Linear and Nonlinear Dynamic Behaviour

2018 ◽  
Author(s):  
Ray Huffaker ◽  
Marco Bittelli ◽  
Rodolfo Rosa
Keyword(s):  
Dynamic Behavior ◽  
Dynamic Systems ◽  
Dynamic Behaviour ◽  
Nonlinear Dynamical Systems ◽  
Logistic Map ◽  
Nonlinear Dynamical ◽  
Deterministic Dynamic ◽  
Random Behavior ◽  
Single Solution ◽  
Equivalent Systems

In this chapter, we describe how highly erratic dynamic behavior can arise from a nonlinear logistic map, and how this apparently random behavior is governed by a surprising order. With this lesson in mind, we should not be overly surprised that highly erratic and random appearing observed data might also be generated by parsimonious deterministic dynamic systems. At a minimum, we contend that researchers should apply NLTS to test for this possibility. We also introduced tools to analyze dynamic behavior that form the foundation for NLTS. In particular, we have stressed the quite unexpected capability to achieve some form of predictability even with only one trajectory at hand. In subsequent chapters, we treat known nonlinear dynamical systems as unknown, and investigate how NLTS methods rely on a single solution (or multiple solutions) generated by them to reconstruct equivalent systems. This is a conventional approach in the literature for seeing how NLTS methods work since we know what needs to be reconstructed.

Periodicity and Nonlinearity of Nonlinear Dynamical Systems Under Periodic Excitation

2009 ◽  
Author(s):  
Lu Han ◽  
Liming Dai ◽  
Huayong Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Dynamic Systems ◽  
Nonlinear Dynamic ◽  
Nonlinear Dynamical Systems ◽  
Research Work ◽  
Nonlinear Dynamic Systems ◽  
Periodic Excitation ◽  
Nonlinear Dynamical

Periodicity and nonlinearity of nonlinear dynamic systems subjected to regular external excitations are studied in this research work. Diagnoses of regular and chaotic responses of nonlinear dynamic systems are performed with the implementation of a newly developed Periodicity Ratio in combining with the application of Lyapunov Exponent. The properties of the nonlinear dynamics systems are classified into four categories: periodic, irregular-nonchaotic, quasiperiodic and chaotic, each corresponding to their Periodicity Ratios. Detailed descriptions about diagnosing the responses of the four categories are presented with utilization of the Periodicity Ratio.

scholarly journals NOMINAL CROSS RECURRENCE AS A GENERALIZED LAG SEQUENTIAL ANALYSIS FOR BEHAVIORAL STREAMS

2011 ◽  
Vol 21 (04) ◽  
pp. 1153-1161 ◽  
Author(s):  
RICK DALE ◽  
ANNE S. WARLAUMONT ◽  
DANIEL C. RICHARDSON
Keyword(s):  
Sequential Analysis ◽  
Nonlinear Dynamical Systems ◽  
Logistic Map ◽  
Coarse Graining ◽  
Human Interaction ◽  
Coarse Grained ◽  
Nonlinear Dynamical ◽  
Movement Data ◽  
Lag Sequential Analysis ◽  
Quantification Analysis

We briefly present lag sequential analysis for behavioral streams, a commonly used method in psychology for quantifying the relationships between two nominal time series. Cross recurrence quantification analysis (CRQA) is shown as an extension of this technique, and we exemplify this nominal application of CRQA to eye-movement data in human interaction. In addition, we demonstrate nominal CRQA in a simple coupled logistic map simulation used in previous communication research, permitting the investigation of properties of nonlinear systems such as bifurcation and onset to chaos, even in the streams obtained by coarse-graining a coupled nonlinear model. We end with a summary of the importance of CRQA for exploring the relationship between two behavioral streams, and review a recent theoretical trend in the cognitive sciences that would be usefully informed by this and similar nonlinear methods. We hope this work will encourage scientists interested in general properties of complex, nonlinear dynamical systems to apply emerging methods to coarse-grained, nominal units of measure, as there is an immediate need for their application in the psychological domain.

scholarly journals Superconvergence of topological entropy in the symbolic dynamics of substitution sequences

2019 ◽  
Vol 7 (2) ◽  
Author(s):  
Leon Zaporski ◽  
Felix Flicker
Keyword(s):  
Topological Entropy ◽  
Symbolic Dynamics ◽  
Convergence Rates ◽  
Nonlinear Dynamical Systems ◽  
Logistic Map ◽  
Universality Class ◽  
Accumulation Point ◽  
Nonlinear Dynamical ◽  
Double Exponential ◽  
Symbolic Sequences

We consider infinite sequences of superstable orbits (cascades) generated by systematic substitutions of letters in the symbolic dynamics of one-dimensional nonlinear systems in the logistic map universality class. We identify the conditions under which the topological entropy of successive words converges as a double exponential onto the accumulation point, and find the convergence rates analytically for selected cascades. Numerical tests of the convergence of the control parameter reveal a tendency to quantitatively universal double-exponential convergence. Taking a specific physical example, we consider cascades of stable orbits described by symbolic sequences with the symmetries of quasilattices. We show that all quasilattices can be realised as stable trajectories in nonlinear dynamical systems, extending previous results in which two were identified.

Ljapunov Exponents, Hyperchaos and Hurst Exponent

2005 ◽  
Vol 60 (4) ◽  
pp. 252-254 ◽  
Author(s):  
Willi-Hans Steeb ◽  
Eugenio Cosme Andrieu
Keyword(s):  
Dynamical Systems ◽  
Hurst Exponent ◽  
Nonlinear Dynamical Systems ◽  
Logistic Map ◽  
Two Dimensional ◽  
Nonlinear Dynamical

Abstract We consider nonlinear dynamical systems with chaotic and hyperchaotic behaviour.We investigate the behaviour of the Hurst exponent at the transition from chaos to hyperchaos. A two-dimensional coupled logistic map is studied.

scholarly journals The Study of Identification Method for Dynamic Behavior of High-Dimensional Nonlinear System

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Pan Fang ◽  
Liming Dai ◽  
Yongjun Hou ◽  
Mingjun Du ◽  
Wang Luyou
Keyword(s):  
Numerical Methods ◽  
Nonlinear System ◽  
Dynamic Behavior ◽  
Evaluation Method ◽  
Nonlinear Dynamical Systems ◽  
Autonomous Systems ◽  
High Dimensional ◽  
Nonlinear Dynamical ◽  
System Parameters ◽  
Research Findings

The dynamic behavior of nonlinear systems can be concluded as chaos, periodicity, and the motion between chaos and periodicity; therefore, the key to study the nonlinear system is identifying dynamic behavior considering the different values of the system parameters. For the uncertainty of high-dimensional nonlinear dynamical systems, the methods for identifying the dynamics of nonlinear nonautonomous and autonomous systems are treated. In addition, the numerical methods are employed to determine the dynamic behavior and periodicity ratio of a typical hull system and Rössler dynamic system, respectively. The research findings will develop the evaluation method of dynamic characteristics for the high-dimensional nonlinear system.

Phase Space Transport in a Class of Multi-Degree of-Freedom Systems

1997 ◽  
Author(s):  
Shyh-Leh Chen ◽  
Steven W. Shaw
Keyword(s):  
Phase Space ◽  
Dynamic Systems ◽  
Nonlinear Dynamical Systems ◽  
Quantitative Information ◽  
Fluid Mixing ◽  
Dynamics Model ◽  
Analytical Estimate ◽  
Nonlinear Dynamical ◽  
Qualitative And Quantitative ◽  
Phase Space Transport

Abstract In this paper we describe some recent advances in the basic theory and applications of phase space transport in nonlinear dynamic systems. These methods offer both qualitative and quantitative information about the behavior of solutions near homoclinic and heteroclinic motions in nonlinear dynamical systems. Applications of these ideas are found in fluid mixing and the escape of solutions from potential energy wells under the action of disturbances, for example, in models of ship capsize. In this work the theory is extended to a certain class of higher-order systems in which several time scales are involved. In addition, a new analytical estimate is derived and used for the rate of transport in the case of two-dimensional Poincare maps. Extensive simulation results from a specific ship dynamics model are used to demonstrate and verify these results.

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.

Sign in / Sign up

Export Citation Format

Share Document