Chaos and chaotic transients in an aeroelastic system
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Phase portrait
Aeroelasticity
Poincaré map
Position (finance)
We present the results of the investigation of a family of dynamic systems on a real plane containing reciprocal cubic and square polynomials in their right parts. A Poincare method of serial mappings has been used. All possible for the systems under consideration topologically different types of phase portraits in a Poincare circle have been described and constructed. The total amount of such portraits appears to be over 200. Coefficient criteria of their realization have been outlined.
Phase portrait
Poincaré conjecture
Poincaré map
Realization (probability)
Square (algebra)
Phase plane
Reciprocal
Cubic function
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Abstract In this paper, authors present results of the original investigation of a special class of dynamic systems with the reciprocal polynomial –cubic and square – right parts on a real plane. The global task was to construct all topologically different phase portraits in a Poincare circle with criteria of them. For such an aim a Poincare method of a central and orthogonal mappings has been used. Eventually above the two hundred of different phase portraits were constructed. Each and every portrait has been described in a table. Each line of a table describes one invariant cell of the phase portrait under consideration, its boundary, a source of its phase flow and a sink of it.
Phase portrait
Poincaré conjecture
Portrait
Poincaré map
Square (algebra)
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Phonetic signals are considered from the viewpoint of nonlinear dynamics. Phase portraits of the signals are analyzed in embedding space, dimension and the largest Lyapunov exponent are estimated. It is shown that dimension of speech signals is low and the largest Lyapunov exponent is positive.
Phase portrait
Correlation dimension
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Phase portrait
Harmonic
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A two-degree-of-freedom vibro-impact system was investigated.By using a constant phase surface during the pre-impact instant as Poincare section and introducing a local map,Poincare map was constructed and the corresponding Jacobi matrix was obtained.Using Gram-Schmidt ortho-normalization and the iterative method,the method for calculating the spectra of Lyapunov exponents of the vibro-impact system was obtained.The numerical simulation showed that the sequences of the spectra of Lyapunov exponents of the periodic attractor and the chaotic attractor of the system can converge well.The numerical simulation also showed that the largest Lyapunov exponent agrees with the attractor behavior observed in the corresponding global bifurcation diagram.
Poincaré map
Normalization
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Poincaré map
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Poincaré map
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Using the classical tools of nonlinear dynamics, we study the process of self-organization and the appearance of the chaos in the metabolic process in a cell with the help of a mathematical model of the transformation of steroids by a cell Arthrobacter globiformis. We constructed the phase-parametric diagrams obtained under a variation of the dissipation of the kinetic membrane potential. The oscillatory modes obtained are classified as regular and strange attractors. We calculated the bifurcations, by which the self-organization and the chaos occur in the system, and the transitions "chaos-order", "order-chaos", "order-order", and "chaos-chaos" arise. Feigenbaum's scenarios and the intermittences are found. For some selected modes, the projections of the phase portraits of attractors, Poincaré sections, and Poincaré maps are constructed. The total spectra of Lyapunov indices for the modes under study are calculated. The structural stability of the attractors is demonstrated. A general scenario of the formation of regular and strange attractors in the given metabolic process in a cell is found. The physical nature of their appearance in the metabolic process is studied.
Cell metabolism
Self-Organization
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A new three-dimensional nonlinear autonomous system is presented,and the qualitative behaviors of the system are analyzed according to the stability theories.By calculating the Lyapunov exponents and the Lyapunov dimensions of the system,the dynamical behaviors of the system with evaluating all the parameters are discussed.The results show that with variation of some parameters,the system may be periodic,quasi-periodic and chaotic.And the results are verified by bifurcation diagrams,phase portraits and Poincare maps.
Phase portrait
Poincaré map
Qualitative analysis
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Based on the further evolvement of the improved chaotic system with constant Lyapunov exponent spectrum, by introducing an absolute term in the dynamic equation, a novel chaotic attractor is found in this paper. Firsty, the existence of chaotic attractor is verified by simulation of phase portrait, Poincaré mapping, and Lyapunov exponent spectrum. Secondly, the basic dynamical behaviour of the new system is investigated and expounded. Simulation of Lyapunov exponent spectrum, bifurcation diagram and numerical analysis on amplitude evolvement of state variables show that the state variables of the chaotic system can be modified linearly by a global linear amplitude adjuster while the Lyapunov exponent spectrum keeps on stable and the chaotic attractor displays the same phase portrait. Finally, an analog circuit is designed to implement the new system, the chaotic attractor is observed and the action of global linear amplitude adjuster is verified, all of which show a good agreement between numerical simulation and experimental results.
Phase portrait
Rössler attractor
Constant (computer programming)
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