Coexisting Attractors, Energy Analysis and Boundary of Lü System

2020 ◽  
Vol 30 (03) ◽  
pp. 2050048
Author(s):  
Hongyan Jia ◽  
Wenxin Shi ◽  
Guoyuan Qi
Keyword(s):  
Initial Conditions ◽  
Type System ◽  
Hamiltonian Function ◽  
Casimir Function ◽  
Coexisting Attractors ◽  
External Torque ◽  
Lü System ◽  
Different Types ◽  
Inertial Torque ◽  
Lu System

In this study, first, the phenomenon of multistability in the Lü system is found, which shows the coexistence of two different point attractors and one chaotic attractor. These coexisting attractors are dependent on initial conditions of the system while the parameters of the system are fixed. Then, the Lü system is transformed to a Kolmogorov-type system, which includes the conservative torque consisting of the inertial torque and the internal torque, the dissipative torque, and the external torque. Moreover, by analyzing the combination of different types of torques and investigating the cycling of energy based on the Casimir function and Hamiltonian function, the interaction between the external torque and other torques is found to be the main reason for the Lü system to generate chaos. Finally, by investigating the Casimir function, it is found that the boundary of the Lü system is only related to system parameters.

scholarly journals Multistability and Formation of Spiral Waves in a Fractional-Order Memristor-Based Hyperchaotic Lü System with No Equilibrium Points

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Bo Yan ◽  
Shaobo He ◽  
Shaojie Wang
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Spiral Wave ◽  
Spiral Waves ◽  
High Complexity ◽  
Coexisting Attractors ◽  
Lü System ◽  
Hyperchaotic Lu System ◽  
Lu System

Multistablity analysis and formation of spiral wave in the fractional-order nonlinear systems is a recent hot topic. In this paper, dynamics, coexisting attractors, complexity, and synchronization of the fractional-order memristor-based hyperchaotic Lü system are investigated numerically by means of bifurcation diagram, Lyapunov exponents (LEs), chaos diagram, and sample entropy (SampEn) algorithm. The results show that the system has rich dynamics and high complexity. Meanwhile, coexisting attractors in the system are observed and hidden dynamics are illustrated by changing the initial conditions. Finally, the network based on the system is built, and the emergence of spiral waves is investigated and chimera states are observed.

scholarly journals A New Nonlinear Chaotic Complex Model and Its Complex Antilag Synchronization

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Emad E. Mahmoud ◽  
Fatimah S. Abood
Keyword(s):  
Nonlinear Term ◽  
Time Lag ◽  
Complex Model ◽  
Lag Synchronization ◽  
State Variable ◽  
Lü System ◽  
System A ◽  
Different Types ◽  
Complex Nonlinear System ◽  
Lu System

Another chaotic nonlinear Lü model with complex factors is covered here. We can build this riotous complex system when we add a complex nonlinear term to the third condition of the complex Lü system and think of it as if every one of the factors is mind boggling or complex. This system in real adaptation is a 6-dimensional continuous autonomous chaotic system. Different types of chaotic complex Lü system are developed. Also, another sort of synchronization is presented by us which is simple for anybody to ponder for the chaotic complex nonlinear system. This sort might be called a complex antilag synchronization (CALS). There are irregular properties for CALS and they do not exist in the literature; for example, (i) the CALS contains or fused two sorts of synchronizations (antilag synchronization ALS and lag synchronization LS); (ii) in CALS the attractors of the main and slave systems are moving opposite or similar to each other with time lag; (iii) the state variable of the main system synchronizes with a different state variable of the slave system. A scheme is intended to accomplish CALS of chaotic complex systems in light of Lyapunov function. The acquired outcomes and effectiveness can be represented by a simulation case for our new model.

scholarly journals A Mass- and Energy-Conserving Numerical Model for a Fractional Gross–Pitaevskii System in Multiple Dimensions

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1765
Author(s):  
Adán J. Serna-Reyes ◽  
Jorge E. Macías-Díaz
Keyword(s):  
Initial Conditions ◽  
Type System ◽  
Continuous System ◽  
Mass Conservation ◽  
Continuous Model ◽  
Manuscript Studies ◽  
Multiple Dimensions ◽  
Negative Constant ◽  
Energy Conserving ◽  
Finite Difference Discretization

This manuscript studies a double fractional extended p-dimensional coupled Gross–Pitaevskii-type system. This system consists of two parabolic partial differential equations with equal interaction constants, coupling terms, and spatial derivatives of the Riesz type. Associated with the mathematical model, there are energy and non-negative mass functions which are conserved throughout time. Motivated by this fact, we propose a finite-difference discretization of the double fractional Gross–Pitaevskii system which inherits the energy and mass conservation properties. As the continuous model, the mass is a non-negative constant and the solutions are bounded under suitable numerical parameter assumptions. We prove rigorously the existence of solutions for any set of initial conditions. As in the continuous system, the discretization has a discrete Hamiltonian associated. The method is implicit, multi-consistent, stable and quadratically convergent. Finally, we implemented the scheme computationally to confirm the validity of the mass and energy conservation properties, obtaining satisfactory results.

scholarly journals SATO–CRUTCHFIELD FORMULATION FOR SOME EVOLUTIONARY GAMES

2003 ◽  
Vol 14 (07) ◽  
pp. 963-971 ◽  
Author(s):  
E. AHMED ◽  
A. S. HEGAZI ◽  
A. S. ELGAZZAR
Keyword(s):  
Prisoner's Dilemma ◽  
Initial Conditions ◽  
Evolutionarily Stable Strategy ◽  
Prisoner’S Dilemma ◽  
Evolutionary Games ◽  
Theoretical Explanation ◽  
Battle Of The Sexes ◽  
Different Types ◽  
Evolutionarily Stable ◽  
Rules Of The Game

The Sato–Crutchfield equations are analytically and numerically studied. The Sato–Crutchfield formulation corresponds to losing memory. Then the Sato–Crutchfield formulation is applied for some different types of games including hawk–dove, prisoner's dilemma and the battle of the sexes games. The Sato–Crutchfield formulation is found not to affect the evolutionarily stable strategy of the ordinary games. But choosing a strategy becomes purely random, independent of the previous experiences, initial conditions, and the rules of the game itself. The Sato–Crutchfield formulation for the prisoner's dilemma game can be considered as a theoretical explanation for the existence of cooperation in a population of defectors.

scholarly journals Analysis of the self-similar solutions of a generalized Burger's equation with nonlinear damping

2001 ◽  
Vol 7 (3) ◽  
pp. 253-282 ◽  
Author(s):  
Ch. Srinivasa Rao ◽  
P. L. Sachdev ◽  
Mythily Ramaswamy
Keyword(s):  
Initial Conditions ◽  
Nonlinear Damping ◽  
Nonlinear Ordinary Differential Equation ◽  
The Self ◽  
Similar Reduction ◽  
Generalized Burgers Equation ◽  
Different Types ◽  
Self Similar ◽  
Parameter Ranges ◽  
Similar Solutions

The nonlinear ordinary differential equation resulting from the self-similar reduction of a generalized Burgers equation with nonlinear damping is studied in some detail. Assuming initial conditions at the origin we observe a wide variety of solutions – (positive) single hump, unbounded or those with a finite zero. The existence and nonexistence of positive bounded solutions with different types of decay (exponential or algebraic) to zero at infinity for specific parameter ranges are proved.

Dynamics of Dual-Cell Series Miura-Ori Structures With Different Types of Inter-Cell Connections

2021 ◽  
Author(s):  
Hai Zhou ◽  
Haiping Wu ◽  
Jian Xu ◽  
Hongbin Fang
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Theoretical Basis ◽  
Initial Conditions ◽  
Cell Structure ◽  
Dynamic Responses ◽  
Complex Environment ◽  
Current State ◽  
Different Types ◽  
Cell Series

Abstract Origami-inspired structures and materials have shown remarkable properties and performances originating from the intricate geometries of folding. Origami folding could be a dynamic process and origami structures could possess rich dynamic characteristics under external excitations. However, the current state of dynamics of origami has mostly focused on the dynamics of a single cell. This research has performed numerical simulations on multi-stable dual-cell series Miura-Ori structures with different types of inter-cell connections based on a dynamic model that does not neglect in-plane mass. We introduce a concept of equivalent constraint stiffness k* to distinguish different types of inter-cell connections. Results of numerical simulations reveal the multi-stable dual-cell structure will exhibit a variety of complex nonlinear dynamic responses with the increasing of connection stiffness because of the deeper energy well it has. The connection stiffness has a strong effect on the steady-state dynamic responses under different excitation amplitudes and a variety of initial conditions. This effect makes us able to adjust the dynamic behaviors of dual-cell series Miura-Ori structure to our needs in a complex environment. Furthermore, the results of this research could provide us a theoretical basis for the dynamics of origami folding and serve as guidelines for designing dynamic applications of origami metastructures and metamaterials.

Risk in Practice

2019 ◽  
pp. 59-75
Author(s):  
Louise K. Comfort
Keyword(s):  
Seismic Risk ◽  
Adaptive Systems ◽  
Path Dependence ◽  
Extreme Event ◽  
Initial Conditions ◽  
Cultural Dimension ◽  
Moderate Degree ◽  
Different Types ◽  
New Ideas ◽  
Response Systems

This chapter examines the four different types of response systems that were identified by degree of adaptation to the problem of seismic risk. Auto-adaptive systems are those that are high on technical infrastructure, high on organizational flexibility, and high on cultural openness to new ideas and strategies of action. Operative adaptive systems are those systems that demonstrate awareness of seismic risk and a moderate degree of professional planning and preparedness to reduce risk of losses. Emergent adaptive systems are those systems that are low on technical structure but show some degree of flexibility in organizational processes and beginning openness to new information and new strategies of action in the cultural dimension. Meanwhile, nonadaptive systems are those systems unable to mobilize effective response operations independently after an extreme event, and virtually all assistance comes from external sources. In practice, initial conditions influenced the formation of response systems following earthquakes in all 12 cases, leading to different types of adaptation. The path dependence that follows from each distinctive set of initial conditions illustrates both the promise and the challenge of shaping communities that are resilient to seismic risk.

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