PHASE PORTRAIT OF DYNAMIC SYSTEMS FROM A SECONDARY ROW

2018 ◽  
pp. 138-148
Author(s):  
Kaloyan Yankov
Keyword(s):  
Phase Portrait ◽  
Dynamic Systems ◽  
Phase Plane ◽  
Higher Order ◽  
Graphical Form ◽  
Time Behavior ◽  
Long Time ◽  
Phase Plane Method ◽  
The Stability ◽  
Higher Order Differential Equations

The phase portrait of the second and higher order differential equations presents in graphical form the behavior of the solution set without solving the equation. In this way, the stability of a dynamic system and its long-time behavior can be studied. The article explores the capabilities of Mathcad for analysis of systems by the phase plane method. A sequence of actions using Mathcad's operators to build phase portrait and phase trace analysis is proposed. An example is given by a differential equation of the second order. The approach is also applicable to systems of higher order.

scholarly journals PHASE PORTRAIT OF SECOND-ORDER DYNAMIC SYSTEMS

2018 ◽  
Vol 6 (3) ◽  
pp. 252-262 ◽  
Author(s):  
Kaloyan Yankov
Keyword(s):  
Phase Portrait ◽  
Plasma Renin ◽  
Higher Order ◽  
Second Order ◽  
Graphical Form ◽  
Time Behavior ◽  
Long Time ◽  
Phase Plane Method ◽  
The Stability ◽  
Higher Order Differential Equations

The phase portrait of the second and higher order differential equations presents in graphical form the behavior of the solution set without solving the equation. In this way, the stability of a dynamic system and its long-time behavior can be studied. The article explores the capabilities of Mathcad for analysis of systems by the phase plane method. A sequence of actions using Mathcad's operators to build phase portrait and phase trace analysis is proposed. The approach is illustrated by a model of plasma renin activity after treatment of experimental animals with nicardipine. The identified process is a differential equation of the second order. The algorithm is also applicable to systems of higher order.

scholarly journals Stability and Hopf Bifurcation in a Three-Component Planktonic Model with Spatial Diffusion and Time Delay

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Kejun Zhuang ◽  
Gao Jia ◽  
Dezhi Liu
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Solution ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Time Behavior ◽  
Delayed Reaction ◽  
Constant Equilibrium ◽  
Homogeneous Neumann Boundary Condition ◽  
Long Time ◽  
The Stability

Due to the different roles that nontoxic phytoplankton and toxin-producing phytoplankton play in the whole aquatic system, a delayed reaction-diffusion planktonic model under homogeneous Neumann boundary condition is investigated theoretically and numerically. This model describes the interactions between the zooplankton and two kinds of phytoplanktons. The long-time behavior of the model and existence of positive constant equilibrium solution are first discussed. Then, the stability of constant equilibrium solution and occurrence of Hopf bifurcation are detailed and analyzed by using the bifurcation theory. Moreover, the formulas for determining the bifurcation direction and stability of spatially bifurcating solutions are derived. Finally, some numerical simulations are performed to verify the appearance of the spatially homogeneous and nonhomogeneous periodic solutions.

PAINLEVÉ TEST AND SOME EXACT SOLUTIONS FOR (2+1)-DIMENSIONAL MODIFIED KORTEWEG–DE VRIES–BURGERS EQUATION

2013 ◽  
Vol 10 (03) ◽  
pp. 1250058 ◽  
Author(s):  
ALY MAHER ABOURABIA ◽  
KAWSAR MOHAMED HASSAN ◽  
EHAB SAID SELIMA
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Phase Plane ◽  
Water Model ◽  
Integral Approach ◽  
Painlevé Test ◽  
Wave Nature ◽  
De Vries ◽  
Phase Plane Method ◽  
The Stability

In this paper, we investigate the solitary wave solutions for the two-dimensional modified Korteweg–de Vries–Burgers (mKdV-B) equation in shallow water model. Despite that Painlevé test fails to prove the integrability of the mKdV-B equation by using the WTC-Kruskal algorithm, the Bäcklund transformation is obtained via the truncation expansion. The exact solutions of the mKdV-B equation are found using factorization techniques, Exp-function and energy integral approach of the corresponding ordinary differential equation. We found that the dispersion relation of the linearized mKdV-B equation lies on the complex plane yielding a damping character. By keeping the water height relatively small, we illustrate the resulting solutions in several figures showing the shock and solitary wave nature in the flow. The stability for the mKdV-B equation is analyzed by using the phase plane method.

scholarly journals Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

2021 ◽  
Author(s):  
Annalisa Cesaroni ◽  
Heiko Kröner ◽  
Matteo Novaga
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Time Behavior ◽  
Long Time ◽  
Self Similar ◽  
The Stability ◽  
Lipschitz Graphs ◽  
Similar Solutions ◽  
Anisotropic Mean Curvature

We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the isotropic mean curvature flow.

Dynamics of a Discrete Prey–Predator Model with Mixed Functional Response

2019 ◽  
Vol 29 (14) ◽  
pp. 1950199
Author(s):  
Mohammed Fathy Elettreby ◽  
Aisha Khawagi ◽  
Tamer Nabil
Keyword(s):  
Type I ◽  
Time Behavior ◽  
Stable Fixed Point ◽  
Functional Responses ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Long Time ◽  
Predator Model ◽  
The Stability

In this paper, we propose a discrete Lotka–Volterra predator–prey model with Holling type-I and -II functional responses. We investigate the stability of the fixed points of this model. Also, we study the effects of changing each control parameter on the long-time behavior of the model. This model contains a lot of complex dynamical behaviors ranging from a stable fixed point to chaotic attractors. Finally, we illustrate the analytical results by some numerical simulations.

scholarly journals Long-time behavior of random and nonautonomous Fisher-KPP equations. Part II. Transition fronts

2019 ◽  
Vol 19 (06) ◽  
pp. 1950046 ◽  
Author(s):  
Rachidi B. Salako ◽  
Wenxian Shen
Keyword(s):  
Time Behavior ◽  
Long Time Behavior ◽  
Spreading Speeds ◽  
Kpp Equation ◽  
Random Transition ◽  
Existence And Stability ◽  
Long Time ◽  
Kpp Equations ◽  
The Stability ◽  
Transition Fronts

In the current series of two papers, we study the long-time behavior of the following random Fisher-KPP equation: [Formula: see text] where [Formula: see text], [Formula: see text] is a given probability space, [Formula: see text] is an ergodic metric dynamical system on [Formula: see text], and [Formula: see text] for every [Formula: see text]. We also study the long-time behavior of the following nonautonomous Fisher-KPP equation: [Formula: see text] where [Formula: see text] is a positive locally Hölder continuous function. In the first part of the series, we studied the stability of positive equilibria and the spreading speeds of (1.1) and (1.2). In this second part of the series, we investigate the existence and stability of transition fronts of (1.1) and (1.2). We first study the transition fronts of (1.1). Under some proper assumption on [Formula: see text], we show the existence of random transition fronts of (1.1) with least mean speed greater than or equal to some constant [Formula: see text] and the nonexistence of random transition fronts of (1.1) with least mean speed less than [Formula: see text]. We prove the stability of random transition fronts of (1.1) with least mean speed greater than [Formula: see text]. Note that it is proved in the first part that [Formula: see text] is the infimum of the spreading speeds of (1.1). We next study the existence and stability of transition fronts of (1.2). It is not assumed that [Formula: see text] and [Formula: see text] are bounded above and below by some positive constants. Many existing results in literature on transition fronts of Fisher-KPP equations have been extended to the general cases considered in the current paper. The current paper also obtains several new results.

scholarly journals Stability and Convergence of Solutions to Volterra Integral Equations on Time Scales

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Eleonora Messina ◽  
Antonia Vecchio
Keyword(s):  
Integral Equations ◽  
Time Scales ◽  
Sufficient Conditions ◽  
Volterra Integral Equations ◽  
Stability And Convergence ◽  
Time Behavior ◽  
Convergence Properties ◽  
Convergence Of Solutions ◽  
Long Time ◽  
The Stability

We consider Volterra integral equations on time scales and present our study about the long time behavior of their solutions. We provide sufficient conditions for the stability and investigate the convergence properties when the kernel of the equations vanishes at infinity.

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