Dynamical System Visualization and Analysis Via Performance Maps

Alpigini, James J.

doi:10.1057/palgrave.ivs.9500082

“…One constant throughout the interdisciplinary history of nonlinear dynamical systems' study is that nonlinear systems are extremely difficult to solve analytically because they cannot be broken down into constituent parts, solved individually, then recombined as a solution. Scientists have instead relied heavily on visual and qualitative approaches, a perspective first developed by Henri Poincaré in the late 1800s, to discover and analyze the fascinating dynamics of nonlinearity [28,29]. Information visualization helps analysts detect and examine hidden structure in complex datasets [30].…”

Section: Introductionmentioning

confidence: 99%

Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction

Boeing

¹

2016

Systems

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Nearly all nontrivial real-world systems are nonlinear dynamical systems. Chaos describes certain nonlinear dynamical systems that have a very sensitive dependence on initial conditions. Chaotic systems are always deterministic and may be very simple, yet they produce completely unpredictable and divergent behavior. Systems of nonlinear equations are difficult to solve analytically, and scientists have relied heavily on visual and qualitative approaches to discover and analyze the dynamics of nonlinearity. Indeed, few fields have drawn as heavily from visualization methods for their seminal innovations: from strange attractors, to bifurcation diagrams, to cobweb plots, to phase diagrams and embedding. Although the social sciences are increasingly studying these types of systems, seminal concepts remain murky or loosely adopted. This article has three aims. First, it argues for several visualization methods to critically analyze and understand the behavior of nonlinear dynamical systems. Second, it uses these visualizations to introduce the foundations of nonlinear dynamics, chaos, fractals, self-similarity and the limits of prediction. Finally, it presents Pynamical, an open-source Python package to easily visualize and explore nonlinear dynamical systems' behavior.

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“…One constant throughout the interdisciplinary history of nonlinear dynamical systems' study is that nonlinear systems are extremely difficult to solve analytically because they cannot be broken down into constituent parts, solved individually, then recombined as a solution. Scientists have instead relied heavily on visual and qualitative approaches, a perspective first developed by Henri Poincaré in the late 1800s, to discover and analyze the fascinating dynamics of nonlinearity [28,29]. Information visualization helps analysts detect and examine hidden structure in complex datasets [30].…”

Section: Introductionmentioning

confidence: 99%

Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction

Boeing¹

2017

Preprint

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Nearly all nontrivial real-world systems are nonlinear dynamical systems. Chaos describes certain nonlinear dynamical systems that have a very sensitive dependence on initial conditions. Chaotic systems are always deterministic and may be very simple, yet they produce completely unpredictable and divergent behavior. Systems of nonlinear equations are difficult to solve analytically, and scientists have relied heavily on visual and qualitative approaches to discover and analyze the dynamics of nonlinearity. Indeed, few fields have drawn as heavily from visualization methods for their seminal innovations: from strange attractors, to bifurcation diagrams, to cobweb plots, to phase diagrams and embedding. Although the social sciences are increasingly studying these types of systems, seminal concepts remain murky or loosely adopted. This article has three aims. First, it argues for several visualization methods to critically analyze and understand the behavior of nonlinear dynamical systems. Second, it uses these visualizations to introduce the foundations of nonlinear dynamics, chaos, fractals, self-similarity and the limits of prediction. Finally, it presents Pynamical, an open-source Python package to easily visualize and explore nonlinear dynamical systems’ behavior.

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“…Instead, researchers have long relied on visualization techniques to make system behavior comprehensible (Alpigini 2004;Layek 2015). Such visualization is useful for exploring nonlinear time series data (Bradley and Kantz 2015;Boeing 2016).…”

Section: Discussionmentioning

confidence: 99%

Pynamical: Model and visualize discrete nonlinear dynamical systems, chaos, and fractals

Boeing

¹

2018

Preprint

0

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SummaryPynamical is an educational Python package for introducing the modeling, simulation, and visualization of discrete nonlinear dynamical systems and chaos, focusing on onedimensional maps (such as the logistic map and the cubic map). Pynamical facilitates defining discrete one-dimensional nonlinear models as Python functions with just-in-time compilation for fast simulation. It comes packaged with the logistic map, the Singer map, and the cubic map predefined. The models may be run with a range of parameter values over a set of time steps, and the resulting numerical output is returned as a pandas DataFrame. Pynamical can then visualize this output in various ways, including with bifurcation diagrams (May 1976), two-dimensional phase diagrams (Packard et al. 1980), three-dimensional phase diagrams, and cobweb plots (Hofstadter 1985). These visualizations enable simple qualitative assessments of system behavior including phase transitions (Feigenbaum 1983), bifurcation points (Sander and Yorke 2015), attractors and limit cycles (Grebogi, Ott, and Yorke 1987), basins of attraction (Sprott and Xiong 2015), and fractals (Mandelbrot 1967(Mandelbrot , 1983.Although most real-world systems are nonlinear dynamical systems, their mathematical analysis is notoriously difficult because they cannot be simply broken down into individual parts then recombined linearly (Strogatz 2014). Instead, researchers have long relied on visualization techniques to make system behavior comprehensible (Alpigini 2004;Layek 2015). Such visualization is useful for exploring nonlinear time series data (Bradley and Kantz 2015;Boeing 2016). Pynamical facilitates the modeling, visualization, and exploration of this rich nonlinear behavior, as demonstrated in Figures 1 and 2. Accordingly this package has various applications in research, engineering, and particularly pedagogy. For instance, it allows students to explore and model any well-defined discrete nonlinear dynamical system (with just one line of code) and then simulate its behavior over n generations in time (with just one more line of code). This facilitates easy experimentation in parameter space and sensitivity analysis. Moreover, Pynamical easily produces publication-quality visualizations to illustrate findings without having to reinvent the wheel with ad hoc algorithms to produce, reshape, and plot simulation data. Finally, it enables the animation of such systems, to explore and present their qualitative behavior and evolution.The latest release of the software can be installed via conda or pip and full documentation can be found at https://pynamical.readthedocs.io.

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Dynamical System Visualization and Analysis Via Performance Maps

Cited by 7 publications

References 13 publications

Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction

Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction

Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction

Pynamical: Model and visualize discrete nonlinear dynamical systems, chaos, and fractals

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