Transient Chaos: Complex Dynamics on Finite Time Scales

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The distribution of dominance times has been widely reported to follow a gamma function in humans. Our magnetoencephalography MEG experiments with subjects observing an ambiguous Necker cube flickering image with its two perceptional interpretations yield similar results. The results of the simulations with our model are in a good agreement with our MEG experiments. The simulations reveal that increasing brain noise tends to change the distribution of dominance times from Gaussian to Gamma and then finally to Exponential.

The range of noise values in our model corresponding to Gamma distribution should only be used to compare the model predictions and experimental observations. In the same range of brain noise, we observe a monotonic decrease in the dominance time with increasing brain noise, both in the model and experiment. The results of this work provide interesting and valuable insights into the human brain. This simple yet versatile mechanism may turn out to be radical in understanding human perception.

Keywords: perception, selective adaptation, brain, noise, MEG. We know that the reliable logical response can be extracted from a noisy bistable system at an intermediate value of noise strength when two random, two-level, square waveform serve as the inputs. The asymmetry of the potential plays a very important role and dictates the type of logical operation, such as OR or AND, exhibited by the system.

Here we have shown that one can build logic gates with symmetric bistable potential if the two states of the double-well are thermalized with two different heat baths. We have found that if a given state is kept at a sufficiently low temperature compared to the other, the system shows one type of logic behavior say, OR. Interestingly, the system's response turns into the other kind say, AND if the temperature of the initial low-temperature well is increased slowly and the quality of the logical response first improves and then becomes weak after passing through a maximum at a particular value of temperature.

However, the reliability of the second kind of logical response AND is not as good as the first kind OR and it depends on the amplitude of the inputs. Still one can construct both types of logic gates with maximum reliability by properly choosing the initial low-temperature well. In this poster we will present a collection of intuitive, easy-to-use and performant software packages for nonlinear dynamics and chaos, which compose the fully open source GitHub organization "JuliaDynamics". All implementations were made from the ground up, based on the principles of clarity and intuition, taking full advantage of new programming paradigms.

Examples of features that we will show include Lyapunov exponents, categorizing chaotic behavior, attractor dimensions, recurrence plots, Poincare sections, feature-full billiard evolution, spatiotemporal timeseries prediction and much more. Importantly, obtaining the result for any of these features requires typing typically lines of code examples will be shown on the poster. The software that we will present are: - DynamicalSystems.


  1. Transient Chaos : Complex Dynamics on Finite Time Scales.
  2. PDF Transient Chaos: Complex Dynamics on Finite Time Scales: 173 (Applied Mathematical Sciences).
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Embryonic stem cells are derived from the early blastocyst-stage embryos. Through differentiation programs they can generate every cell type in the body. These differentiation programs are usually driven by extracellular signals.

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We are interested in how the information carried by these external inputs is encoded by the cell in signaling networks activity. We quantitatively analyze signaling dynamics in individual mouse embryonic stem cells in response to an extracellular stimulus.

These time series display some dynamic events that look like pulses of signaling activity, interspersed with intervals of noise. We aim to distinguish noise fluctuations from genuine dynamical activity. To this end we developed a set of local observables employing statistical physics and information theory concepts.


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  • Together with interpeak intervals statistics, we explore how the extracellular stimulus concentrations correlate with signaling dynamic signatures. But this length and precision come at a high cost. Collection of an ice core is very expensive and extraction of proxy data from it is time consumingas well as susceptible to both human and machine error.

    Ensuring the accuracy of these data is as challenging as it is important. However, recent advances in information theory and ice-core measurement technology have provided us the means to begin tackling this problem.

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    In this talk, we demonstrate that estimates of the Shannon entropy rate of the water-isotope data from the West Antarctica Ice Sheet Divide ice core, calculated using permutation entropy techniques, identify regions of the core that merit further investigation. To date this approach has flagged regions containing missing ice, data-post processing errors, instrumentation irregularities, signatures of geothermal heating and identified several intervals in the data that may be of direct relevance to paleoclimate interpretationincluding periods of abrupt climate change.

    Phase oscillators evolving by a Moebius transformation of the unit circle according to the Watanabe-Strogatz theory in principle cannot contract in phase space to form more than one cluster. In numerical simulation, however, Kuramoto-Sakaguchi phase oscillators under common multiplicative noise, which alone leads to the synchronization of the oscillators, and under repulsive coupling, which alone leads to the desynchronization of the oscillators, have been shown to form stable multicluster states.

    Using Fokker-Planck formulation, we show that two-cluster states under such a model are non-attractive. Integrating with various time steps reveals that clustering is a numerical artifact, explained by the existence of higher order Fourier terms in the errors of the employed numerical integration schemes. Monitoring the induced change in certain integrals of motion we quantify these errors.

    On the other hand, in ensembles of general limit cycle oscillators, such as Van der Pol oscillators, due to an anharmonic phase response function, as well as additional amplitude dynamics, multiclusters can occur naturally. Thus that common noise can induce clustering in general oscillator systems under repulsive coupling remains valid. In recent years, data-driven discovery of mathematical models has emerged as a promising alternative to more traditional modeling approaches. While promising, existing approaches have several major weaknesses. Most notably, they break down for data with high levels of noise and have to be tuned empirically to produce meaningful results, making them ill-suited for analyzing experimental data.

    We show how these weaknesses can be addressed using a combination of sparse regression and a weak formulation of the model PDE. The weak formulation has substantial freedom that makes it quite powerful and flexible, but a question arises how this freedom can be used to robustly obtain the most accurate model. Using the 4th-order Kuramoto-Sivashinsky equation for illustration, we show that the approach can be optimized in the limits of low and high noise. In particular, we derive the scaling that relates the accuracy of the model, the parameters of our method, and the properties of the data such as its spatial and temporal resolution or the level of noise.

    The system exhibits superextreme events which denote that the sudden expansion of the system state variables more than the standard extreme events qualifier threshold. In the absence of time-varying parameter, the system is in the chaotic or periodic state. When we include the time-varying parameter into the system, superextreme events have emerged. The influence of parametric perturbation is studied in the different parameters of the model system.

    We explored its dynamical origin and emerging mechanism using different characterization techniques. References: 1.

    Manifestations of the intermittency route to chaos in the physics of condensed matter and of c...

    Kingston, K. Thamilmaran, P. Pal, U. Feudel, and S. Dana, Phys. E 96, Bonatto and A. Endler, Phys.

    Albeverio, V. Jentsch, and H. Kantz, Extreme events in nature and society Springer, Han, Q. Bi, P.

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    Ji, and J. Kurths, Phys. E 92 Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, Phys. X 2, The continuously increasing number of newly discovered worlds outside of our own solar system requires as precise as possible parameter estimations such as planetary masses, orbital characteristics, bulk density, etc.

    Transient Chaos: Complex Dynamics on Finite-Time Scales

    Comprehensive statistical methods and inverse dynamical analyses have been worked out to obtain system parameters from astronomical observations. Nevertheless, the time domain measurements as scalar time series transformed into complex networks serve a powerful tool to investigate dynamical systems via network topology. Many recent works make significant effort to explore the causality relations and coupling directions between connected dynamical systems. In this study a new estimation procedure of planetary masses is presented making use of eclipse time variation in multi-planetary systems.

    Due to the gravitational coupling the motion of planets differs from pure Keplerian ellipse resulting in variable orbital periods. Measuring this tiny effect for nearly co-planar planets one is able to reconstruct the trajectories sharing the same phase space. Transforming then the obtained state vectors of the entangled dynamical systems into network representation, it can be shown that the coupling directions between the interacting sub-networks are related to planetary masses relative to each other.

    Real world dynamic networks are open systems receiving inputs from their environment. There can be many origins for these inputs, including systems malfunction, attacks or model errors. In general, these inputs affect the state of the dynamic system and need to be taken into account for state estimation, data assimilation or prediction. However, in many cases not even the state nodes of the system targeted by these unknown inputs are known.

    In this poster, we present criteria to decide, whether the state nodes targeted by unknown inputs can be identified from output measurements.

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    In cases were the exact localisation of the inputs is impossible, we present algorithms to localise the region in the dynamic network, where the unknown input applies. Our results provide a principled way for error localisation in complex dynamic networks with potential applications in many areas including network biology, engineering and communications. Experimental study of networks of coupled oscillators attracts the attention of many researchers. Convenient objects for the experimental investigation of collective dynamics in networks are coupled electronic oscillators.

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