Concept: Logistic map
The mathematical structure of Sudoku puzzles is akin to hard constraint satisfaction problems lying at the basis of many applications, including protein folding and the ground-state problem of glassy spin systems. Via an exact mapping of Sudoku into a deterministic, continuous-time dynamical system, here we show that the difficulty of Sudoku translates into transient chaotic behavior exhibited by this system. We also show that the escape rate κ, an invariant of transient chaos, provides a scalar measure of the puzzle’s hardness that correlates well with human difficulty ratings. Accordingly, η = -log(10)κ can be used to define a “Richter”-type scale for puzzle hardness, with easy puzzles having 0 < η ≤ 1, medium ones 1 < η ≤ 2, hard with 2 < η ≤ 3 and ultra-hard with η > 3. To our best knowledge, there are no known puzzles with η > 4.
- Journal of the Royal Society, Interface / the Royal Society
- Published over 6 years ago
A new tool for visualization and analysis of system dynamics is introduced: the phasegram. Its application is illustrated with both classical nonlinear systems (logistic map and Lorenz system) and with biological voice signals. Phasegrams combine the advantages of sliding-window analysis (such as the spectrogram) with well-established visualization techniques from the domain of nonlinear dynamics. In a phasegram, time is mapped onto the x-axis, and various vibratory regimes, such as periodic oscillation, subharmonics or chaos, are identified within the generated graph by the number and stability of horizontal lines. A phasegram can be interpreted as a bifurcation diagram in time. In contrast to other analysis techniques, it can be automatically constructed from time-series data alone: no additional system parameter needs to be known. Phasegrams show great potential for signal classification and can act as the quantitative basis for further analysis of oscillating systems in many scientific fields, such as physics (particularly acoustics), biology or medicine.
Extreme events are ubiquitous in a wide range of dynamical systems, including turbulent fluid flows, nonlinear waves, large-scale networks, and biological systems. We propose a variational framework for probing conditions that trigger intermittent extreme events in high-dimensional nonlinear dynamical systems. We seek the triggers as the probabilistically feasible solutions of an appropriately constrained optimization problem, where the function to be maximized is a system observable exhibiting intermittent extreme bursts. The constraints are imposed to ensure the physical admissibility of the optimal solutions, that is, significant probability for their occurrence under the natural flow of the dynamical system. We apply the method to a body-forced incompressible Navier-Stokes equation, known as the Kolmogorov flow. We find that the intermittent bursts of the energy dissipation are independent of the external forcing and are instead caused by the spontaneous transfer of energy from large scales to the mean flow via nonlinear triad interactions. The global maximizer of the corresponding variational problem identifies the responsible triad, hence providing a precursor for the occurrence of extreme dissipation events. Specifically, monitoring the energy transfers within this triad allows us to develop a data-driven short-term predictor for the intermittent bursts of energy dissipation. We assess the performance of this predictor through direct numerical simulations.
- IEEE transactions on pattern analysis and machine intelligence
- Published over 6 years ago
We consider the problem of categorizing video sequences of dynamic textures, i.e., nonrigid dynamical objects such as fire, water, steam, flags, etc. This problem is extremely challenging because the shape and appearance of a dynamic texture continuously change as a function of time. State-of-the-art dynamic texture categorization methods have been successful at classifying videos taken from the same viewpoint and scale by using a Linear Dynamical System (LDS) to model each video, and using distances or kernels in the space of LDSs to classify the videos. However, these methods perform poorly when the video sequences are taken under a different viewpoint or scale. In this paper, we propose a novel dynamic texture categorization framework that can handle such changes. We model each video sequence with a collection of LDSs, each one describing a small spatiotemporal patch extracted from the video. This Bag-of-Systems (BoS) representation is analogous to the Bag-of-Features (BoF) representation for object recognition, except that we use LDSs as feature descriptors. This choice poses several technical challenges in adopting the traditional BoF approach. Most notably, the space of LDSs is not euclidean; hence, novel methods for clustering LDSs and computing codewords of LDSs need to be developed. We propose a framework that makes use of nonlinear dimensionality reduction and clustering techniques combined with the Martin distance for LDSs to tackle these issues. Our experiments compare the proposed BoS approach to existing dynamic texture categorization methods and show that it can be used for recognizing dynamic textures in challenging scenarios which could not be handled by existing methods.
Following the publication of the Task Force document on heart rate variability (HRV) in 1996, a number of articles have been published to describe new HRV methodologies and their application in different physiological and clinical studies. This document presents a critical review of the new methods. A particular attention has been paid to methodologies that have not been reported in the 1996 standardization document but have been more recently tested in sufficiently sized populations. The following methods were considered: Long-range correlation and fractal analysis; Short-term complexity; Entropy and regularity; and Nonlinear dynamical systems and chaotic behaviour. For each of these methods, technical aspects, clinical achievements, and suggestions for clinical application were reviewed. While the novel approaches have contributed in the technical understanding of the signal character of HRV, their success in developing new clinical tools, such as those for the identification of high-risk patients, has been rather limited. Available results obtained in selected populations of patients by specialized laboratories are nevertheless of interest but new prospective studies are needed. The investigation of new parameters, descriptive of the complex regulation mechanisms of heart rate, has to be encouraged because not all information in the HRV signal is captured by traditional methods. The new technologies thus could provide after proper validation, additional physiological, and clinical meaning. Multidisciplinary dialogue and specialized courses in the combination of clinical cardiology and complex signal processing methods seem warranted for further advances in studies of cardiac oscillations and in the understanding normal and abnormal cardiac control processes.
- Conservation biology : the journal of the Society for Conservation Biology
- Published almost 7 years ago
United States and Canadian governments have responded to legal requirements to reduce human-induced whale mortality via vessel strikes and entanglement in fishing gear by implementing a suite of regulatory actions. We analyzed the spatial and temporal patterns of mortality of large whales in the Northwest Atlantic (23.5°N to 48.0°N), 1970 through 2009, in the context of management changes. We used a multinomial logistic model fitted by maximum likelihood to detect trends in cause-specific mortalities with time. We compared the number of human-caused mortalities with U.S. federally established levels of potential biological removal (i.e., species-specific sustainable human-caused mortality). From 1970 through 2009, 1762 mortalities (all known) and serious injuries (likely fatal) involved 8 species of large whales. We determined cause of death for 43% of all mortalities; of those, 67% (502) resulted from human interactions. Entanglement in fishing gear was the primary cause of death across all species (n = 323), followed by natural causes (n = 248) and vessel strikes (n = 171). Established sustainable levels of mortality were consistently exceeded in 2 species by up to 650%. Probabilities of entanglement and vessel-strike mortality increased significantly from 1990 through 2009. There was no significant change in the local intensity of all or vessel-strike mortalities before and after 2003, the year after which numerous mitigation efforts were enacted. So far, regulatory efforts have not reduced the lethal effects of human activities to large whales on a population-range basis, although we do not exclude the possibility of success of targeted measures for specific local habitats that were not within the resolution of our analyses. It is unclear how shortfalls in management design or compliance relate to our findings. Analyses such as the one we conducted are crucial in critically evaluating wildlife-management decisions. The results of these analyses can provide managers with direction for modifying regulated measures and can be applied globally to mortality-driven conservation issues. Evaluación del Manejo para Mitigar Efectos Antropogénicos sobre Ballenas Mayores.
In this paper, we characterize major depression (MD) as a complex dynamic system in which symptoms (e.g., insomnia and fatigue) are directly connected to one another in a network structure. We hypothesize that individuals can be characterized by their own network with unique architecture and resulting dynamics. With respect to architecture, we show that individuals vulnerable to developing MD are those with strong connections between symptoms: e.g., only one night of poor sleep suffices to make a particular person feel tired. Such vulnerable networks, when pushed by forces external to the system such as stress, are more likely to end up in a depressed state; whereas networks with weaker connections tend to remain in or return to a non-depressed state. We show this with a simulation in which we model the probability of a symptom becoming ‘active’ as a logistic function of the activity of its neighboring symptoms. Additionally, we show that this model potentially explains some well-known empirical phenomena such as spontaneous recovery as well as accommodates existing theories about the various subtypes of MD. To our knowledge, we offer the first intra-individual, symptom-based, process model with the potential to explain the pathogenesis and maintenance of major depression.
Antibiotics as emerging environmental contaminants, are widely used in both human and veterinary medicines. A solid-state nanopore sensing method is reported in this article to detect Tetracycline, which is based on Tet-off and Tet-on systems. rtTA (reverse tetracycline-controlled trans-activator) and TRE (Tetracycline Responsive Element) could bind each other under the action of Tetracycline to form one complex. When the complex passes through nanopores with 8 ~ 9 nanometers in diameter, we could detect the concentrations of Tet from 2 ng/mL to 2000 ng/mL. According to the Logistic model, we could define three growth zones of Tetracycline for rtTA and TRE. The slow growth zone is 0-39.5 ng/mL. The rapid growth zone is 39.5-529.7 ng/mL. The saturated zone is > 529.7 ng/mL. Compared to the previous methods, the nanopore sensor could detect and quantify these different kinds of molecule at the single-molecule level.
Many biological variables sampled from human subjects show a diurnal pattern, which poses special demands on the techniques used to analyze such data. Furthermore, most biological variables belong to nonlinear dynamical systems, which may make linear statistical techniques less suitable to analyze their dynamics. The current study investigates the usefulness of two analysis techniques based on nonlinear lagged vector embeddings: sequentially weighted global linear maps (SMAP), and bundle embeddings.
Immune reconstitution kinetics and subsequent clinical outcomes in HLA-matched recipients of allogeneic stem cell transplantation (SCT) are variable and difficult to predict. Considering SCT as a dynamical system, may allow sequence differences across the exomes of the transplant donors and recipients to be used to simulate an alloreactive T cell response, which may allow better clinical outcome prediction. To accomplish this, whole exome sequencing was performed on 34 HLA matched SCT donor-recipient pairs (DRP), and the nucleotide sequence differences translated to peptides. The binding affinity of the peptides to the relevant HLA in each DRP was determined. The resulting array of peptide-HLA binding affinity values in each patient was considered as an operator modifying a hypothetical T cell repertoire vector, in which each T cell clone proliferates in accordance to the logistic equation of growth. Using an iterating system of matrices, each simulated T cell clone’s growth was calculated with the steady state population being proportional to the magnitude of the binding affinity of the driving HLA-peptide complex. Incorporating competition between T cell clones responding to different HLA-peptide complexes reproduces a number of features of clinically observed T cell clonal repertoire in the simulated repertoire. These include, sigmoidal growth kinetics of individual T cell clones and overall repertoire, Power Law clonal frequency distribution, increase in repertoire complexity over time with increasing clonal diversity and finally, alteration of clonal dominance when a different antigen array is encountered, such as in stem cell transplantation. The simulated, alloreactive T cell repertoire was markedly different in HLA matched DRP. The patterns were differentiated by rate of growth, and steady state magnitude of the simulated T cell repertoire and demonstrate a possible correlation with survival. In conclusion, exome wide sequence differences in DRP may allow simulation of donor alloreactive T cell response to recipient antigens and may provide a quantitative basis for refining donor selection and titration of immunosuppression following SCT.