BIFURCATION STRUCTURE AT 1/3-SUBHARMONIC RESONANCE IN AN ASYMMETRIC NONLINEAR ELASTIC OSCILLATOR

1996 ◽  
Vol 06 (08) ◽  
pp. 1529-1546 ◽  
Author(s):  
G. REGA ◽  
A. SALVATORI
Keyword(s):  
Invariant Manifolds ◽  
Initial Conditions ◽  
Basins Of Attraction ◽  
Subharmonic Resonance ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Bifurcation Structure ◽  
Response Curves ◽  
Nonlinear Elastic ◽  
Insight Into

The attractor-basin bifurcation structure in an asymmetric nonlinear oscillator representative of the planar finite forced dynamics of elastic structural systems with initial curvature is studied at the 1/3-subharmonic resonance regime. Local and global analyses are made by means of different computational tools to obtain frequency-response curves of coexisting regular solutions, bifurcation diagrams ensuing from different sets of initial conditions, manifolds structure of direct and inverse saddles corresponding to unstable periodic solutions, basins of attraction at different values of the control parameter. Deep insight into the global dynamics of the system and its evolution is achieved through the analysis of synthetic attractor-basin-manifold phase portraits. The topological mechanisms which entail onset and disappearance of various attractors, and the main and secondary evolutions to chaos, are identified. Special attention is devoted to the analysis of sudden bifurcational events characterizing the system global dynamics, associated with the topological behavior of the invariant manifolds of several direct and inverse saddles. Features of basin metamorphosis, attractor-basin accessibility, and window occurrence are examined. The approach followed, consisting in combined bifurcation analysis of the attractor-basin structure and of the manifold structure, is thought to be useful for a variety of dynamical systems.

scholarly journals Global Dynamics of Delayed Sigmoid Beverton–Holt Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Toufik Khyat ◽  
M. R. S. Kulenović
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Global Dynamics ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The Difference

In this paper, certain dynamic scenarios for general competitive maps in the plane are presented and applied to some cases of second-order difference equation xn+1=fxn,xn−1, n=0,1,…, where f is decreasing in the variable xn and increasing in the variable xn−1. As a case study, we use the difference equation xn+1=xn−12/cxn−12+dxn+f, n=0,1,…, where the initial conditions x−1,x0≥0 and the parameters satisfy c,d,f>0. In this special case, we characterize completely the global dynamics of this equation by finding the basins of attraction of its equilibria and periodic solutions. We describe the global dynamics as a sequence of global transcritical or period-doubling bifurcations.

Multistability and Bubbling Route to Chaos in a Deterministic Model for Geomagnetic Field Reversals

2019 ◽  
Vol 29 (12) ◽  
pp. 1930034
Author(s):  
Paulo C. Rech ◽  
Sudarshan Dhua ◽  
N. C. Pati
Keyword(s):  
Fixed Point ◽  
Geomagnetic Field ◽  
Deterministic Model ◽  
Initial Conditions ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Multiple Attractors ◽  
System A ◽  
Two Parameters

We report coexisting multiple attractors and birth of chaos via period-bubbling cascades in a model of geomagnetic field reversals. The model system comprises a set of three coupled first-order quadratic nonlinear equations with three control parameters. Up to seven kinds of multistable attractors, viz. fixed point-periodic, fixed point-chaotic, periodic–periodic, periodic-chaotic, chaotic–chaotic, fixed point-periodic–periodic, fixed point-periodic-chaotic are obtained depending on the initial conditions for critical parameter sets. Antimonotonicity is a striking characteristic feature of nonlinear systems through which a full Feigenbaum tree corresponding to creation and annihilation of period-doubling cascades is developed. By analyzing the two-parameters dependent dynamics of the system, a critical biparameter zone is identified, where antimonotonicity comes into existence. The complex dynamical behaviors of the system are explored using phase portraits, bifurcation diagrams, Lyapunov exponents, isoperiodic diagram, and basins of attraction.

Hidden Attractor and Multistability in a Novel Memristor-Based System Without Symmetry

2021 ◽  
Vol 31 (11) ◽  
pp. 2150168
Author(s):  
Musha Ji’e ◽  
Dengwei Yan ◽  
Lidan Wang ◽  
Shukai Duan
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Control Unit ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Dynamic Behaviors ◽  
Potential Applications

Memristor, as a typical nonlinear element, is able to produce chaotic signals in chaotic systems easily. Chaotic systems have potential applications in secure communications, information encryption, and other fields. Therefore, it is of importance to generate abundant dynamic behaviors in a single chaotic system. In this paper, a novel memristor-based chaotic system without equilibrium points is proposed. One of the essential features is the absence of symmetry in this system, which increases the complexity of the new system. Then, the nonlinear dynamic behaviors of the system are analyzed in terms of chaos diagrams, bifurcation diagrams, Poincaré maps, Lyapunov exponent spectra, the sum of Lyapunov exponents, phase portraits, 0–1 test, recurrence analysis and instantaneous phase. The results of the sum of Lyapunov exponents show that the given system is a quasi-Hamiltonian system with certain initial conditions (IC) and parameters. Next, other critical phenomena, such as hidden multi-scroll attractors, abundant coexistence characteristics, are found characterized through basins of attraction and others. Especially, it reveals some rare phenomena in other systems that multiple hidden hyperchaotic attractors coexist. Finally, the circuit implementation based on Micro Control Unit (MCU) confirms theoretical analysis and the numerical simulation.

scholarly journals Global Dynamics of Certain Homogeneous Second-Order Quadratic Fractional Difference Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
M. Garić-Demirović ◽  
M. R. S. Kulenović ◽  
M. Nurkanović
Keyword(s):  
Difference Equation ◽  
Unique Feature ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Minimal Period ◽  
Fractional Difference ◽  
The Difference

We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the formxn+1=xn-12/(axn2+bxnxn-1+cxn-12),n=0,1,2,…,where the parameters  a,  b, and  c  are positive numbers and the initial conditionsx-1andx0are arbitrary nonnegative numbers. The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable.

Intrinsic Localized Modes of Harmonic Oscillations in Nonlinear Oscillator Arrays

2013 ◽  
Vol 8 (4) ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Keisuke Nishimura
Keyword(s):  
Frequency Response ◽  
Initial Conditions ◽  
Excitation Frequency ◽  
Basins Of Attraction ◽  
Localized Modes ◽  
Response Curves ◽  
Harmonic Oscillations ◽  
Intrinsic Localized Modes ◽  
Mdof Systems ◽  
Oscillator Arrays

Intrinsic localized modes (ILMs) are investigated in an array with N Duffing oscillators that are weakly coupled with each other when each oscillator is subjected to sinusoidal excitation. The purpose of this study is to investigate the behavior of ILMs in nonlinear multi-degree-of-freedom (MDOF) systems. In the theoretical analysis, van der Pol's method is employed to determine the expressions for the frequency response curves for fundamental harmonic oscillations. In the numerical calculations, the frequency response curves are shown for N = 2 and 3 and compared with the results of the numerical simulations. Basins of attraction are shown for a two-oscillator array with hard-type nonlinearities to examine the possibility of appearance of ILMs when an oscillator is disturbed. The influences of the connecting springs for both hard- and soft-type nonlinearities on the appearance of the ILMs are examined. Increasing the values of the connecting spring constants may cause Hopf bifurcation followed by amplitude modulated motion (AMM) including chaotic vibrations. The influence of the imperfection of an oscillator is also investigated. Bifurcation sets are calculated to show the influence of the system parameters on the excitation frequency range of ILMs. Furthermore, time histories are shown for the case of N = 10, and many patterns of ILMs may appear depending on the initial conditions.

scholarly journals On the genetic divergence of two adjacent populations living in a homogeneous habitat

2021 ◽  
Vol 29 (5) ◽  
pp. 706-726
Author(s):  
Efim Frisman ◽  
◽  
Matvej Kulakov ◽  
Keyword(s):  
Growth Rates ◽  
Genetic Divergence ◽  
Initial Conditions ◽  
Pitchfork Bifurcation ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
First Case ◽  
Fundamental Possibility ◽  
Mean Fitness ◽  
Population Ratio

The purpose is to study the mechanisms leading to the genetic divergence, i.e. stable genetic differences between two adjacent populations coupled by migration of individuals. We considered the case when the fitness of individuals is strictly determined genetically by a single diallelic locus with alleles A and a, the population is panmictic and Mendel's laws of inheritance hold. The dynamic model contains three phase variables: concentration of allele A in each population and fraction (weight) of the first population in the total population size. We assume that the numbers of coupled populations change independently or strictly synchronously. In the first case, the growth rates are determined by fitness of homo- and heterozygotes, the mean fitness of the each population and the initial concentrations of alleles. In the second case, the growth rates are the same. Methods. To study the model, we used the qualitative theory of differential equations studies, including the construction of parametric and phase portraits, basins of attraction and bifurcation diagrams. We studied local bifurcations that provide the fundamental possibility of genetic divergence. Results. If heterozygote fitness is higher than homozygotes, then both populations are polymorphic with the same concentration of homologous alleles. If the heterozygotes fitness is reduced, then over time the populations will have the same monomorphism in one allele, regardless of the type of population changes. In this case, the dynamics is bistable. We showed that the divergence in the model is a result of subcritical pitchfork bifurcation of an unstable polymorphic state. As a result, the genetic divergent state is unstable and exists as part of the transient process to one of monomorphic state. Conclusion. Divergence is stable only for populations that maintain a population ratio in a certain way. In this case, it is preceded by a saddle-node bifurcation and dynamics is quad-stable, i.e. depending on the initial conditions, two types of stable monomorphism and divergence are possible simultaneously.

Synchronization and Global Dynamics of a Cournot Model with Nonlinear Demand and R&D Spillovers

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Wei Zhou ◽  
Mengfan Cui
Keyword(s):  
Invariant Manifolds ◽  
Global Bifurcation ◽  
Basins Of Attraction ◽  
Feasible Region ◽  
Global Dynamics ◽  
Cournot Model ◽  
One Dimensional ◽  
Critical Curves ◽  
Dynamical Behaviors ◽  
The One

In this paper, a dynamical Cournot model with nonlinear demand and R&D spillovers is established. The system is symmetric when the duopoly firms have same economic environments, and it is proved that both the diagonal and the coordinate axes are the one-dimensional invariant manifolds of system. The results show that Milnor attractor of system can be found through calculating the transverse Lyapunov exponents. The synchronization phenomenon is verified through basins of attraction. The effects of adjusting speed and R&D spillovers on the dynamical behaviors of the system are discussed. The topological structures of basins of attraction are analyzed through critical curves, and the evolution process of “holes” in the feasible region is numerically simulated. In addition, various global bifurcation behaviors, such as two kinds of contact bifurcation and the blowout bifurcation, are shown.

Dynamical Effects of Offset Terms on a Modified Chua’s Oscillator and Its Circuit Implementation

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
Léandre Kamdjeu Kengne ◽  
Karthikeyan Rajagopal ◽  
Nestor Tsafack ◽  
Paul Didier Kamdem Kuate ◽  
Balamurali Ramakrishnan ◽  
...  
Keyword(s):  
Initial Conditions ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Nonlinear Phenomena ◽  
Hidden Attractor ◽  
Chua’S Oscillator ◽  
Kirchhoff’S Laws ◽  
Attraction Basins ◽  
The Mathematical Model

This paper addresses the effects of offset terms on the dynamics of a modified Chua’s oscillator. The mathematical model is derived using Kirchhoff’s laws. The model is analyzed with the help of the maximal Lyapunov exponent, bifurcation diagrams, phase portraits, and basins of attraction. The investigations show that the offset terms break the symmetry of the system, generating more complex nonlinear phenomena like coexisting asymmetric bifurcations, coexisting asymmetric attractors, asymmetric double-scroll chaotic attractors and asymmetric attraction basins. Also, a hidden attractor (period-1 limit cycle) is found when varying the initial conditions. More interestingly, this latter attractor coexists with all other self-excited ones. A microcontroller-based implementation of the circuit is carried out to verify the numerical investigations.

scholarly journals THE LONG RUN OUTCOMES AND GLOBAL DYNAMICS OF A DUOPOLY GAME WITH MISSPECIFIED DEMAND FUNCTIONS

2004 ◽  
Vol 06 (03) ◽  
pp. 343-379 ◽  
Author(s):  
GIAN-ITALO BISCHI ◽  
CARL CHIARELLA ◽  
MICHAEL KOPEL
Keyword(s):  
Demand Function ◽  
Initial Conditions ◽  
Basins Of Attraction ◽  
Steady States ◽  
Model Parameters ◽  
Global Dynamics ◽  
Market Demand ◽  
Economic Significance ◽  
Reaction Functions ◽  
Long Run

In this paper we study a model of a quantity-setting duopoly market where firms lack knowledge of the market demand. Using a misspecified demand function firms determine their profit-maximizing choices of their corresponding perceived market game. For illustrative purposes we assume that the (true) demand function is linear and that the reaction functions of the players are quadratic. We then investigate the global dynamics of this game and characterize the number of steady states and their welfare properties. We study the basins of attraction of these steady states and present situations in which global bifurcations of their basins occur when model parameters are varied. The economic significance of our result is to show that in situations where players choose their actions based on a misspecified model of the environment, additional self-confirming steady states may emerge, despite the fact that the Nash-equilibrium of the game under perfect knowledge is unique. As a consequence the long run outcome of the game and overall welfare is highly dependent upon initial conditions.

BASINS OF ATTRACTION OF THE TWO-DIMENSIONAL "POOR MAN'S NAVIER–STOKES EQUATION"

2004 ◽  
Vol 14 (07) ◽  
pp. 2381-2386 ◽  
Author(s):  
S. A. BIBLE ◽  
J. M. MCDONOUGH
Keyword(s):  
Turbulent Flows ◽  
Stokes Equation ◽  
Initial Conditions ◽  
Power Spectra ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Large Eddy

This research is part of an ongoing effort to construct "synthetic velocity" subgrid-scale (SGS) models using discrete dynamical systems (DDSs) for use in large-eddy simulations of turbulent flows. Here we will outline the derivation of the two-dimensional (2-D) "Poor Man's Navier–Stokes" (PMNS) equation from the 2-D, incompressible Navier–Stokes equation to be used in such models and report results from subsequent numerical investigations. In our results emphasis is placed on the effects of initial conditions on the dynamics of the 2-D PMNS equation, using such modes of investigation as regime maps, basins of attraction diagrams, phase portraits, time series and power spectra. The most important findings of this investigation concern applicable ranges of bifurcation parameters, causes and effects of symmetries seen in solutions of the PMNS equation, and the suitability of the methods of investigation used here.

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