scholarly journals Neimark-Sacker Bifurcation in Demand-Inventory Model with Stock-Level-Dependent Demand

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Piotr Hachuła ◽  
Magdalena Nockowska-Rosiak ◽  
Ewa Schmeidel
Keyword(s):  
Equilibrium Points ◽  
Three Dimensional ◽  
Inventory Model ◽  
Phase Portraits ◽  
Dimensional System ◽  
Planar System ◽  
Stock Level ◽  
Lyapunov Coefficient ◽  
Analysis Of Dynamics ◽  
Dependent Demand

An analysis of dynamics of demand-inventory model with stock-level-dependent demand formulated as a three-dimensional system of difference equations with four parameters is considered. By reducing the model to the planar system with five parameters, an analysis of one-parameter bifurcation of equilibrium points is presented. By the analytical method, we prove that nondegeneracy conditions for the existence of Neimark-Sacker bifurcation for the planar system are fulfilled. To check the sign of the first Lyapunov coefficient of Neimark-Sacker bifurcation, we use numerical simulations. We give phase portraits of the planar system to confirm the previous analytical results and show new interesting complex dynamical behaviours emerging in it. Finally, the economical interpretation of the system is given.

THE CUSP–HOPF BIFURCATION

2007 ◽  
Vol 17 (08) ◽  
pp. 2547-2570 ◽  
Author(s):  
J. HARLIM ◽  
W. F. LANGFORD
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Points ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Planar System ◽  
Bursting Oscillations ◽  
Codimension Two ◽  
Truncated Normal ◽  
Comprehensive Study

The coalescence of a Hopf bifurcation with a codimension-two cusp bifurcation of equilibrium points yields a codimension-three bifurcation with rich dynamic behavior. This paper presents a comprehensive study of this cusp-Hopf bifurcation on the three-dimensional center manifold. It is based on truncated normal form equations, which have a phase-shift symmetry yielding a further reduction to a planar system. Bifurcation varieties and phase portraits are presented. The phenomena include all four cases that occur in the codimension-two fold–Hopf bifurcation, in addition to bistability involving equilibria, limit cycles or invariant tori, and a fold–heteroclinic bifurcation that leads to bursting oscillations. Uniqueness of the torus family is established locally. Numerical simulations confirm the prediction from the bifurcation analysis of bursting oscillations that are similar in appearance to those that occur in the electrical behavior of neurons and other physical systems.

scholarly journals Hopf Bifurcation, Positively Invariant Set, and Physical Realization of a New Four-Dimensional Hyperchaotic Financial System

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
G. Kai ◽  
W. Zhang ◽  
Z. C. Wei ◽  
J. F. Wang ◽  
A. Akgul
Keyword(s):  
Hopf Bifurcation ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Financial System ◽  
Invariant Set ◽  
Phase Portraits ◽  
Lyapunov Coefficient ◽  
Stable Periodic ◽  
Analytical Proof ◽  
Positively Invariant Set

This paper introduces a new four-dimensional hyperchaotic financial system on the basis of an established three-dimensional nonlinear financial system and a dynamic model by adding a controller term to consider the effect of control on the system. In terms of the proposed financial system, the sufficient conditions for nonexistence of chaotic and hyperchaotic behaviors are derived theoretically. Then, the solutions of equilibria are obtained. For each equilibrium, its stability and existence of Hopf bifurcation are validated. Based on corresponding first Lyapunov coefficient of each equilibrium, the analytical proof of the existence of periodic solutions is given. The ultimate bound and positively invariant set for the financial system are obtained and estimated. There exists a stable periodic solution obtained near the unstable equilibrium point. Finally, the dynamic behaviors of the new system are explored from theoretical analysis by using the bifurcation diagrams and phase portraits. Moreover, the hyperchaotic financial system has been simulated using a specially designed electronic circuit and viewed on an oscilloscope, thereby confirming the results of the numerical integrations and its real contribution to engineering.

Phase-plane geometries in enzyme kinetics

1994 ◽  
Vol 72 (3) ◽  
pp. 800-812 ◽  
Author(s):  
Simon J. Fraser ◽  
Marc R. Roussel
Keyword(s):  
Steady State ◽  
Phase Plane ◽  
Competitive Inhibition ◽  
Three Dimensional ◽  
Product Inhibition ◽  
Slow Manifold ◽  
Dimensional System ◽  
Planar System ◽  
State Behaviour ◽  
Kinetics Of

The transient and steady-state behaviour of the reversible Michaelis–Menten mechanism [R] and Competitive Inhibition (CI) mechanism is studied by analysis in the phase plane. Usually, the kinetics of both mechanisms is simplified to give a modified Michaelis–Menten velocity expression; this applies to the CI mechanism with excess inhibitor and to mechanism [R] in the product inhibition limit. In this paper, [R] is treated exactly as a plane autonomous system of differential equations and its true (dynamical) steady state is described by a line-like slow manifold M. Initial velocity experiments for [R] no longer strictly correspond to the hyperbolic law (as in the irreversible Michaelis–Menten mechanism) and this leads to corrections to the standard integrated rate law. Using a new analysis, the slow dynamics of the CI mechanism is reduced from a three-dimensional system to a planar system. In this mechanism transient decay collapses the trajectory flow onto a two-dimensional "slow" surface Σ; motion on Σ can be treated exactly as projected dynamics in the plane. This projected flow may differ in important ways from that of two-step mechanisms, e.g., it may lack a proper steady state. The relevance of these more accurate dynamical descriptions is discussed in relation to experimental design and metabolic function.

scholarly journals Interspecific density-dependent model of predator–prey relationship in the chemostat

2020 ◽  
pp. 2050086
Author(s):  
Tahani Mtar ◽  
Radhouane Fekih-Salem ◽  
Tewfik Sari
Keyword(s):  
Three Dimensional ◽  
Steady States ◽  
Dimensional System ◽  
Planar System ◽  
Predator Species ◽  
Predator Prey ◽  
Density Dependent ◽  
Existence And Stability ◽  
Dependent Growth ◽  
Transcritical Bifurcations

The objective of this study is to analyze a model of competition for one resource in the chemostat with general interspecific density-dependent growth rates, taking into account the predator–prey relationship. This relationship is characterized by the fact that the prey species promotes the growth of the predator species which in turn inhibits the growth of the first species. The model is a three-dimensional system of ordinary differential equations. With the same dilution rates, the model can be reduced to a planar system where the two models have the same local and even global behavior. The existence and stability conditions of all steady states of the reduced model in the plane are determined according to the operating parameters. Using the nullcline method, we present a geometric characterization of the existence and stability of all equilibria showing the multiplicity of coexistence steady states. The bifurcation diagrams illustrate that the steady states can appear or disappear only through saddle-node or transcritical bifurcations. Moreover, the operating diagrams describe the asymptotic behavior of this system by varying the control parameters and show the effect of the inhibition of predation on the emergence of the bistability region and the reduction until the disappearance of the coexistence region by increasing this inhibition parameter.

A Novel Hyperchaotic System With Adaptive Control, Synchronization, and Circuit Simulation

2018 ◽  
pp. 382-419 ◽  
Author(s):  
Sundarapandian Vaidyanathan ◽  
Ahmad Taher Azar ◽  
Aceng Sambas ◽  
Shikha Singh ◽  
Kammogne Soup Tewa Alain ◽  
...  
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Circuit Simulation ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Dissipative System ◽  
Phase Portraits ◽  
Hyperchaotic System ◽  
Qualitative Properties ◽  
New Hyperchaotic System

This chapter announces a new four-dimensional hyperchaotic system having two positive Lyapunov exponents, a zero Lyapunov exponent, and a negative Lyapunov exponent. Since the sum of the Lyapunov exponents of the new hyperchaotic system is shown to be negative, it is a dissipative system. The phase portraits of the new hyperchaotic system are displayed with both two-dimensional and three-dimensional phase portraits. Next, the qualitative properties of the new hyperchaotic system are dealt with in detail. It is shown that the new hyperchaotic system has three unstable equilibrium points. Explicitly, it is shown that the equilibrium at the origin is a saddle-point, while the other two equilibrium points are saddle-focus equilibrium points. Thus, it is shown that all three equilibrium points of the new hyperchaotic system are unstable. Numerical simulations with MATLAB have been shown to validate and demonstrate all the new results derived in this chapter. Finally, a circuit design of the new hyperchaotic system is implemented in MultiSim to validate the theoretical model.

A Chaotic System with Different Shapes of Equilibria

2016 ◽  
Vol 26 (04) ◽  
pp. 1650069 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Sajad Jafari ◽  
Xiong Wang ◽  
Jun Ma
Keyword(s):  
Infinite Number ◽  
Chaotic Attractor ◽  
Chaotic Systems ◽  
Current Knowledge ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Dimensional System ◽  
Different Shapes ◽  
Number Of Equilibria ◽  
Three Dimensional System

Although many chaotic systems have been introduced in the literature, a few of them possess uncountably infinite equilibrium points. The aim of our short work is to widen the current knowledge of the chaotic systems with an infinite number of equilibria. A three-dimensional system with special properties, for example, exhibiting chaotic attractor with circular equilibrium, chaotic attractor with ellipse equilibrium, chaotic attractor with square-shaped equilibrium, and chaotic attractor with rectangle-shaped equilibrium, is proposed.

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