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

2018 ◽  
Author(s):  
Geoff Boeing
Keyword(s):  
Dynamical Systems ◽  
Phase Diagrams ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Models ◽  
Logistic Map ◽  
Three Dimensional ◽  
Basins Of Attraction ◽  
Nonlinear Dynamical ◽  
One Dimensional ◽  
One Dimensional Maps

Pynamical is an educational Python package for introducing the modeling, simulation, and visualization of discrete nonlinear dynamical systems and chaos, focusing on one-dimensional 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, two-dimensional phase diagrams, three-dimensional phase diagrams, and cobweb plots. These visualizations enable simple qualitative assessments of system behavior including phase transitions, bifurcation points, attractors and limit cycles, basins of attraction, and fractals.

Application of a Generalized Frobenius-Perron Operator Scheme to Multivariate Nonlinear Dynamical Systems

1995 ◽  
Vol 50 (12) ◽  
pp. 1123-1127
Author(s):  
R. Stoop ◽  
W.-H. Steeb
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Free Energies ◽  
Nonlinear Dynamical ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Operator Scheme

Abstract The concept of generalized Frobenius-Perron operators is applied to multivariante nonlinear dynamical systems, and the associated generalized free energies are investigated. As important applications, diffusion-related free energies obtained from normally and superlinearly diffusive one-dimensional maps are discussed.

An implicit class of continuous dynamical system with data-sample outputs: a robust approach

2019 ◽  
Vol 37 (2) ◽  
pp. 589-606
Author(s):  
Raymundo Juarez ◽  
Vadim Azhmyakov ◽  
A Tadeo Espinoza ◽  
Francisco G Salas
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Models ◽  
Differential Algebraic Equations ◽  
Linear Feedback ◽  
Continuous Systems ◽  
Algebraic Equations ◽  
Nonlinear Dynamical ◽  
Ellipsoid Method ◽  
Invariant Ellipsoid

Abstract This paper addresses the problem of robust control for a class of nonlinear dynamical systems in the continuous time domain. We deal with nonlinear models described by differential-algebraic equations (DAEs) in the presence of bounded uncertainties. The full model of the control system under consideration is completed by linear sampling-type outputs. The linear feedback control design proposed in this manuscript is created by application of an extended version of the conventional invariant ellipsoid method. Moreover, we also apply some specific Lyapunov-based descriptor techniques from the stability theory of continuous systems. The above combination of the modified invariant ellipsoid approach and descriptor method makes it possible to obtain the robustness of the designed control and to establish some well-known stability properties of dynamical systems under consideration. Finally, the applicability of the proposed method is illustrated by a computational example. A brief discussion on the main implementation issue is also included.

UNIFORMITY AND SPATIAL CHAOS OF SPATIAL PHYSICS KINEMATIC SYSTEM

2010 ◽  
Vol 24 (28) ◽  
pp. 5495-5503
Author(s):  
SHUTANG LIU ◽  
FUYAN SUN ◽  
JIE SUN
Keyword(s):  
Dynamical Systems ◽  
Lattice Model ◽  
Nonlinear Dynamical Systems ◽  
Coupled Map Lattice ◽  
Nonlinear Dynamical ◽  
One Dimensional ◽  
Uniform System ◽  
Spatial Chaos ◽  
Kinematic System ◽  
Bifurcation Behavior

This article summarizes the uniformity law of spatial physics kinematic systems, and studies the chaos and bifurcation behavior of the uniform system in space. In particular, it also fully explains the relation among the uniform system, the coupled map lattice model which has attracted considerable interest currently, and one-dimensional nonlinear dynamical systems.

scholarly journals MEASURING AND CONTROLLING THE CHAOTIC MOTION OF PROFITS

2009 ◽  
Vol 19 (11) ◽  
pp. 3593-3604 ◽  
Author(s):  
CRISTINA JANUÁRIO ◽  
CLARA GRÁCIO ◽  
DIANA A. MENDES ◽  
JORGE DUARTE
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonlinear Dynamical Systems ◽  
Control Technique ◽  
Stable Steady State ◽  
Chaotic Motions ◽  
Nonlinear Dynamical ◽  
Firm Model ◽  
One Dimensional Maps

The study of economic systems has generated deep interest in exploring the complexity of chaotic motions in economy. Due to important developments in nonlinear dynamics, the last two decades have witnessed strong revival of interest in nonlinear endogenous business chaotic models. The inability to predict the behavior of dynamical systems in the presence of chaos suggests the application of chaos control methods, when we are more interested in obtaining regular behavior. In the present article, we study a specific economic model from the literature. More precisely, a system of three ordinary differential equations gather the variables of profits, reinvestments and financial flow of borrowings in the structure of a firm. Firstly, using results of symbolic dynamics, we characterize the topological entropy and the parameter space ordering of kneading sequences, associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The analysis of the variation of this numerical invariant, in some realistic system parameter region, allows us to quantify and to distinguish different chaotic regimes. Finally, we show that complicated behavior arising from the chaotic firm model can be controlled without changing its original properties and the dynamics can be turned into the desired attracting time periodic motion (a stable steady state or into a regular cycle). The orbit stabilization is illustrated by the application of a feedback control technique initially developed by Romeiras et al. [1992]. This work provides another illustration of how our understanding of economic models can be enhanced by the theoretical and numerical investigation of nonlinear dynamical systems modeled by ordinary differential equations.

The Necessary Condition for the Existence of Periodic Solutions of Certain Three Dimensional Nonlinear Dynamical Systems

2012 ◽  
Vol 252 ◽  
pp. 40-43
Author(s):  
Ting Ting Quan ◽  
Jing Li ◽  
Min Sun
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Nonlinear Dynamical Systems ◽  
Three Dimensional ◽  
Basic Research ◽  
Necessary Condition ◽  
Nonlinear Dynamical ◽  
Closed Orbits ◽  
Existence Of Periodic Solutions ◽  
Important Significance

In this paper, we investigate a class of three dimensional nonlinear dynamical systems whose unperturbed systems have a family of periodic orbits. Firstly, we establish the moving Frenet Frame on these closed orbits. Secondly, the successor functions are defined by the orbits which go through the normal plane. Finally, by judging the existence of solutions of the equations obtained from the Successor functions, we obtain the necessary condition for the existence of periodic solutions of these three dimensional nonlinear dynamical systems. The result has important significance for the basic research of applied mechanics.

scholarly journals Computation of the topological entropy in chaotic biophysical bursting models for excitable cells

2006 ◽  
Vol 2006 ◽  
pp. 1-18 ◽  
Author(s):  
Jorge Duarte ◽  
Luís Silva ◽  
J. Sousa Ramos
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Excitable Cells ◽  
Physiological Models ◽  
Nonlinear Dynamical ◽  
Chaotic Bursting ◽  
Oscillatory State ◽  
One Dimensional Maps

One of the interesting complex behaviors in many cell membranes is bursting, in which a rapid oscillatory state alternates with phases of relative quiescence. Although there is an elegant interpretation of many experimental results in terms of nonlinear dynamical systems, the dynamics of bursting models is not completely described. In the present paper, we study the dynamical behavior of two specific three-variable models from the literature that replicate chaotic bursting. With results from symbolic dynamics, we characterize the topological entropy of one-dimensional maps that describe the salient dynamics on the attractors. The analysis of the variation of this important numerical invariant with the parameters of the systems allows us to quantify the complexity of the phenomenon and to distinguish different chaotic scenarios. This work provides an example of how our understanding of physiological models can be enhanced by the theory of dynamical systems.

ON STRONG AND WEAK CHAOTIC PARTIAL SYNCHRONIZATION

2000 ◽  
Vol 10 (01) ◽  
pp. 179-203 ◽  
Author(s):  
Yu. MAISTRENKO ◽  
O. POPOVYCH ◽  
M. HASLER
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Chaotic Behavior ◽  
Chaotic Synchronization ◽  
State Variables ◽  
Large Parameter ◽  
Nonlinear Dynamical ◽  
One Dimensional ◽  
Rich Variety ◽  
Parameter Ranges ◽  
One Dimensional Maps

We study coupled nonlinear dynamical systems with chaotic behavior in the case when two or more (but not all) state variables synchronize, i.e. converge to each other asymptotically in time. It is shown that for symmetrical systems, such partial chaotic synchronization is usually only weak, whereas with nonsymmetrical coupling it can be strong in large parameter ranges. These facts are illustrated with systems of three coupled one-dimensional maps, for which a rich variety of different "partial chaotic synchronizing" phenomena takes place.

The Sufficient Condition for the Existence of Periodic Solution of Certain Three Dimensional Nonlinear Dynamical Systems

2012 ◽  
Vol 249-250 ◽  
pp. 147-152
Author(s):  
Jing Li ◽  
Ting Ting Quan ◽  
Xiao Dong Jin
Keyword(s):  
Periodic Solution ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Closed Orbit ◽  
Three Dimensional ◽  
Existence Of Solution ◽  
Sufficient Condition ◽  
Frenet Frame ◽  
Nonlinear Dynamical ◽  
Important Significance

In this paper, the sufficient condition for the existence of periodic solution of a class of three dimensional nonlinear dynamical systems is investigated. The moving Frenet frame is established on the closed orbit and the successor functions near the closed orbit are defined. According to the study of the existence of solution of the equation which obtained from the successor functions with the implicit function theorem, the sufficient condition for the existence of periodic solution of these systems is obtained. The results in this paper have important significance to decide the existence of periodic solution of three dimensional nonlinear dynamical systems.

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