffing equation is treated in detail.

A possible relationship between continuous and discontinuous events in biological systems: Multiple stationary states in chemical systems

The possibility of multiple stationary states arising in a wide class of biological control systems is investigated. Eleven different cases are considered, and examples illustrating the temporal and phase plane behavior of systems involving multiple stationary states are presented. It is illustrated that the addition of a little positive feedback to an otherwise purely negative biochemical feedback control system imparts stability to stationary states that would otherwise be unstable.