Numerical solutions of the generalized Rosenau–Kawahara-RLW equation arising in fluid mechanics via B-spline collocation method

2018 ◽  
Vol 29 (11) ◽  
pp. 1850116 ◽  
Author(s):  
Turgut Ak ◽  
Sharanjeet Dhawan ◽  
Bilge İnan
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Wave Interaction ◽  
Spline Collocation ◽  
Nonlinear Dynamical ◽  
Rlw Equation ◽  
Nonlinear Physical Phenomena ◽  
Physical Phenomena ◽  
Solitary Wave Interaction

Present study reports the solution of generalized Rosenau–Kawahara-RLW equation. It includes motion of single solitary wave, interaction of two solitary waves along with the calculated invariants and error norms. Gaussian and undular bore initial conditions are studied to show evolution of solitons. Developed train of solitons and conservation of invariants are shown via figures and tables in the respective sections. Various case studies are presented to demonstrate the efficiency of the proposed numerical scheme. Solutions so produced may be helpful for explaining various nonlinear physical phenomena in nonlinear dynamical systems.

DYNAMICAL BEHAVIOR OF q-DEFORMED HENON MAP

2011 ◽  
Vol 21 (05) ◽  
pp. 1349-1356 ◽  
Author(s):  
VINOD PATIDAR ◽  
G. PUROHIT ◽  
K. K. SUD
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Hénon Map ◽  
Chaotic Motions ◽  
Henon Map ◽  
Nonlinear Dynamical ◽  
Complete Deformation ◽  
Nonlinear Maps ◽  
Nonlinear Physical Phenomena ◽  
Physical Phenomena

A recent proposal of q-deformation scheme for the nonlinear maps has stimulated new directions in the studies of various nonlinear dynamical systems. Such studies are advantageous in the analytical modeling of several nonlinear physical phenomena which cannot be perfectly represented by the standard or canonical models. This paper attempts to numerically analyze the behavior of q-deformed version of Henon map, which is one of the prototypical models exhibiting strange chaotic attractor. We particularly investigate the effect of q-deformation on the various kinds of long-term asymptotic dynamics of the canonical Henon map and also characterize the complete deformation parameter space into different regions corresponding to the periodic and strange chaotic motions. We also identify the route to chaos and new structures in the chaotic strange attractor of q-deformed Henon map.

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.

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.

scholarly journals Chaotic Systems and Their Recent Implementations on Improving Intelligent Systems

Economics ◽  
2015 ◽  
pp. 1167-1200
Author(s):  
Utku Köse ◽  
Ahmet Arslan
Keyword(s):  
Artificial Intelligence ◽  
Chaos Theory ◽  
Intelligent Systems ◽  
Chaotic Systems ◽  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Research Effort ◽  
Nonlinear Dynamical ◽  
Main Research ◽  
Reference Guide

Chaos Theory is a kind of a scientific approach/research effort which is based on examining behaviors of nonlinear dynamical systems which are highly sensitive to their initial conditions. Currently, there are many different scientific studies based on the Chaos Theory and the related solution approaches, methods, or techniques for problems of this theory. Additionally, the theory is used for improving the introduced studies of different fields in order to get more effective, efficient, and accurate results. At this point, this chapter aims to provide a review-based study introducing recent implementations of the Chaos Theory on improving intelligent systems, which can be examined in the context of the Artificial Intelligence field. In this sense, the main research way is directed into the works performed or introduced mostly in years between 2008 and 2013. By providing a review-based study, the readers are enabled to have ideas on Chaos Theory, Artificial Intelligence, and the related works that can be examined within intersection of both fields. At this point, the chapter aims to discuss not only recent works, but also express ideas regarding future directions within the related implementations of chaotic systems to improve intelligent systems. The chapter is generally organized as a reference guide for academics, researchers, and scientists tracking the literature of the related fields: Artificial Intelligence and the Chaos Theory.

scholarly journals Bifurcations, Hidden Chaos and Control in Fractional Maps

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 879 ◽  
Author(s):  
Adel Ouannas ◽  
Othman Abdullah Almatroud ◽  
Amina Aicha Khennaoui ◽  
Mohammad Mossa Alsawalha ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Discrete Fractional Calculus ◽  
Hidden Attractors ◽  
Nonlinear Dynamical ◽  
Control Schemes ◽  
Stable Equilibria ◽  
And Control

Recently, hidden attractors with stable equilibria have received considerable attention in chaos theory and nonlinear dynamical systems. Based on discrete fractional calculus, this paper proposes a simple two-dimensional and three-dimensional fractional maps. Both fractional maps are chaotic and have a unique equilibrium point. Results show that the dynamics of the proposed fractional maps are sensitive to both initial conditions and fractional order. There are coexisting attractors which have been displayed in terms of bifurcation diagrams, phase portraits and a 0-1 test. Furthermore, control schemes are introduced to stabilize the chaotic trajectories of the two novel systems.

DETERMINISTIC LEARNING OF NONLINEAR DYNAMICAL SYSTEMS

2009 ◽  
Vol 19 (04) ◽  
pp. 1307-1328 ◽  
Author(s):  
CONG WANG ◽  
TIANRUI CHEN ◽  
GUANRONG CHEN ◽  
DAVID J. HILL
Keyword(s):  
Dynamical Systems ◽  
System Dynamics ◽  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Broad Class ◽  
Excitation Condition ◽  
Accurate Identification ◽  
Nonlinear Dynamical ◽  
Persistent Excitation ◽  
Deterministic Learning

In this paper, we investigate the problem of identifying or modeling nonlinear dynamical systems undergoing periodic and period-like (recurrent) motions. For accurate identification of nonlinear dynamical systems, the persistent excitation condition is normally required to be satisfied. Firstly, by using localized radial basis function networks, a relationship between the recurrent trajectories and the persistence of excitation condition is established. Secondly, for a broad class of recurrent trajectories generated from nonlinear dynamical systems, a deterministic learning approach is presented which achieves locally-accurate identification of the underlying system dynamics in a local region along the recurrent trajectory. This study reveals that even for a random-like chaotic trajectory, which is extremely sensitive to initial conditions and is long-term unpredictable, the system dynamics of a nonlinear chaotic system can still be locally-accurate identified along the chaotic trajectory in a deterministic way. Numerical experiments on the Rossler system are included to demonstrate the effectiveness of the proposed approach.

scholarly journals Analysis of optical solitons solutions of two nonlinear models using analytical technique

2021 ◽  
Vol 6 (12) ◽  
pp. 13258-13271
Author(s):  
Naeem Ullah ◽  
◽  
Muhammad Imran Asjad ◽  
Azhar Iqbal ◽  
Hamood Ur Rehman ◽  
...  
Keyword(s):  
Nonlinear Models ◽  
Nonlinear Partial Differential Equations ◽  
Optical Solitons ◽  
Nonlinear Dynamical ◽  
Analytical Technique ◽  
The Core ◽  
Generalized Kudryashov Method ◽  
Kudryashov Method ◽  
Nonlinear Physical Phenomena ◽  
Physical Phenomena

<abstract><p>Looking for the exact solutions in the form of optical solitons of nonlinear partial differential equations has become very famous to analyze the core structures of physical phenomena. In this paper, we have constructed some various type of optical solitons solutions for the Kaup-Newell equation (KNE) and Biswas-Arshad equation (BAE) via the generalized Kudryashov method (GKM). The conquered solutions help to understand the dynamic behavior of different physical phenomena. These solutions are specific, novel, correct and may be beneficial for edifying precise nonlinear physical phenomena in nonlinear dynamical schemes. Graphical recreations for some of the acquired solutions are offered.</p></abstract>

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