QUENCHING LORENZIAN CHAOS

How fluctuations can be eliminated or attenuated is a matter of general interest in the study of steadily-forced dissipative nonlinear dynamical systems. Here, we extend previous work on "nonlinear quenching" [Hide, 1997] by investigating the phenomenon in systems governed by the novel autonomous set of nonlinear ordinary differential equations (ODE's) [Formula: see text], ẏ=-xzq+bx-y and ż=xyq-cz (where (x, y, z) are time(t)-dependent dimensionless variables and [Formula: see text], etc.) in representative cases when q, the "quenching function", satisfies q=1-e+ey with 0≤e≤1. Control parameter space based on a,b and c can be divided into two "regions", an S-region where the persistent solutions that remain after initial transients have died away are steady, and an F-region where persistent solutions fluctuate indefinitely. The "Hopf boundary" between the two regions is located where b=bH(a, c; e) (say), with the much studied point (a, b, c)=(10, 28, 8/3), where the persistent "Lorenzian" chaos that arises in the case when e=0 was first found lying close to b=bH(a, c; 0). As e increases from zero the S-region expands in total "volume" at the expense of F-region, which disappears altogether when e=1 leaving persistent solutions that are steady throughout the entire parameter space.

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INTERACTIVE GRAPHICAL EXPLORATION OF MULTIDIMENSIONAL NONLINEAR DYNAMICAL SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127492000264 ◽

1992 ◽

Vol 02 (02) ◽

pp. 251-261 ◽

Cited By ~ 1

Author(s):

JUDY CHALLINGER

Keyword(s):

Dynamical System ◽

Parameter Space ◽

Nonlinear Dynamical Systems ◽

Nonlinear Dynamical System ◽

Three Dimensional ◽

Initial Point ◽

Unit Cube ◽

Dimensional Parameter ◽

Data Set ◽

Nonlinear Dynamical

This paper discusses the application of an inherently three-dimensional graphical representation tool, isosurfaces, as a means to interactively explore and visualize the attractors of a nonlinear dynamical system with a fifteen-dimensional parameter space. A program has been written which allows the scientist to interactively select and visualize three-dimensional sub-spaces of the fifteen-dimensional parameter space. The dynamical system used to illustrate these concepts is a discrete-time, nonlinear, three-nation Richardson model with economic constraints. This dynamical system, which models the shifting alliances of nations in an arms race, maps an initial point in the unit cube to another point in the unit cube after multiple iterations of the model functions. Using an isosurface function on the resulting volumetric data set, surfaces indicating the changing alliances of nations are generated and rendered.

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The Negative Side of Chua's Circuit Parameter Space: Stability Analysis, Period-Adding, Basin of Attraction Metamorphoses, and Experimental Investigation

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414300250 ◽

2014 ◽

Vol 24 (09) ◽

pp. 1430025 ◽

Cited By ~ 9

Author(s):

Rene O. Medrano-T ◽

Ronilson Rocha

Keyword(s):

Parameter Space ◽

Nonlinear Dynamical Systems ◽

Basin Of Attraction ◽

Circuit Parameter ◽

Chua’S Circuit ◽

Nonlinear Dynamical ◽

Chua's Circuit ◽

Describing Functions ◽

Theoretical Investigations ◽

Analysis Of Stability

Although Chua's circuit is one of the most studied nonlinear dynamical systems, its version with negative parameters remains practically untouched. This work reports an interesting and rich dynamic scenery that was hidden in this almost unexplored region. The study is focused on 2D parameter space and presents an analysis of stability based on describing functions. Numerical investigations present a gallery of period-adding cascades and a strong presence of basin boundary metamorphoses. The key to this new scenario is that for negative parameters, Chua's system does not satisfy the Shilnikov condition and it is shown that the homoclinic orbit organizes the parameter space completely different from as known. The obtained experimental results corroborate with the numerical and theoretical investigations.

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VISUALIZING BIFURCATIONS IN HIGH DIMENSIONAL SYSTEMS: THE SPECTRAL BIFURCATION DIAGRAM

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127403008387 ◽

2003 ◽

Vol 13 (10) ◽

pp. 3015-3027 ◽

Cited By ~ 21

Author(s):

DAVID ORRELL ◽

LEONARD A. SMITH

Keyword(s):

Control Parameter ◽

Nonlinear Dynamical Systems ◽

The Other ◽

High Dimensional ◽

Behavior Changes ◽

Bifurcation Diagrams ◽

Nonlinear Dynamical ◽

The Third ◽

Alternative Approach ◽

Lorenz 96

This paper presents methods to visualize bifurcations in flows of nonlinear dynamical systems, using the Lorenz '96 systems as examples. Three techniques are considered; the first two, density and max/min diagrams, are analagous to the bifurcation diagrams used for maps, which indicate how the system's behavior changes with a control parameter. However the diagrams are generally harder to interpret than the corresponding diagrams of maps, due to the continuous nature of the flow. The third technique takes an alternative approach: by calculating the power spectrum at each value of the control parameter, a plot is produced which clearly shows the changes between periodic, quasi-periodic, and chaotic states, and reveals structure not shown by the other methods.

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Interplay of symmetries and other integrability quantifiers in finite-dimensional integrable nonlinear dynamical systems

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2015.0847 ◽

2016 ◽

Vol 472 (2190) ◽

pp. 20150847

Author(s):

R. Mohanasubha ◽

V. K. Chandrasekar ◽

M. Senthilvelan ◽

M. Lakshmanan

Keyword(s):

Dynamical Systems ◽

Integrable Systems ◽

Analytical Methods ◽

Nonlinear Dynamical Systems ◽

Nonlinear Ordinary Differential Equations ◽

Lie Point Symmetries ◽

Darboux Polynomials ◽

Nonlinear Dynamical ◽

Finite Dimensional ◽

Integrating Factors

In this work, we establish a connection between the extended Prelle–Singer procedure and other widely used analytical methods to identify integrable systems in the case of n th-order nonlinear ordinary differential equations (ODEs). By synthesizing these methods, we bring out the interlink between Lie point symmetries, contact symmetries, λ-symmetries, adjoint symmetries, null forms, Darboux polynomials, integrating factors, the Jacobi last multiplier and generalized λ-symmetries corresponding to the n th-order ODEs. We also prove these interlinks with suitable examples. By exploiting these interconnections, the characteristic quantities associated with different methods can be deduced without solving the associated determining equations.

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Sequential sampling strategy for extreme event statistics in nonlinear dynamical systems

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.1813263115 ◽

2018 ◽

Vol 115 (44) ◽

pp. 11138-11143 ◽

Cited By ~ 20

Author(s):

Mustafa A. Mohamad ◽

Themistoklis P. Sapsis

Keyword(s):

Dynamical Systems ◽

Parameter Space ◽

Extreme Event ◽

Nonlinear Dynamical Systems ◽

Computational Cost ◽

Optimization Procedure ◽

Irregular Waves ◽

Dimensional System ◽

Nonlinear Dynamical ◽

Quantity Of Interest

We develop a method for the evaluation of extreme event statistics associated with nonlinear dynamical systems from a small number of samples. From an initial dataset of design points, we formulate a sequential strategy that provides the “next-best” data point (set of parameters) that when evaluated results in improved estimates of the probability density function (pdf) for a scalar quantity of interest. The approach uses Gaussian process regression to perform Bayesian inference on the parameter-to-observation map describing the quantity of interest. We then approximate the desired pdf along with uncertainty bounds using the posterior distribution of the inferred map. The next-best design point is sequentially determined through an optimization procedure that selects the point in parameter space that maximally reduces uncertainty between the estimated bounds of the pdf prediction. Since the optimization process uses only information from the inferred map, it has minimal computational cost. Moreover, the special form of the metric emphasizes the tails of the pdf. The method is practical for systems where the dimensionality of the parameter space is of moderate size and for problems where each sample is very expensive to obtain. We apply the method to estimate the extreme event statistics for a very high-dimensional system with millions of degrees of freedom: an offshore platform subjected to 3D irregular waves. It is demonstrated that the developed approach can accurately determine the extreme event statistics using a limited number of samples.

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