/pdf/dynamical-systems-with-applications-using-maple-6glx3v43l8.pdf

Dynamical Systems with Applications Using MAPLE

Chaotic phenomena and presence of complexity in various nonlinear dynamical systems extensively discussed in the context of recent researches. Discrete as well as continuous dynamical systems both considered here. Visualization of regularity and chaotic motion presented through bifurcation diagrams by varying a parameter of the system while keeping other parameters constant. In the processes, some perfect indicator of regularity and chaos discussed with appropriate examples. Measure of chaos in terms of Lyapunov exponents and that of complexity as increase in topological entropies discussed. The methodology to calculate these explained in details with exciting examples. Regular and chaotic attractors emerging during the study are drawn and analyzed. Correlation dimension, which provides the dimensionality of a chaotic attractor discussed in detail and calculated for different systems. Results obtained presented through graphics and in tabular form. Two techniques of chaos control, pulsive feedback control and asymptotic stability analysis, discussed and applied to control chaotic motion for certain cases. Finally, a brief discussion held for the concluded investigation.

/pdf/chaos-and-complexity-dynamics-of-evolutionary-systems-13gngkkocr.pdf

Chaos and Complexity Dynamics of Evolutionary Systems

In this paper, we have considered Time-Frequency Analysis (TFA) and Poincare Surface of Section (PSS) for the study of the phase space structure of nonlinear dynamical system. We have examined a sample of orbits taken in the framework of Circular Restricted Three-Body Problem (CRTBP). We have computed ridge-plots (i.e. time-frequency landscape) using the phase of the continuous wavelet transform. Clear visualization of resonance trappings and the transitions is an important feature of this method, which is presented using ridge-plots. The identification between periodic and quasi-periodic, chaotic sticky and nonsticky and regular and chaotic orbits are done in comparatively less time and with less computational effort. The spatial case of Circular Restricted Three-Body problem is considered to show the strength of Time-Frequency Analysis to higher dimensional systems. Also, with the help of ridge-plots, we can visualize the phenomenon of transient chaos.

Characterization of the Phase Space Structure of Circular Restricted Three-Body Problem: An Alternative Approach

/pdf/complexity-dynamics-of-gumowski-mira-map-2wm7e0o4q1.pdf

Complexity Dynamics of Gumowski-Mira Map

Correlation dimension and topological entropy in discrete maps

The, “Complex systems”, stands as a broad term for many diverse disciplines of science and engineering including natural & medical sciences Complexities appearing in various dynamical systems during evolution are now interesting subjects of studies Chaos appearing in various dynamical systems can also be viewed as a form of complexity For some cases nonlinearities within the systems and for other cases piecewise continuity property of the system are responsible for such complexity Dynamical systems represented by mathematical models having piecewise continuous properties show strange complexity character during evolution Interesting recent articles explain widely on complexities in various systems Observable quantities for complexity are measurement of Lyapunov exponents (LCEs), topological entropies, correlation dimension etc The present article is related to study of complexity in systems having piecewise continuous properties Some mathematical models are considered here in this regard including famous Lozi map, a discrete mathematical model and Chua circuit, a continuous model Investigations have been carried forward to obtain various attractors of these maps appearing during evolution in diverse and interesting pattern for different set of values of parameters and for different initial conditions Numerical investigations extended to obtain bifurcation diagrams, calculations of LCEs, topological entropies and correlation dimension together with their graphical representation

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Complexity Studies in Some Piece wise Continuous Dynamical Systems

In this paper, A discrete-time food chain model of three species has been introduced comprising of a set of three nonlinear difference equations. Evolutionary dynamics of this system have investigated and regular as well as chaotic attractors have been obtained for certain parameter values. Bifurcation diagrams have been drawn by varying parameters along coordinate axes and the motion has been analyzed. Numerical calculations have been carried out for calculations of Lyapunov exponents, topological entropies and correlation dimension. The investigation is then further extended to obtain FLI, SALI and DLI for regular and chaotic evolution of the food chain model.

On dynamics of chaotic evolution in food chain model

We discuss the role of convective zones in the treatment of coupled pulsation and convection. In particular, we have investigated behaviors of the convective depth, , and of the ratio between the convective and total luminosities, c. In deep convection, additional fixed points appear, the values of which increase with the depth of zone, and become larger than unity. A diagram of stability criteria derived from the linearized model has been drawn, which shows stable regions fitting each criterion. Then, various flow patterns have been obtained by means of numerical simulations for sets of c and , and the structure of the mode has been discussed. The results of simulations are shown in tables and figures with a fixed set of parameters other than the set of c and , which show that deep convective zones cause chaotic solutions and long-period oscillations. Some suggestions are made based on these findings in stellar pulsation within the convective zone.

Nonlinear Behaviors of Pulsating Stars with Convective Zones

Asymptotic stability analysis applied to stabilize unstable fixed points and to control chaotic motions in two and threedimensional discrete dynamical systems. A new set of parameter values obtained which stabilizes an unstable fixed point and control the chaotic evolution to regularity. The output of the considered model and that of the adjustable system continuously compared by a typical feedback and the difference used by the adaptation mechanism to modify the parameters. Suitable numerical simulation which are used thoroughly discussed and parameter values are adjusted. The findings are significant and interesting. This strategy has some advantages over many other chaos control methods in discrete systems but, however it can be applied within some limitations. KeywordsAsymptotic stability, Control parameter, Chaos, Lyapunov exponents.

https://www.ijmems.in/volumes/volume6/number2/42-IJMEMS-20-115-6-2-677-687-2021.pdf

L. M. Saha

Papers

Correlation dimension and topological entropy in discrete maps

Complexity Studies in Some Piece wise Continuous Dynamical Systems

On dynamics of chaotic evolution in food chain model

Nonlinear Behaviors of Pulsating Stars with Convective Zones

Asymptotic Stability Analysis Applied in Two and Three-Dimensional Discrete Systems to Control Chaos