scholarly journals New Conservation Laws and Exact Cosmological Solutions in Brans–Dicke Cosmology with an Extra Scalar Field

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1364
Author(s):  
Antonios Mitsopoulos ◽  
Michael Tsamparlis ◽  
Genly Leon ◽  
Andronikos Paliathanasis
Keyword(s):  
Scalar Field ◽  
Cosmological Model ◽  
Conservation Laws ◽  
Nonlinear Dynamical Systems ◽  
Scalar Fields ◽  
Jordan Frame ◽  
Nonlinear Dynamical ◽  
New Exact Solutions ◽  
The Stability ◽  
Variational Symmetries

The derivation of conservation laws and invariant functions is an essential procedure for the investigation of nonlinear dynamical systems. In this study, we consider a two-field cosmological model with scalar fields defined in the Jordan frame. In particular, we consider a Brans–Dicke scalar field theory and for the second scalar field we consider a quintessence scalar field minimally coupled to gravity. For this cosmological model, we apply for the first time a new technique for the derivation of conservation laws without the application of variational symmetries. The results are applied for the derivation of new exact solutions. The stability properties of the scaling solutions are investigated and criteria for the nature of the second field according to the stability of these solutions are determined.

Is There a Relation Between Synchronization Stability and Bifurcation Type?

2020 ◽  
Vol 30 (08) ◽  
pp. 2050123
Author(s):  
Zahra Faghani ◽  
Zhen Wang ◽  
Fatemeh Parastesh ◽  
Sajad Jafari ◽  
Matjaž Perc
Keyword(s):  
Complex Networks ◽  
Nonlinear Dynamical Systems ◽  
General Relation ◽  
Dynamical Behavior ◽  
Stability Function ◽  
Nonlinear Dynamical ◽  
Practical Applications ◽  
Stability Functions ◽  
Stability And Bifurcation ◽  
The Stability

Synchronization in complex networks is an evergreen subject with many practical applications across the natural and social sciences. The stability of synchronization is thereby crucial for determining whether the dynamical behavior is stable or not. The master stability function is commonly used to that effect. In this paper, we study whether there is a relation between the stability of synchronization and the proximity to certain bifurcation types. We consider four different nonlinear dynamical systems, and we determine their master stability functions in dependence on key bifurcation parameters. We also calculate the corresponding bifurcation diagrams. By means of systematic comparisons, we show that, although there are some variations in the master stability functions in dependence on bifurcation proximity and type, there is in fact no general relation between synchronization stability and bifurcation type. This has important implication for the restrained generalizability of findings concerning synchronization in complex networks for one type of node dynamics to others.

scholarly journals Quadratic integrals of a multi-scalar cosmological model

2020 ◽  
Vol 35 (10) ◽  
pp. 2050068 ◽  
Author(s):  
Sameerah Jamal
Keyword(s):  
Scalar Field ◽  
Cosmological Model ◽  
Conservation Laws ◽  
Analytical Solutions ◽  
Physical System ◽  
Noether Symmetries ◽  
Spatial Curvature ◽  
Special Case ◽  
Friedmann Robertson Walker ◽  
Field Integral

In the context of Friedmann–Robertson–Walker (FRW) spacetime with zero spatial curvature, we consider a multi-scalar tensor cosmology model under the pretext of obtaining quadratic conservation laws. We propose two new interaction potentials of the scalar field. Integral to this task is the existence of dynamical Noether symmetries which are Lie–Bäcklund transformations of the physical system. Finally, analytical solutions of the field are found corresponding to each new model. In one of the models, we find that the scale factor mimics [Formula: see text]-cosmology in a special case.

Adaptive Fuzzy Sliding Mode Control for a Class of Nonlinear Dynamical Systems

2011 ◽  
Vol 71-78 ◽  
pp. 4309-4312 ◽  
Author(s):  
Wen Da Zheng ◽  
Gang Liu ◽  
Jie Yang ◽  
Hong Qing Hou ◽  
Ming Hao Wang
Keyword(s):  
Sliding Mode Control ◽  
Nonlinear Dynamical Systems ◽  
Sliding Mode ◽  
Adaptive Controller ◽  
Adaptive Sliding Mode Control ◽  
Nonlinear Dynamical ◽  
Mode Control ◽  
Fuzzy Sliding Mode ◽  
The Stability ◽  
Theory Of Lyapunov Stability

This paper presents a FBFN-based (Fuzzy Basis Function Networks) adaptive sliding mode control algorithm for nonlinear dynamic systems. Firstly, we designed an perfect control law according to the nominal plant. However, there always exists discrepancy between nominal and actual mode, and the FBFN was applied to approximate the uncertainty. After that, the adaptive law was designed to update the parameters of FBFN to alleviate the approximating errors. Based on the theory of Lyapunov stability, the stability of the adaptive controller was given with a sufficient condition. Simulation example was also given to illustrate the effectiveness of the method.

Bifurcation Control of Ro¨ssler System

2003 ◽  
Author(s):  
Z. Q. Wu ◽  
P. Yu
Keyword(s):  
Feedback Control ◽  
Nonlinear Dynamical Systems ◽  
Equilibrium Points ◽  
New Method ◽  
Control Law ◽  
Equilibrium Solutions ◽  
Nonlinear Dynamical ◽  
The Stability ◽  
Parameter Values ◽  
Feasible System

In this paper, a new method is proposed for controlling bifurcations of nonlinear dynamical systems. This approach employs the idea used in deriving the transition variety sets of bifurcations with constraints to find the stability region of equilibrium points in parameter space. With this method, one can design, via a feedback control, appropriate parameter values to delay either static, or dynamic or both bifurcations as one wishes. The approach is applied to consider controlling bifurcations of the Ro¨ssler system. The uncontrolled Ro¨ssler has two equilibrium solutions, one of which exhibits static bifurcation while the other has Hopf bifurcation. When a feedback control based on the new method is applied, one can delay the bifurcations and even change the type of bifurcations. An optimal control law is obtained to stabilize the Ro¨ssler system using all feasible system parameter values. Numerical simulations are used to verify the analytical results.

scholarly journals State Observation for Lipschitz Nonlinear Dynamical Systems Based on Lyapunov Functions and Functionals

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1424 ◽  
Author(s):  
Angelo Alessandri ◽  
Patrizia Bagnerini ◽  
Roberto Cianci
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Lyapunov Functions ◽  
Nonlinear Dynamical Systems ◽  
Estimation Error ◽  
Stability Conditions ◽  
Nonlinear Dynamical ◽  
State Observation ◽  
State Observers ◽  
The Stability

State observers for systems having Lipschitz nonlinearities are considered for what concerns the stability of the estimation error by means of a decomposition of the dynamics of the error into the cascade of two systems. First, conditions are established in order to guarantee the asymptotic stability of the estimation error in a noise-free setting. Second, under the effect of system and measurement disturbances regarded as unknown inputs affecting the dynamics of the error, the proposed observers provide an estimation error that is input-to-state stable with respect to these disturbances. Lyapunov functions and functionals are adopted to prove such results. Third, simulations are shown to confirm the theoretical achievements and the effectiveness of the stability conditions we have established.

ON THE BIRTH OF MULTIPLE LIMIT CYCLES IN NONLINEAR SYSTEMS

1996 ◽  
Vol 06 (12b) ◽  
pp. 2587-2603 ◽  
Author(s):  
JORGE L. MOIOLA ◽  
GUANRONG CHEN
Keyword(s):  
Limit Cycles ◽  
Nonlinear Dynamical Systems ◽  
Approximation Technique ◽  
Hopf Bifurcations ◽  
Systems Methodology ◽  
Nonlinear Dynamical ◽  
Parameter Perturbations ◽  
Feedback Control Systems ◽  
Stability Indexes ◽  
The Stability

Degenerate (or singular) Hopf bifurcations of a certain type determine the appearance of multiple limit cycles under system parameter perturbations. In the study of these degenerate Hopf bifurcations, computational formulas for the stability indexes (i.e., curvature coefficients) are essential. However, such formulas are very difficult to derive, and so are usually computed by different approximation methods. Inspired by the feedback control systems methodology and the harmonic balance approximation technique, higher-order approximate formulas for such curvature coefficients are derived in this paper in the frequency domain setting. The results obtained are then applied to a study of nonlinear dynamical systems within the region of one periodic solution, bypassing a direct investigation of the multiple limit cycles and some tedious discussion of the complex multiplicity issue. Finally, we will show that several types of stability bifurcations can be controlled based on the results obtained in this paper.

scholarly journals Tracking Nonlinear Oscillations with Time-Delayed Feedback

2006 ◽  
Vol 5-6 ◽  
pp. 417-424
Author(s):  
Jan Sieber ◽  
B. Krauskopf
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Oscillations ◽  
Stability Boundary ◽  
Nonlinear Dynamical ◽  
Friction Oscillator ◽  
Long Time ◽  
Basic Ideas ◽  
The Stability

We demonstrate a method for tracking the onset of oscillations (Hopf bifurcation) in nonlinear dynamical systems. Our method does not require a mathematical model of the dynamical system but instead relies on feedback controllability. This makes the approach potentially applicable in an experiment. The main advantage of our method is that it allows one to vary parameters directly along the stability boundary. In other words, there is no need to observe the transient oscillations of the dynamical system for a long time to determine their decay or growth. Moreover, the procedure automatically tracks the change of the critical frequency along the boundary and is able to continue the Hopf bifurcation curve into parameter regions where other modes are unstable.We illustrate the basic ideas with a numerical realization of the classical autonomous dry friction oscillator.

Semi-Explicit Composition Methods in Memcapacitor Circuit Simulation

2019 ◽  
Vol 10 (2) ◽  
pp. 37-52 ◽  
Author(s):  
Denis N. Butusov ◽  
Valerii Y. Ostrovskii ◽  
Artur I. Karimov ◽  
Valery S. Andreev
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Circuit Simulation ◽  
Simulation Software ◽  
Continuous Systems ◽  
Implicit Midpoint Rule ◽  
Nonlinear Dynamical ◽  
Composition Methods ◽  
The Stability ◽  
New Generation ◽  
Embedded Applications

Composition algorithms make up a prospective class of methods for solving ordinary differential equations. Their main advantage is an ability to retain some properties of the simulated continuous systems, e.g. phase space volume. Meanwhile, computational costs of composition solvers for non-Hamiltonian systems are high because the implicit midpoint rule should be used as a basic method. This also complicates the development of embedded applications based on the numerical solution of ODEs, such as hardware chaos generators. In this article, a new semi-explicit composition methods are proposed. The stability regions for different composition algorithms were plotted and a memcapacitor circuit was studied as a test problem. Computational experiments reveal the superior properties of semi-explicit composition algorithms as a hardware-targeted ODE solvers. The obtained results imply that the development of semi-explicit composition algorithms is a step towards construction a new generation of simulation software for nonlinear dynamical systems and embedded chaos generators.

scholarly journals On damping in the vicinity of critical points

2013 ◽  
Vol 371 (1993) ◽  
pp. 20120426 ◽  
Author(s):  
L. N. Virgin ◽  
R. Wiebe
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Damping Ratio ◽  
Transcritical Bifurcation ◽  
Nonlinear Dynamical ◽  
Free Decay ◽  
Generic Bifurcation ◽  
Primary Focus ◽  
The Stability ◽  
Transcritical Bifurcations ◽  
Bifurcation Behaviour

The effect of damping on the behaviour of oscillations in the vicinity of bifurcations of nonlinear dynamical systems is investigated. Here, our primary focus is single degree-of-freedom conservative systems to which a small linear viscous energy dissipation has been added. Oscillators with saddle–node, pitchfork and transcritical bifurcations are shown analytically to exhibit several interesting characteristics in the free decay response near a bifurcation. A simple mechanical oscillator with a transcritical bifurcation is used to experimentally verify the analytical results. A transcritical bifurcation was selected because it may be used to represent generic bifurcation behaviour. It is shown that the damping ratio can be used to predict a change in the stability with respect to changing system parameters.

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