scholarly journals The Bifurcation of Two Invariant Closed Curves in a Discrete Model

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Yingying Zhang ◽  
Yicang Zhou
Keyword(s):  
Discrete Model ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Euler Method ◽  
Flip Bifurcation ◽  
Closed Curves ◽  
Dynamical Behaviors ◽  
Discrete Population ◽  
Forward Euler ◽  
Forward Euler Method

A discrete population model integrated using the forward Euler method is investigated. The qualitative bifurcation analysis indicates that the model exhibits rich dynamical behaviors including the existence of the equilibrium state, the flip bifurcation, the Neimark-Sacker bifurcation, and two invariant closed curves. The conditions for existence of these bifurcations are derived by using the center manifold and bifurcation theory. Numerical simulations and bifurcation diagrams exhibit the complex dynamical behaviors, especially the occurrence of two invariant closed curves.

scholarly journals Error estimates for forward Euler shock capturing finite element approximations of the one-dimensional Burgers' equation

2015 ◽  
Vol 25 (11) ◽  
pp. 2015-2042
Author(s):  
Erik Burman
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Burgers Equation ◽  
Euler Method ◽  
Second Order ◽  
Shock Capturing ◽  
Discrete Solution ◽  
A Priori Bound ◽  
Forward Euler ◽  
Forward Euler Method

We propose an error analysis for a shock capturing finite element method for the Burgers' equation using the duality theory due to Tadmor. The estimates use a one-sided Lipschitz stability (Lip+-stability) estimate on the discrete solution and are obtained in a weak norm, but thanks to a total variation a priori bound on the discrete solution and an interpolation inequality, error estimates in Lp-norms (1 ≤ p < ∞) are deduced. Both first-order artificial viscosity and a nonlinear shock capturing term that formally is of second order are considered. For the discretization in time we use the forward Euler method. In the numerical section we verify the convergence order of the nonlinear scheme using the forward Euler method and a second-order strong stability preserving Runge–Kutta method. We also study the Lip+-stability property numerically and give some examples of when it holds strictly and when it is violated.

scholarly journals Model Formulation Over Lie Groups and Numerical Methods to Simulate the Motion of Gyrostats and Quadrotors

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 935 ◽  
Author(s):  
Simone Fiori
Keyword(s):  
Lie Groups ◽  
Conservative System ◽  
Euler Method ◽  
Model Formulation ◽  
Dynamical Equations ◽  
System Equation ◽  
D’Alembert Principle ◽  
Rotation Groups ◽  
Forward Euler ◽  
Forward Euler Method

The present paper recalls a formulation of non-conservative system dynamics through the Lagrange–d’Alembert principle expressed through a generalized Euler–Poincaré form of the system equation on a Lie group. The paper illustrates applications of the generalized Euler–Poincaré equations on the rotation groups to a gyrostat satellite and a quadcopter drone. The numerical solution of the dynamical equations on the rotation groups is tackled via a generalized forward Euler method and an explicit Runge–Kutta integration method tailored to Lie groups.

scholarly journals Bifurcation and Patterns Analysis for a Spatiotemporal Discrete Gierer-Meinhardt System

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 243
Author(s):  
Biao Liu ◽  
Ranchao Wu
Keyword(s):  
Bifurcation Theory ◽  
Center Manifold ◽  
Turing Instability ◽  
Flip Bifurcation ◽  
Center Manifold Theory ◽  
Pattern Dynamics ◽  
Instability Theory ◽  
Pattern Formations ◽  
Manifold Theory ◽  
Theoretical Results

The Gierer-Meinhardt system is one of the prototypical pattern formation models. The bifurcation and pattern dynamics of a spatiotemporal discrete Gierer-Meinhardt system are investigated via the couple map lattice model (CML) method in this paper. The linear stability of the fixed points to such spatiotemporal discrete system is analyzed by stability theory. By using the bifurcation theory, the center manifold theory and the Turing instability theory, the Turing instability conditions in flip bifurcation and Neimark–Sacker bifurcation are considered, respectively. To illustrate the above theoretical results, numerical simulations are carried out, such as bifurcation diagram, maximum Lyapunov exponents, phase orbits, and pattern formations.

Multiple Bifurcations and Complex Responses of Nonlinear Time-Delay Oscillators

2021 ◽  
Author(s):  
Xiaochen Mao ◽  
Fuchen Lei ◽  
Xingyong Li ◽  
Weijie Ding ◽  
Tiantian Shi
Keyword(s):  
Time Delay ◽  
Time Delays ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Electronic Circuit ◽  
Dynamic Features ◽  
Nonlinear Circuit ◽  
Amplitude Death ◽  
Van Der Pol ◽  
Dynamical Behaviors

Abstract In this paper, the dynamical properties of multiple van der Pol-Duffing oscillators with time delays are studied. The amplitude death and bifurcation curves in the parameter plane are determined by using the space decomposition method. Different patterns of bifurcated solutions are given on the basis of the symmetric bifurcation theory. The properties of bifurcated solutions are shown by using the norm forms on the center manifold. The interactions of bifurcations are discussed and their dynamical behaviors are shown. An electronic circuit platform is implemented by means of nonlinear circuit and time delay circuit. The revealed behaviors of the circuit reach an agreement with the obtained results. It is shown that the nonlinearity and time delays have great effects on the system performance and can induce interesting and abundant dynamic features.

scholarly journals Dynamic complexity and bifurcation analysis of a host–parasitoid model with Allee effect and Holling type III functional response

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hua Liu ◽  
Kai Zhang ◽  
Yong Ye ◽  
Yumei Wei ◽  
Ming Ma
Keyword(s):  
Local Stability ◽  
Bifurcation Theory ◽  
Allee Effect ◽  
Center Manifold ◽  
Discrete Time System ◽  
Time System ◽  
Dynamic Complexity ◽  
Center Manifold Theorem ◽  
Dynamical Behaviors ◽  
Local Stability Analysis

AbstractIn this paper, we focus on dynamics in a basic discrete-time system of host–parasitoid interaction. We perform local stability analysis of this system. Furthermore, both flip and Neimark–Sacker bifurcations are also analyzed in the interior of $R_{ +}^{2}$R+2 by using center manifold theorem and bifurcation theory. Finally, numerical simulations are deployed to validate our results with theoretical analysis and to exhibit the dynamical behaviors.

scholarly journals Bifurcation and Chaotic Behavior of a Discrete-Time SIS Model

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Junhong Li ◽  
Ning Cui
Keyword(s):  
Hopf Bifurcation ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Euler Method ◽  
Flip Bifurcation ◽  
Bifurcation Diagrams ◽  
Bifurcation And Chaos ◽  
Sis Model ◽  
Dynamical Behaviors ◽  
The Rich

The discrete-time epidemic model is investigated, which is obtained using the Euler method. It is verified that there exist some dynamical behaviors in this model, such as transcritical bifurcation, flip bifurcation, Hopf bifurcation, and chaos. The numerical simulations, including bifurcation diagrams and computation of Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behaviors.

scholarly journals Bifurcations, Complex Behaviors, and Dynamic Transition in a Coupled Network of Discrete Predator-Prey System

2019 ◽  
Vol 2019 ◽  
pp. 1-22 ◽  
Author(s):  
Tousheng Huang ◽  
Huayong Zhang ◽  
Shengnan Ma ◽  
Ge Pan ◽  
Zhaodeng Wang ◽  
...  
Keyword(s):  
Center Manifold ◽  
Period Doubling ◽  
Flip Bifurcation ◽  
Closed Curves ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Period 2 ◽  
New Perspective ◽  
Matrix Center

The nonlinear dynamics of predator-prey systems coupled into network is an important issue in recent biological advances. In this research, we consider each node of the coupled network represents a discrete predator-prey system, and the network dynamics is investigated. By applying Jacobian matrix, center manifold theorem and bifurcation theorems, stability of fixed points, flip bifurcation and Neimark-Sacker bifurcation of the discrete predator-prey system are analyzed. Via the method of Lyapunov exponents, the nonchaos-chaos transition of the coupled network along the routes to chaos induced by bifurcations is determined. Numerical simulations are performed to demonstrate the bifurcations, various attractors and dynamic transitions of the coupled network. Via comparison, we find that the coupled network exhibits far richer and more complex behaviors than single predator-prey system, including period-doubling cascades in orbits of period-2, period-4, period-8, invariant closed curves, dynamic windows for periodic orbits and invariant curves, quasiperiodic orbits, tori, and chaotic sets. Moreover, the attractors of the coupled network show more diverse and complicated structures. These results may provide a new perspective on the predator-prey dynamics in complex networks.

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