A geometric interpretation of maximal Lyapunov exponent based on deviation curvature

2021 ◽  
pp. 2150092
Author(s):  
Takahiro Yajima ◽  
Shintaro Nakase
Keyword(s):  
Lyapunov Exponent ◽  
Nonlinear Dynamical Systems ◽  
Geometric Interpretation ◽  
Maximal Lyapunov Exponent ◽  
Average Deviation ◽  
Nonlinear Dynamical ◽  
Nonlinear Connection ◽  
Second Order Differential Equations ◽  
Nonlinear Pendulum ◽  
Theoretical Results

In this study, we discuss a relationship between the behavior of nonlinear dynamical systems and geometry of a system of second-order differential equations based on the Jacobi stability analysis. We consider how a maximal Lyapunov exponent is related to the geometric quantities. As a result of a theoretical investigation, the maximal Lyapunov exponent can be represented by a nonlinear connection and a deviation curvature. Thus, this means that the Jacobi stability given by the sign of the deviation curvature affects the change of the maximal Lyapunov exponent. Additionally, for an equation of nonlinear pendulum, we numerically confirm the theoretical results. We observe that a change of the maximal Lyapunov exponent is related to a change of an average deviation curvature. These results indicate that the deviation curvature and Jacobi stability are essential for considering the change of maximal Lyapunov exponent.

Sparse Recovery and Dictionary Learning to Identify the Nonlinear Dynamical Systems: One Step Toward Finding Bifurcation Points in Real Systems

2019 ◽  
Vol 29 (03) ◽  
pp. 1950030 ◽  
Author(s):  
Fahimeh Nazarimehr ◽  
Aboozar Ghaffari ◽  
Sajad Jafari ◽  
Seyed Mohammad Reza Hashemi Golpayegani
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Dictionary Learning ◽  
Nonlinear Dynamical Systems ◽  
Sparse Recovery ◽  
Governing Equations ◽  
Nonlinear Dynamical ◽  
Bifurcation Points ◽  
Long Time

Modeling real dynamical systems is an important challenge in many areas of science. Extracting governing equations of systems from their time-series is a possible solution for such a challenge. In this paper, we use the sparse recovery and dictionary learning to extract governing equations of a system with parametric basis functions. In this algorithm, the assumption of sparsity in the functions of dynamical equations is used. The proposed algorithm is applied to different types of discrete and continuous nonlinear dynamical systems to show the generalization ability of this method. On the other hand, transition from one dynamical regime to another is an important concept in studying real world complex systems like biological and climate systems. Lyapunov exponent is an early warning index. It can predict bifurcation points in dynamical systems. Computation of Lyapunov exponent is a major challenge in its application in real systems, since it needs long time data to be accurate. In this paper, we use the predicted governing equation to generate long time-series, which is needed for Lyapunov exponent calculation. So the proposed method can help us to predict bifurcation points by accurate calculation of Lyapunov exponents.

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.

A Terminal Sliding Mode Control of Disturbed Nonlinear Second-Order Dynamical Systems

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
Pawel Skruch
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Sliding Mode ◽  
Second Order ◽  
Control Law ◽  
Sliding Surface ◽  
Terminal Sliding Mode ◽  
Nonlinear Dynamical ◽  
Second Order Differential Equations ◽  
Bounded Disturbance

The paper presents a terminal sliding mode controller for a certain class of disturbed nonlinear dynamical systems. The class of such systems is described by nonlinear second-order differential equations with an unknown and bounded disturbance. A sliding surface is defined by the system state and the desired trajectory. The control law is designed to force the trajectory of the system from any initial condition to the sliding surface within a finite time. The trajectory of the system after reaching the sliding surface remains on it. A computer simulation is included as an example to verify the approach and to demonstrate its effectiveness.

On the Identification of Chaos From Frequency Content

2011 ◽  
Author(s):  
Richard Wiebe ◽  
Lawrence N. Virgin
Keyword(s):  
Lyapunov Exponent ◽  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Quantitative Measure ◽  
Nonlinear Dynamical ◽  
Data Intensive ◽  
Extreme Sensitivity ◽  
Deterministic Dynamical System ◽  
Lyapunov Exponent Spectrum

The characterization of chaos as a random-like response from a deterministic dynamical system with an extreme sensitivity to initial conditions is well-established, and has provided a stimulus to research in nonlinear dynamical systems in general. In a formal sense, the computation of the Lyapunov Exponent spectrum establishes a quantitative measure, with at least one positive Lyapunov Exponent (and generally bounded motion) indicating a local exponential divergence of adjacent trajectories. However, although the extraction of Lyapunov Exponents can be accomplished with (necessarily noisy) experimental data, this is still a relatively data-intensive and sensitive endeavor. We present here an alternative, pragmatic approach to identifying chaos as a function of system parameters using response frequency characteristics and extending the concept of the spectrogram.

THE ONSET OF CHAOS IN NONLINEAR DYNAMICAL SYSTEMS DETERMINED WITH A NEW FRACTAL TECHNIQUE

Fractals ◽  
2005 ◽  
Vol 13 (01) ◽  
pp. 19-31 ◽  
Author(s):  
DORA E. MUSIELAK ◽  
ZDZISLAW E. MUSIELAK ◽  
KENNY S. KENNAMER
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Lyapunov Method ◽  
Nonlinear Dynamical Systems ◽  
Computational Time ◽  
Nonlinear Dynamical ◽  
Onset Of Chaos ◽  
Experimental Time Series ◽  
Method Agreement

A new fractal technique is used to investigate the onset of chaos in nonlinear dynamical systems. A comparison is made between this fractal technique and the commonly used Lyapunov exponent method. Agreement between the results obtained by both methods indicates that this technique may be used in a manner analogous to the Lyapunov exponents to predict onset of chaos. It is found that the fractal technique is much easier to implement than the Lyapunov method and it requires much less computational time. This fractal technique can easily be adopted to investigate the onset of chaos in many nonlinear dynamical systems and can be used to analyze theoretical and experimental time series.

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