A chaotic limit cycle paradox

Grantham, Walter J.; Lee, Byoung-Soo

doi:10.1007/bf01968529

Cited by 14 publications

(8 citation statements)

References 18 publications

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“…That is why FCM theory is a very convenient approach for performing modeling and simulation tasks. However, when the system reaches a limit cycle or a chaotic behavior, decisionmaking is practically impossible [30]. In the following Section 3 we review some theoretical results that attempts to provide better stability features on FCM-based models.…”

Section: G Nápoles Et Al / How To Improve the Convergence On Sigmoimentioning

confidence: 99%

How to improve the convergence on sigmoid Fuzzy Cognitive Maps?

Nápoles

Bello

Vanhoof

2014

IDA

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Fuzzy Cognitive Maps (FCM) may be defined as Recurrent Neural Networks that allow causal reasoning. According to the transformation function used for updating the activation value of concepts they can be characterized as discrete or continuous. It is remarkable that FCM having discrete neurons never exhibit chaotic states, but this premise cannot be guaranteed for FCM having continuous concepts. On the other hand, complex Sigmoid FCM resulting from experts or learning algorithms often show chaotic or cyclic patterns, therefore leading to confusing interpretation of the investigated system. The first contribution of this paper is focused on explaining why most studies on FCM stability are not applicable to FCM used on classification or decision-making tasks. Next we describe a non-direct learning methodology based on Swarm Intelligence for improving the system stability once the causal weight estimation is done. The objective here is to find a specific threshold function for each map neuron simulating an external stimulus, instead of using the same transformation function for all concepts. At the end, we can compute more stable maps, so better consistency in hidden patterns is achieved. G. Nápoles et al. / How to improve the convergence on sigmoid FCM?decision-making tasks, risk analysis, prediction, text categorization, pattern recognition, management, and classification. However, estimating parameters that characterize the whole system (e.g. the causal weight matrix) may be tedious for humans, leading to inefficient models. In order to increase the reliability of FCM-based models several learning algorithms for tuning such parameters have been introduced [10], although most of them are focused on computing causal relations between map concepts.On the other hand, existing approaches suppose that FCM are closed systems and they do not consider external influences, while other factors such as the FCM stability are frequently ignored. As far as known, there is no existence of any learning method for enhancing the system stability once the system causality is established. For example, let us suppose a FCM resulting from experts where causal connections may be partially modified (e.g. we know the direction of causalities and an approximation of their values that should be preserved). Can we expect lineal stability in the final map? If not, how to improve the stability of the system without affecting causal connections estimated by experts?In the literature a few researches concerning FCM convergence have been proposed. For instance, Kosko [4] developed an analytic method based on Liapounov functions for reaching stable solutions on Feedback Standard Additive Models (SAM -which share several of the FCM characteristics). Unfortunately, Kosko concluded that such conditions cannot be extended to FCM due to the large number of feedback links involved in FCM-based models. More recently, other analytical methods were introduced (see Section 3), but we believe that such approaches are mostly useful for stabilizing FCM used in...

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Section: G Nápoles Et Al / How To Improve the Convergence On Sigmoimentioning

confidence: 99%

How to improve the convergence on sigmoid Fuzzy Cognitive Maps?

Nápoles

Bello

Vanhoof

2014

IDA

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“…We obtained more precise results by using sigmoid function with extreme value of γ as defined in [26]. One other possible demonstration dealing with discontinuity in the forced system can be found in [27]. …”

Section: Overall Numerical Analysismentioning

confidence: 99%

Dynamical Tangles in Third-Order Oscillator with Single Jump Function

Petržela

Götthans

Guzan

2014

The Scientific World Journal

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This contribution brings a deep and detailed study of the dynamical behavior associated with nonlinear oscillator described by a single third-order differential equation with scalar jump nonlinearity. The relative primitive geometry of the vector field allows making an exhaustive numerical analysis of its possible solutions, visualizations of the invariant manifolds, and basins of attraction as well as proving the existence of chaotic motion by using the concept of both Shilnikov theorems. The aim of this paper is also to complete, carry out and link the previous works on simple Newtonian dynamics, and answer the question how individual types of the phenomenon evolve with time via understandable notes.

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“…For example, if parameters are = 1.4, = 1, = 1.1, = 1, then system (1) is a limit cycle, while the largest Lyapunov exponent is positive. Obviously, it is incorrect, which was named a chaotic limit cycle paradox in [24]. To calculate the Largest Lyapunov exponent of system (1) correctly, the signum function should be replaced by continuous hyperbolic tangent function [25]:…”

Section: Generalized Ueda Oscillatormentioning

confidence: 99%

Multiple Coexisting Attractors and Hysteresis in the Generalized Ueda Oscillator

Sun

Li-kun

Dong

et al. 2013

Mathematical Problems in Engineering

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A periodically forced nonlinear oscillator called the generalized Ueda oscillator is proposed. The restoring force term of this equation consists of a nonlinear functionsgn(x)and an absolute function with a variant power. Dynamics is investigated by detailed numerical analysis as well as dynamic simulation, including the largest Lyapunov exponent, phase diagrams, and bifurcation diagrams. Multiple coexisting attractors and complex hysteresis phenomenon are observed. The results show that this system has rich dynamical behaviors, and it has a promising application in the fields of science and engineering.

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A chaotic limit cycle paradox

Cited by 14 publications

References 18 publications

How to improve the convergence on sigmoid Fuzzy Cognitive Maps?

How to improve the convergence on sigmoid Fuzzy Cognitive Maps?

Dynamical Tangles in Third-Order Oscillator with Single Jump Function

Multiple Coexisting Attractors and Hysteresis in the Generalized Ueda Oscillator

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