Fuzzy Responses and Bifurcations of a Forced Duffing Oscillator with a Triple-Well Potential

2015 ◽  
Vol 25 (01) ◽  
pp. 1550005 ◽  
Author(s):  
Ling Hong ◽  
Jun Jiang ◽  
Jian-Qiao Sun
Keyword(s):  
Steady State ◽  
Master Equation ◽  
State Space ◽  
Invariant Manifolds ◽  
The State ◽  
Invariant Sets ◽  
Generalized Cell Mapping ◽  
Potential Wells ◽  
Potential System ◽  
Fuzzy Noise

Responses and bifurcations of a forced triple-well potential system with fuzzy uncertainty are studied by means of the Fuzzy Generalized Cell Mapping (FGCM) method. A rigorous mathematical foundation of the FGCM is established as a discrete representation of the fuzzy master equation for the possibility transition of continuous fuzzy processes. The FGCM offers a very effective approach for solutions to the fuzzy master equation based on the min–max operator of fuzzy logic. A fuzzy response is characterized by its topology in the state space and its possibility measure of membership distribution functions (MDFs). A fuzzy bifurcation implies a sudden change both in the topology and in the MDFs. The response topology is obtained based on the qualitative analysis of the FGCM involving the Boolean operation of 0 and 1. The MDFs are determined by the quantitative analysis of the FGCM with the min–max calculations. With an increase of the intensity of fuzzy noise, noise-induced escape from each of the potential wells defines two types of bifurcations, namely catastrophe and explosion. This paper focuses on the evolution of transient and steady-state MDFs of the fuzzy response. As the intensity of fuzzy noise increases, steady-state MDFs cover a bigger area in the state space with higher membership values spreading out to a larger area. The previous conjectures are further confirmed that steady-state MDFs are dependent on initial possibility distributions due to the nonsmooth and nonlinear nature of the min–max operation. It is found that as time goes on, transient MDFs spread around three potential wells. The evolutionary orientation of transient MDFs aligns with unstable invariant manifolds leading to stable invariant sets. Two examples of additive and multiplicative fuzzy noise are given.

scholarly journals Novel domain expansion methods to improve the computational efficiency of the Chemical Master Equation solution for large biological networks

2020 ◽  
Vol 21 (1) ◽  
Author(s):  
Rahul Kosarwal ◽  
Don Kulasiri ◽  
Sandhya Samarasinghe
Keyword(s):  
Master Equation ◽  
State Space ◽  
Performance Management ◽  
Numerical Solutions ◽  
De Novo ◽  
Exact Analytical Solution ◽  
Chemical Master Equation ◽  
The State ◽  
Biochemical System ◽  
Biochemical Systems

Abstract Background Numerical solutions of the chemical master equation (CME) are important for understanding the stochasticity of biochemical systems. However, solving CMEs is a formidable task. This task is complicated due to the nonlinear nature of the reactions and the size of the networks which result in different realizations. Most importantly, the exponential growth of the size of the state-space, with respect to the number of different species in the system makes this a challenging assignment. When the biochemical system has a large number of variables, the CME solution becomes intractable. We introduce the intelligent state projection (ISP) method to use in the stochastic analysis of these systems. For any biochemical reaction network, it is important to capture more than one moment: this allows one to describe the system’s dynamic behaviour. ISP is based on a state-space search and the data structure standards of artificial intelligence (AI). It can be used to explore and update the states of a biochemical system. To support the expansion in ISP, we also develop a Bayesian likelihood node projection (BLNP) function to predict the likelihood of the states. Results To demonstrate the acceptability and effectiveness of our method, we apply the ISP method to several biological models discussed in prior literature. The results of our computational experiments reveal that the ISP method is effective both in terms of the speed and accuracy of the expansion, and the accuracy of the solution. This method also provides a better understanding of the state-space of the system in terms of blueprint patterns. Conclusions The ISP is the de-novo method which addresses both accuracy and performance problems for CME solutions. It systematically expands the projection space based on predefined inputs. This ensures accuracy in the approximation and an exact analytical solution for the time of interest. The ISP was more effective both in predicting the behavior of the state-space of the system and in performance management, which is a vital step towards modeling large biochemical systems.

Dependencies in Markovian networks

1995 ◽  
Vol 27 (1) ◽  
pp. 226-254 ◽  
Author(s):  
H. Daduna ◽  
R. Szekli
Keyword(s):  
Steady State ◽  
State Space ◽  
Queueing Network ◽  
Negative Association ◽  
The State ◽  
Transient Phase ◽  
Positive Correlations ◽  
Exact Bounds ◽  
Joint Probabilities ◽  
Birth Death

Monotonicity and correlation results for queueing network processes, generalized birth–death processes and generalized migration processes are obtained with respect to various orderings of the state space. We prove positive (e.g. association) and negative (e.g. negative association) correlations in space and positive correlations in time for different situations, in steady state as well as in the transient phase of the system. This yields exact bounds for joint probabilities in terms of their independent versions.

scholarly journals On the Properties of the Reaction Counts Chemical Master Equation

Entropy ◽  
2019 ◽  
Vol 21 (6) ◽  
pp. 607
Author(s):  
Sunkara
Keyword(s):  
Master Equation ◽  
State Space ◽  
Continuous Time ◽  
A Priori ◽  
Chemical Master Equation ◽  
The State ◽  
High Dimensional ◽  
Population Counts ◽  
Theoretical Results ◽  
Over Time

The reaction counts chemical master equation (CME) is a high-dimensional variant ofthe classical population counts CME. In the reaction counts CME setting, we count the reactionswhich have fired over time rather than monitoring the population state over time. Since a reactioneither fires or not, the reaction counts CME transitions are only forward stepping. Typically thereare more reactions in a system than species, this results in the reaction counts CME being higher indimension, but simpler in dynamics. In this work, we revisit the reaction counts CME frameworkand its key theoretical results. Then we will extend the theory by exploiting the reactions counts’forward stepping feature, by decomposing the state space into independent continuous-time Markovchains (CTMC). We extend the reaction counts CME theory to derive analytical forms and estimatesfor the CTMC decomposition of the CME. This new theory gives new insights into solving hittingtimes-, rare events-, and a priori domain construction problems.

BIFURCATIONS OF A FORCED DUFFING OSCILLATOR IN THE PRESENCE OF FUZZY NOISE BY THE GENERALIZED CELL MAPPING METHOD

2006 ◽  
Vol 16 (10) ◽  
pp. 3043-3051 ◽  
Author(s):  
LING HONG ◽  
JIAN-QIAO SUN
Keyword(s):  
Steady State ◽  
Duffing Oscillator ◽  
Mapping Method ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
Deterministic Systems ◽  
Fuzzy Noise ◽  
Topological Changes

Bifurcations of a forced Duffing oscillator in the presence of fuzzy noise are studied by means of the fuzzy generalized cell mapping (FGCM) method. The FGCM method is first introduced. Two categories of bifurcations are investigated, namely catastrophic and explosive bifurcations. Fuzzy bifurcations are characterized by topological changes of the attractors of the system, represented by the persistent groups of cells in the context of the FGCM method, and by changes of the steady state membership distribution. The fuzzy noise-induced bifurcations studied herein are not commonly seen in the deterministic systems, and can be well described by the FGCM method. Furthermore, two conjectures are proposed regarding the condition under which the steady state membership distribution of a fuzzy attractor is invariant or not.

scholarly journals Novel domain expansion methods to improve the computational efficiency of the solution of Chemical Master Equation for large biological networks.

2020 ◽  
Author(s):  
Rahul Kosarwal ◽  
Don Kulasiri ◽  
Sandhya Samarasinghe
Keyword(s):  
Master Equation ◽  
State Space ◽  
Performance Management ◽  
Numerical Solutions ◽  
De Novo ◽  
Exact Analytical Solution ◽  
Chemical Master Equation ◽  
The State ◽  
Biochemical System ◽  
Biochemical Systems

Abstract Background: Numerical solutions of the chemical master equation (CME) are important to understand the stochasticity of biochemical systems. However, solving CMEs is a formidable task due to the nonlinear nature of the reactions and size of the networks that result in different realisations and, most importantly, the exponential growth of the size of the state-space with respect to the number of different species in the system. When the size of the biochemical system is very large in terms of the number of variables, the solution to the CME becomes intractable. Therefore, we introduce the intelligent state projection (𝐼𝑆𝑃) method to use in the stochastic analysis of these systems. For any biochemical reaction network, it is important to capture more than one moment to describe the dynamic behaviour of the system. 𝐼𝑆𝑃 is based on a state-space search and the data structure standards of artificial intelligence (𝐴𝐼) to explore and update the states of a biochemical system. To support the expansion in 𝐼𝑆𝑃, we also develop a Bayesian likelihood node projection (𝐵𝐿𝑁𝑃) function to predict the likelihood of the states. Results: To show the acceptability and effectiveness of our method, we apply the 𝐼𝑆𝑃 to several biological models previously discussed in the literature. According to the results of our computational experiments, we show that 𝐼𝑆𝑃 is effective in terms of speed and accuracy of the expansion, accuracy of the solution, and provides a better understanding of the state-space of the system in terms of blueprint patterns. Conclusions: The 𝐼𝑆𝑃 is the de-novo method to address the accuracy as well as the performance problems for the solution of the CME. It systematically expands the projection space based on predefined inputs, which are useful in providing accuracy in the approximation and an exact analytical solution at the time of interest. The 𝐼𝑆𝑃 was more effective in terms of predicting the behaviour of the state-space of the system and in performance management, which is a vital step towards modelling large biochemical systems.

Dependencies in Markovian networks

1995 ◽  
Vol 27 (01) ◽  
pp. 226-254 ◽  
Author(s):  
H. Daduna ◽  
R. Szekli
Keyword(s):  
Steady State ◽  
State Space ◽  
Queueing Network ◽  
Negative Association ◽  
The State ◽  
Transient Phase ◽  
Positive Correlations ◽  
Exact Bounds ◽  
Joint Probabilities ◽  
Birth Death

Monotonicity and correlation results for queueing network processes, generalized birth–death processes and generalized migration processes are obtained with respect to various orderings of the state space. We prove positive (e.g. association) and negative (e.g. negative association) correlations in space and positive correlations in time for different situations, in steady state as well as in the transient phase of the system. This yields exact bounds for joint probabilities in terms of their independent versions.

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