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.

Numerical characterization of nonlinear dynamical systems using parallel computing: The role of GPUs approach

2016 ◽  
Vol 37 ◽  
pp. 143-162 ◽  
Author(s):  
Filipe I. Fazanaro ◽  
Diogo C. Soriano ◽  
Ricardo Suyama ◽  
Marconi K. Madrid ◽  
José Raimundo de Oliveira ◽  
...  
Keyword(s):  
Parallel Computing ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Numerical Characterization

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.

Dynamics of Human Spine in Jumps Using Lyapunov Exponents

2009 ◽  
Author(s):  
Eliza A. Banu ◽  
Dan Marghitu ◽  
P. K. Raju
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Largest Lyapunov Exponent ◽  
Kinematic Data ◽  
Finite Time Lyapunov Exponents ◽  
Human Spine ◽  
Chaotic Nature ◽  
Male Subjects ◽  
Lyapunov Exponent Spectrum

Characterizing and quantifying the local dynamics of the human spine during various athletic exercises is important for therapeutic and physiotherapy reasons or athletic related effects on the human spine. In this study we computed the largest finite-time Lyapunov exponents for the spine of a human subject during an athletic exercise for several volunteers. The kinematic data was collected using the 3D motion capture system (Motion Realty Inc.). Four healthy male subjects were asked to perform two sets of 30 jumps. In order to draw a conclusion about the chaotic nature of the dynamics of the lumbar spine Lyapunov exponent spectrum was calculated for each volunteer which included four Lyapunov exponents, one negative, one very close to zero and two positive. Subjects 1, 2 and 4 registered a significant increase in the largest Lyapunov exponent from the first set jumps to the next. Subject 3 registered no change in the values of Lyapunov exponents. Nonlinear analysis of the human spine demonstrates the chaotic nature of the system. The computation of the largest Lyapunov exponents enables a more precise characterization of the dynamics of the human spine.

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

2017 ◽  
Author(s):  
Geoff Boeing
Keyword(s):  
Dynamical Systems ◽  
Visual Analysis ◽  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Systems Of Nonlinear Equations ◽  
Self Similarity ◽  
Nonlinear Dynamical ◽  
Qualitative Approaches ◽  
Divergent Behavior ◽  
Behavior Systems

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.

scholarly journals iLQR-VAE : control-based learning of input-driven dynamics with applications to neural data

2021 ◽  
Author(s):  
Marine Schimel ◽  
Ta-Chu Kao ◽  
Kristopher T. Jensen ◽  
Guillaume Hennequin
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Linear Quadratic Regulator ◽  
External Input ◽  
Learning Approaches ◽  
Linear Quadratic ◽  
Nonlinear Dynamical ◽  
Neural Data ◽  
Recognition Model ◽  
Variational Autoencoder

Understanding how neural dynamics give rise to behaviour is one of the most fundamental questions in systems neuroscience. To achieve this, a common approach is to record neural populations in behaving animals, and model these data as emanating from a latent dynamical system whose state trajectories can then be related back to behavioural observations via some form of decoding. As recordings are typically performed in localized circuits that form only a part of the wider implicated network, it is important to simultaneously learn the local dynamics and infer any unobserved external input that might drive them. Here, we introduce iLQR-VAE, a novel control-based approach to variational inference in nonlinear dynamical systems, capable of learning both latent dynamics, initial conditions, and ongoing external inputs. As in recent deep learning approaches, our method is based on an input-driven sequential variational autoencoder (VAE). The main novelty lies in the use of the powerful iterative linear quadratic regulator algorithm (iLQR) in the recognition model. Optimization of the standard evidence lower-bound requires differentiating through iLQR solutions, which is made possible by recent advances in differentiable control. Importantly, having the recognition model implicitly defined by the generative model greatly reduces the number of free parameters and allows for flexible, high-quality inference. This makes it possible for instance to evaluate the model on a single long trial after training on smaller chunks. We demonstrate the effectiveness of iLQR-VAE on a range of synthetic systems, with autonomous as well as input-driven dynamics. We further show state-of-the-art performance on neural and behavioural recordings in non-human primates during two different reaching tasks.

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.

Domains of Attraction for an Autoparametric Mass-Pendulum System: A Lyapunov Exponent Map Approach

2003 ◽  
Author(s):  
Khalid El-Rifai ◽  
George Haller ◽  
Anil K. Bajaj
Keyword(s):  
Lyapunov Exponent ◽  
Finite Time ◽  
Initial Conditions ◽  
Transient Behavior ◽  
System Response ◽  
Pendulum System ◽  
Domains Of Attraction ◽  
Nonlinear Dynamical ◽  
Finite Time Lyapunov Exponents ◽  
Finite Time Lyapunov Exponent

Many recent studies have been performed on resonantly excited mass-pendulum systems with autoparametric (internal) resonance capturing interesting local steady state phenomena. The objective of this work is to explore the transient behavior in such systems. The domains of attraction of the time-periodic system provide some help in understanding the transient dynamics, and these are sought using a recently developed algorithm that solves for the finite-time Lyapunov exponent over a grid of initial conditions. Though the use of finite-time Lyapunov exponents in nonlinear dynamical analyses is not novel, its application to multi-degree-offreedom forced nonlinear systems has not been reported in the literature. In addition to identifying regions of different final states, the technique used captures different levels of attraction within a domain. This sheds some light on the role played by other modes present in a multi-degree-of-freedom system in shaping the overall system response.

scholarly journals Identification of nonlinear dynamical systems with time delay

2021 ◽  
Author(s):  
Ghazaale Leylaz ◽  
Shuo Wang ◽  
Jian-Qiao Sun
Keyword(s):  
Time Delay ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Algebraic Operation ◽  
Computationally Efficient ◽  
Nonlinear Dynamical ◽  
Automatic Model Selection ◽  
Robust To Noise

AbstractThis paper proposes a technique to identify nonlinear dynamical systems with time delay. The sparse optimization algorithm is extended to nonlinear systems with time delay. The proposed algorithm combines cross-validation techniques from machine learning for automatic model selection and an algebraic operation for preprocessing signals to filter the noise and for removing the dependence on initial conditions. We further integrate the bootstrapping resampling technique with the sparse regression to obtain the statistical properties of estimation. We use Taylor expansion to parameterize time delay. The proposed algorithm in this paper is computationally efficient and robust to noise. A nonlinear Duffing oscillator is simulated to demonstrate the efficiency and accuracy of the proposed technique. An experimental example of a nonlinear rotary flexible joint is presented to further validate the proposed method.

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