Help Of Lyapunov Exponents Research Articles

Study of Fixed Points and Chaos in Wave Propagation for the Generalized Damped Forced Korteweg-de Vries Equation using Bifurcation Analysis

In this article, we consider the Generalized Damped Forced Korteweg-de Vries (GDFKdV) equation. The forcing term considered is of the form $F(U)=U(U-v_1)(U-v_2)$, where $v_1$ and $v_2$ are free parameters. We investigate the behaviour of fixed points evaluated for the corresponding dynamical system of our model problem. With respect to these fixed points, we investigate the effects of a few significant parameters involved in the model, namely, the free parameters $v_1$ and $v_2$, the nonlinear, dispersion and damping coefficients using the tools from bifurcation analysis. We also obtain the wave plots for the critical values of the nonlinear and dispersion coefficients for which the system becomes unstable and exhibit chaotic behaviour. We confirm the chaos in our dynamical system under various conditions with the help of Lyapunov exponents.

Controlling subharmonic resonance and chaos by a fast forcing in a van der Pol–Duffing oscillator with parametrically excited damping

A periodically damped van der Pol oscillator is analysed with emphasis on the formation of sub harmonic resonance and the corresponding transition of its motion to the chaotic domain. The slow fast separation technique is used to analytically ascertain the domain of such resonances by the removal of secular terms. The same method is then used in conjunction of multiple scale technique to identify the existence of Hopf bifurcation and the generation of limit cycle. Furthermore the Hamiltonian nature of the system has been utilized in the Melnikov function to determine the threshold of the parameter values for the inset of chaos. Over and above the change in the hysteresis structure due to the periodic damping is explicitly determined. In the chaotic regime the characterization is done with the help of Lyapunov exponent, Poincare map and phase space attractor structure.

Coexistence of triple, quadruple attractors and Wada basin boundaries in a predator–prey model with additional food for predators

In biological control programs, the provision of additional food is regarded as one of the most useful tools for species conservation and pest management. In a system, providing additional food for predators can distract the predators from the overconsumption of prey (short term) or enhance the predation rate (long term). In the present article, we consider a predator–prey model with Holling type-II functional response by incorporating additional food for the predator species in a discrete-time setup. Here, we intend to explore the impact of additional food on the growth of prey as well as the overall dynamics of the underlying system. We analyze the system dynamics by varying two control parameters, viz, the biomass of additional food and the intrinsic growth rate of prey species simultaneously with the help of Lyapunov exponent and isoperiodic diagrams. Through the simultaneous variation of two parameters, very rich and complex dynamical behaviors of a simple deterministic system can come into view with the presence of structurally stable periodic patterns in the transitional and chaotic zones, which is not looked on only by the variability of one parameter. In this study, we notice the presence of infinite families of different organized periodic structures, like Arnold tongues, shrimp-shaped domains, a new kind of ‘double fishhook’ structures, etc. in the quasiperiodic and chaotic zones of the parameter planes with various kinds of period-adding phenomena. The study also discloses the transition to chaos via shrimp-induced period-bubbling process. We observe the coexistence of double and triple attractors in several different sets of attractors. The most fascinating finding in this study is the coexistence of quadruple attractors, one of the rarest phenomena in ecological systems. The basin boundaries of all these coexisting double, triple, and quadruple attractors are of a very complex nature, they are either fractal basin boundaries or Wada basin boundaries.

Dynamics of chaotic waterwheel model with the asymmetric flow within the frame of Caputo fractional operator

The chaotic waterwheel model is a mechanical model that exhibits chaos and is also a practical system that justifies the Lorenz system. The chaotic waterwheel model (or Malkus waterwheel model) is modified with the addition of asymmetric water inflow to the system. The hereditary property of the modified chaotic waterwheel model is analyzed to determine the system’s stability and identify the parameter that contributes to the stability We also examine the factor that leads to the bifurcation. We determine the well-posed nature of the modified system. The modified chaotic waterwheel model is defined with the Caputo fractional operator. The existence and uniqueness, boundedness, stability, Lyapunov stability, and numerical simulation are studied for the modified fractional waterwheel model. The bifurcation parameter and Lyapunov exponent are examined to study the chaotic nature of the system with respect to the fractional order. The nature of the system is captured with the help of the efficient numerical approach Adams–Bashforth–Moulton Method. The numerical approach demonstrates that the chaotic nature of the modified chaotic waterwheel is changed into unstable nature, which could further reduce to the stable case with suitable values of the parameter. This analysis is justified with the help of Lyapunov exponent. We consider irrational order (π,e) in the present work to illustrate the reliability of fractional order.

Dynamics of particles around the time conformal quantum corrected Schwarzschild black hole

Abstract In this paper, we explore the dynamics of neutral and charged particles in presence/absence of magnetic field around the quantum corrected-Schwarzschild black hole as well as the time conformal quantum corrected-Schwarzschild black hole. We investigate the conditions under which the particles move towards or away from the vicinity of black hole when they collide with other particles. We study the effective potential and effective force in presence/absence of magnetic field graphically. Recently many researchers took interest in point particle dynamics and one can initiate the study of a more complex motion of extended celestial bodies. A major area of interest among these important fields, is the particle motions (neutral and charged) around black hole and naked singularities. In this work we want to show that the quantum corrected-Schwarzschild black hole admits the time conformal factor e ϵ g ( t ) without violating the symmetry structure of the black hole, where ϵ is a small parameter that causes perturbation in the proposed spacetime and g ( t ) is a function of time t . The inception of time conformal factor, re-scales all the conservation laws that hold in the given spacetime. This type of study is important because it tells about the formation of gravitational waves and Hawking radiation. The study of thermal stability, horizons and energy conditions corresponding to the time conformal quantum corrected Schwarzschild spacetime is also explored. Furthermore, we analyze and compare the dynamics of charged and neutral particle around time conformal quantum corrected-Schwarzschild black hole with the time conformal Schwarzschild black hole. The effective force and effective potential with magnetic field and angular momentum are given graphically. The stability of the proposed spacetime is also discussed with the help of Lyapunov exponent.

Numerical and Dynamic Errors Analysis of Planar Multibody Mechanical Systems With Adjustable Clearance Joints Based on Lagrange Equations and Experiment.

For theoretical study and engineering application, it is necessary to provide an accurate and simple dynamical model to simulate the multibody mechanical systems with clearance joints and it is also the subject of this article. Based on Lagrange equations of the first kind, a different numerical methodology, the length and rotation angle of the clearance joints are looked as independent coordinates for the first time, is presented in detail. The slider-crank mechanism, with a single or double adjustable revolute clearance joints, is used as a numerical model. A test rig and a simulink model, fully in accordance with the numerical model, are used to measure the velocity, displacement, and acceleration. The numerical results tally with experimental and simulink results reveal that the new methodology, presented in this paper, provides a correct approach to build the dynamical equations of mechanism with clearance joints. Lyapunov exponent is used to analyze the motion status, chaotic or periodic, of the slider. Based on data points, mean absolute deviation (MAD) is applied to judge the dynamical errors, displacement, velocity, and acceleration, of the slider due to clearance joints. With the help of Lyapunov exponent and MAD, the results indicated that various clearance sizes and drive speeds can change the dynamical behaviors of the slider, which is complex but can be predicted in some way.

Hyperchaos and Coexisting Attractors in a Modified van der Pol–Duffing Oscillator

This paper deals with a new modified hyperchaotic van der Pol–Duffing (MVPD) snap oscillator. Various dynamical properties of the proposed system are investigated with the help of Lyapunov exponents, stability analysis of the equilibrium points and bifurcation plots. The existence of the Hopf bifurcation is established by analyzing the corresponding characteristic equation. It is also proved that the MVPD oscillator shows multistability with coexisting attractors. Various numerical simulations are conducted and presented to show the dynamical behavior of the MVPD system. To show that the system is hardware realizable, we derive the discrete model of the MVPD system using the Euler’s method and using the hardware–software cosimulation, the proposed MVPD system is implemented in Field Programmable Gate Arrays. It is shown that the output of the digital implementations of the MVPD systems matches the numerical analysis.

Characterization of Chaotic Electro Optic Modulator with the Helpof Lyapunov Exponent and Bifurcation Analysis Technique.

Chaos is a fundamental property that comes out of nonlinearity and a sensitive dependence on initial condition. A nonlinear dynamic system that displays a chaotic steady state behaviour is purely deterministic, but its long-term behaviour cannot be predicted because of the property of sensitivity to initial conditions. One of the most promising applications of chaos theory, which exploits both the deterministic and unpredictable aspects of chaotic behaviour, is chaos-based secure communications. The sensitivity to initial condition is quantified with the help of the entropy named Lyapunov Exponent. Generation of chaos from nonlinear optical devices have been explored from several years. In this paper an EO modulator system of cos2 type of nonlinearity is selected as chaos producing device and characterized with the help of Lyapunov Exponent. The Lyapunov Exponent is computed here to determine the dependence of a chaotic signal on the initial conditions and also on the system parameters. The period doubling behaviour in this Electro optic modulator system have also been studied here. The transition from periodic to chaotic behaviours in this one-dimensional discrete dynamical system have been controlled through variation of a single parameter out of different system parameters. With the help of Lyapunov Exponent and bifurcation analysis technique an estimation of the system parameters has been carried out to predict the nature of the output generated at the terminal of the EO cell.

Dynamic bifurcation and strange nonchaos in a two-frequency parametrically driven nonlinear oscillator

A periodically forced series LCR circuit with Chua’s diode as a nonlinear element exhibits slow passage through Hopf bifurcation. This slow passage leads to a delay in the Hopf bifurcation. The delay in this bifurcation is a unique quantity and it can be predicted using various numerical analysis. We find that when an additional periodic force is added to the system, the delay in bifurcation becomes chaotic which leads to an unpredictability in bifurcation delay. Further, we study the bifurcation of the periodic delay to chaotic delay in the slow passage effect through strange nonchaotic delay. We also report the occurrence of strange nonchaotic dynamics while varying the parameter of the additional force included in the system. We observe that the system exhibits a hitherto unknown dynamical transition to a strange nonchaotic attractor. With the help of Lyapunov exponent, we explain the new transition to strange nonchaotic attractor and its mechanism is studied by making use of rational approximation theory. The birth of SNA has also been confirmed numerically, using Poincaré maps, phase sensitivity exponent, the distribution of finite-time Lyapunov exponents and singular continuous spectrum analysis.

Ergodic properties for some non-expanding non-reversible systems

We study several properties of invariant measures obtained from preimages, for non-invertible maps on fractal sets which model non-reversible dynamical systems. We give two ways to describe the distribution of all preimages for endomorphisms which are not necessarily expanding on a basic set Λ . We give a topological dynamics condition which guarantees that the corresponding measures converge to a unique conformal ergodic borelian measure; this helps in estimating the unstable dimension a.e. with respect to this measure with the help of Lyapunov exponents. When there exist negative Lyapunov exponents of this limit measure, we study the conditional probabilities induced on the non-uniform local stable manifolds by the limit measure, and also its pointwise dimension on stable manifolds.

On the analysis of spatial chaos and Hopf bifurcation in an optical open flow

Abstract Spatial chaos in a system of copropagating nonlinear beam, interacting mutually, is explored. The case of four beams belonging to two weakly degenerate mode in a multimode fibre is analysed. The modulation of the waves due to mutual energy exchange is shown to give rise to chaos. We characterize the chaos with the help of Lyapunov exponent and phase space analysis. The case of centre manifold with imaginary eigenvalue is separated with the help of normal form analysis, when the system is reduced to a two component one. The reduced problem is then analysed with the help of a tangent field diagram.