Numerical Bifurcation Analysis of Maps

Kuznetsov, Yuri A.; Meijer, Hil Gaétan Ellart

doi:10.1017/9781108585804

Cited by 65 publications

(43 citation statements)

References 129 publications

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“…Furthermore, higher dimensional systems can possess bifurcations of multiple codimensions, which as significantly more difficult to study. Bifurcations of codimension 3 are so exotic that they are not well studied [ 23 , 24 ]. Software to handle such calculations has only recently been developed [ 24 ].…”

Section: Discussionmentioning

confidence: 99%

“…Bifurcations of codimension 3 are so exotic that they are not well studied [ 23 , 24 ]. Software to handle such calculations has only recently been developed [ 24 ]. In practical terms this means that the convergence properties can only be studied numerically for models with a small number of parameters.…”

Section: Discussionmentioning

confidence: 99%

“…The coordinate ascent variational inference for the Ising model is a discrete-time autonomous dynamical system. Before giving a complete proof of the dynamical properties of the CAVI algorithm for the Ising model in dimension 2, we first give a basic introduction to the theory of discrete time dynamical systems and bifurcations following [ 23 , 24 , 25 , 38 ]. Formally, a dynamical system is triple

where T is a time set, X is the state space and

is a family of evolution operators parameterized by

satisfying

and

for all

.…”

Section: Appendix A1 Notationmentioning

confidence: 99%

“…Global bifurcations are much harder to analyze and since we do not attempt to investigate them in this paper we will not expand upon them further. More detailed results on bifurcations in codimension 1 and 2 can be found in [ 23 , 24 ]. We will now formally define the sufficient conditions for a system to undergo a period doubling or a pitchfork bifurcation.…”

Section: Appendix A1 Notationmentioning

confidence: 99%

“…For sake of concreteness, we focus on the 2D Ising model. While dynamical systems theory possesses the tools [ 23 , 24 , 25 ] necessary to analyze higher dimensional systems, they were mainly developed for non-sequential systems. The general theory for n -dimensional discrete dynamical systems is dependent on having the evolution function in the form

.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Dynamics of Coordinate Ascent Variational Inference: A Case Study in 2D Ising Models

Plummer

Pati

Bhattacharya

2020

Entropy

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Variational algorithms have gained prominence over the past two decades as a scalable computational environment for Bayesian inference. In this article, we explore tools from the dynamical systems literature to study the convergence of coordinate ascent algorithms for mean field variational inference. Focusing on the Ising model defined on two nodes, we fully characterize the dynamics of the sequential coordinate ascent algorithm and its parallel version. We observe that in the regime where the objective function is convex, both the algorithms are stable and exhibit convergence to the unique fixed point. Our analyses reveal interesting discordances between these two versions of the algorithm in the region when the objective function is non-convex. In fact, the parallel version exhibits a periodic oscillatory behavior which is absent in the sequential version. Drawing intuition from the Markov chain Monte Carlo literature, we empirically show that a parameter expansion of the Ising model, popularly called the Edward–Sokal coupling, leads to an enlargement of the regime of convergence to the global optima.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

where T is a time set, X is the state space and

is a family of evolution operators parameterized by

satisfying

and

for all

.…”

Section: Appendix A1 Notationmentioning

confidence: 99%

Section: Appendix A1 Notationmentioning

confidence: 99%

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Dynamics of Coordinate Ascent Variational Inference: A Case Study in 2D Ising Models

Plummer

Pati

Bhattacharya

2020

Entropy

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Developing a continuous SIR epidemic model and its discrete version using Euler method: Analyzing dynamics with analytical and numerical methods

Sahaminejad,

Nyamoradi,

Eskandari

2024

Math Methods in App Sciences

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In this paper, we develop a continuous SIR epidemic model and its discrete version using the Euler method. The continuous SIR epidemic model is a widely used mathematical tool for understanding the dynamics of infectious diseases. We derive the system of differential equations that describe the rates of change in the number of individuals in each compartment and discuss the key parameters that influence the dynamics of the model. Next, we use the Euler method to discretize the continuous SIR epidemic model and obtain a discrete‐time version. We analyze the dynamics of the discrete SIR epidemic model using both analytical and numerical methods. We derive the positively invariant set of the model and discuss the initial conditions of the model and how they relate to the total population. We compare the behavior of the continuous and discrete SIR epidemic models and show that the discretized model captures the essential features of the continuous model. We also investigate the impact of key parameters such as the transmission rate, recovery rate, and death rate on the dynamics of the model. Our findings suggest that the discrete SIR epidemic model is a powerful mathematical tool for understanding the dynamics of infectious diseases and developing effective strategies for their control. By studying the behavior of the model under different scenarios, researchers can gain insights into the transmission and prevalence of infectious diseases and develop interventions that reduce their impact on public health.

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Bifurcation analysis of a two‐dimensional discrete‐time predator–prey model

Hamada

El‐Azab

El-Metwally

2022

Math Methods in App Sciences

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The dynamical behavior of a discrete predator-prey system with a nonmonotonic functional response is investigated in this work. We study the local asymptotic stability of the positive equilibrium of the system by examining the characteristic equation of the linearized system corresponding to the model.By choosing the growth rate as a bifurcation parameter, the existence of Neimark-Sacker and period-doubling bifurcations at the positive equilibrium is established. Furthermore, the effects of perturbations on the system dynamics are investigated. Finally, examples are presented to illustrate our main results.

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Numerical Bifurcation Analysis of Maps

Cited by 65 publications

References 129 publications

Dynamics of Coordinate Ascent Variational Inference: A Case Study in 2D Ising Models

Dynamics of Coordinate Ascent Variational Inference: A Case Study in 2D Ising Models

Developing a continuous SIR epidemic model and its discrete version using Euler method: Analyzing dynamics with analytical and numerical methods

Bifurcation analysis of a two‐dimensional discrete‐time predator–prey model

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