scholarly journals Bifurcation analysis of a discrete-time four-dimensional cubic autocatalator chemical reaction model with coupling through uncatalysed reactant

2021 ◽  
Vol 87 (2) ◽  
pp. 415-439
Author(s):  
Muhammad Salman Khan ◽  
Keyword(s):  
Fixed Point ◽  
Chemical Reaction ◽  
Discrete Time ◽  
Hybrid Control ◽  
Continuous System ◽  
Reaction Model ◽  
Dimensional System ◽  
Chemical Reaction Model ◽  
Cubic Autocatalator ◽  
Positive Fixed Point

In this manuscript, we discuss a four-dimensional cubic autocatalator chemical reaction model in continuous form. We investigate the existence of one and only positive fixed point and then we have obtained some parametric conditions for local stability of continuous system by using Routh-Hurwitz stability criteria. Moreover, we discretize the four-dimensional continuous cubic autocatalator chemical reaction model by using Euler’s forward method and then by using a nonstandard difference scheme we obtained a consistent discrete-time counterpart of four-dimensional cubic autocatalator chemical reaction model. Parametric conditions for local asymptotic stability of one and only positive fixed point of obtained system are also discussed. It is shown that the obtained system experiences the Neimark-Sacker bifurcation at one and only positive fixed point by using a general standard for Neimark-Sacker bifurcation. The discrete-time counterpart of genuine four-dimensional system displays chaotic dynamics at different standards of bifurcation parameter. Furthermore, the control of Neimark-Sacker bifurcation and chaos is also deliberated by using a generalized hybrid control scheme, which is based on parameter perturbation and feedback control. Finally, some numerical examples are given to strengthen our theoretical results.

scholarly journals A cubic autocatalator chemical reaction model with limit cycle analysis and consistency preserving discretization

2021 ◽  
Vol 87 (2) ◽  
pp. 441-462
Author(s):  
Qamar Din ◽  
◽  
Muhammad Sajjad Shabir ◽  
Muhammad Asif Khan
Keyword(s):  
Chemical Reaction ◽  
Discrete Time ◽  
Continuous System ◽  
Reaction Model ◽  
Qualitative Properties ◽  
Time Model ◽  
Interior Equilibrium ◽  
Chemical Reaction Model ◽  
Scheme Analysis ◽  
Cubic Autocatalator

This article deals with the study of some qualitative properties of a cubic autocatalator chemical reaction model. Particularly, we obtain a dynamically consistent cubic autocatalator discrete-time model by applying a nonstandard difference scheme. Analysis of the existence of equilibria and their stability is carried out. It is proved that a continuous system undergoes the Hopf bifurcation at its interior equilibrium, whereas the discrete-time version undergoes Neimark-Sacker bifurcation at its interior fixed point. Moreover, numerical simulation is provided to strengthen our theoretical discussion.

scholarly journals Complex dynamics of a discrete-time prey-predator system with Allee effect

2021 ◽  
Author(s):  
Özlem Gümüş
Keyword(s):  
Fixed Point ◽  
Discrete Time ◽  
Allee Effect ◽  
Control Method ◽  
Complex Dynamics ◽  
Hybrid Control ◽  
Chaotic Behavior ◽  
The Stability ◽  
Predator System ◽  
Positive Fixed Point

In this paper, we investigate the stability and bifurcation of a discrete-time prey-predator system which is subject to an Allee effect on prey population. It is concluded that the system undergoes flip and Neimark- Sacker bifurcations in a small neigborhood of the unique positive fixed point which depends on the number of prey-predator. The chaotic behavior that emerges with Neimark-Sacker bifurcation is controlled by the OGY method and hybrid control method. Moreover, the numerical simulations are done to demonsrate the theoratical results.

scholarly journals Development of an Uncertainty Quantification Predictive Chemical Reaction Model for Syngas Combustion

2017 ◽  
Vol 31 (3) ◽  
pp. 2274-2297 ◽  
Author(s):  
N. A. Slavinskaya ◽  
M. Abbasi ◽  
J. H. Starcke ◽  
R. Whitside ◽  
A. Mirzayeva ◽  
...  
Keyword(s):  
Uncertainty Quantification ◽  
Chemical Reaction ◽  
Reaction Model ◽  
Syngas Combustion ◽  
Chemical Reaction Model

scholarly journals Qualitative analysis of an autocatalytic chemical reaction model with decay

2014 ◽  
Vol 144 (2) ◽  
pp. 427-446 ◽  
Author(s):  
Jun Zhou ◽  
Junping Shi
Keyword(s):  
Steady State ◽  
Chemical Reaction ◽  
Reaction Diffusion ◽  
Steady State Solution ◽  
Reaction Model ◽  
Existence Of A Solution ◽  
Positive Steady State ◽  
Chemical Reaction Model ◽  
The One ◽  
Necessary And Sufficient

In this paper, we revisit a reaction—diffusion autocatalytic chemical reaction model with decay. For higher-order reactions, we prove that the system possesses at least two positive steady-state solutions; hence, it has bistable dynamics similar to the system without decay. For the linear reaction, we determine the necessary and sufficient condition to ensure the existence of a solution. Moreover, in the one-dimensional case, we prove that the positive steady-state solution is unique. Our results demonstrate the drastic difference in dynamics caused by the order of chemical reactions.

scholarly journals Discrete-Time Predator-Prey Model with Bifurcations and Chaos

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
K. S. Al-Basyouni ◽  
A. Q. Khan
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Discrete Time ◽  
Control Method ◽  
Linear Stability Theory ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Positive Fixed Point

In this paper, local dynamics, bifurcations and chaos control in a discrete-time predator-prey model have been explored in ℝ + 2 . It is proved that the model has a trivial fixed point for all parametric values and the unique positive fixed point under definite parametric conditions. By the existing linear stability theory, we studied the topological classifications at fixed points. It is explored that at trivial fixed point model does not undergo the flip bifurcation, but flip bifurcation occurs at the unique positive fixed point, and no other bifurcations occur at this point. Numerical simulations are performed not only to demonstrate obtained theoretical results but also to tell the complex behaviors in orbits of period-4, period-6, period-8, period-12, period-17, and period-18. We have computed the Maximum Lyapunov exponents as well as fractal dimension numerically to demonstrate the appearance of chaotic behaviors in the considered model. Further feedback control method is employed to stabilize chaos existing in the model. Finally, existence of periodic points at fixed points for the model is also explored.

Coarse-Grained Chemical Reaction Model

2004 ◽  
Vol 108 (6) ◽  
pp. 1815-1821 ◽  
Author(s):  
Yaroslava G. Yingling ◽  
Barbara J. Garrison
Keyword(s):  
Chemical Reaction ◽  
Reaction Model ◽  
Coarse Grained ◽  
Chemical Reaction Model
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