Naimark Sacker Bifurcation Research Articles

Research on Stability and Bifurcation for Two-Dimensional Two-Parameter Squared Discrete Dynamical Systems

This study investigates a class of two-dimensional, two-parameter squared discrete dynamical systems. It determines the conditions for local stability at the fixed points for these proposed systems. Theoretical and numerical analyses are conducted to examine the bifurcation behavior of the proposed systems. Conditions for the existence of Naimark–Sacker bifurcation, transcritical bifurcation, and flip bifurcation are derived using center manifold theorem and bifurcation theory. Results of the theoretical analyses are validated by numerical simulation studies. Numerical simulations also reveal the complex bifurcation behaviors exhibited by the proposed systems and their advantage in image encryption.

Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation

We investigate the global asymptotic stability of the following second order rational difference equation of the form xn+1=Bxnxn-1+F/bxnxn-1+cxn-12, n=0,1,…, where the parameters B, F, b, and c and initial conditions x-1 and x0 are positive real numbers. The map associated with this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric space. In some cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. Also, we show that considered equation exhibits the Naimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period.

Organization of the periodicity in the parameter-space of a glycolysis discrete-time mathematical model

In this paper, we report on the dynamics of a discrete-time mathematical model, which is obtained by the forward Euler method from the continuous-time mathematical model of the glycolysis process. More specifically, here we investigate the parameter-space of a two-dimensional map resulting from this discretization process. Different places where period-doubling and Naimark–Sacker bifurcations occur are determined. We also investigate the organization of typical periodic structures embedded in a quasiperiodic region which is a result of a Naimark–Sacker bifurcation. We identify period-adding, Farey, and Fibonacci sequences of periodic structures embedded in this quasiperiodic region.

Nonlinear vibrations characteristics of single-walled carbon nanotubes by nonlocal elastic shell model

The model of single-walled carbon nanotubes nonlinear vibrations, which is based on shell theory, is derived. The system of partial differential equations with respect to three displacements projections is obtained using the Sanders–Koiter shell theory and nonlocal elasticity. The system of nonlinear ordinary differential equations, which describes the nanotubes vibrations, is obtained by the Galerkin method. The harmonic balanced method is used to calculate free nonlinear vibrations. The bifurcation behavior of CNT vibrations owing to the Naimark–Sacker bifurcation is analyzed. Almost periodic vibrations of carbon nanotubes are investigated numerically. The nanotube periodic and almost periodic vibrations have three displacements projections u,v,w, which are commensurable. This is essential difference of the nanotube vibrations from macroshells.

Analysis of Integrodifference Equations with a Separable Dispersal Kernel

Integrodifference equations are a class of infinite-dimensional dynamical systems in discrete time that have recently received great attention as mathematical models of population dynamics in spatial ecology. The dispersal of individuals between generations is described by a ‘dispersal kernel’, a probability density function for the distance that an individual moves within a season. Previous authors recognized that the dynamics are reduced to a finite-dimensional problem when the dispersal kernel is separable. We prove some open questions from their work on the dynamics of a single population and then extend the idea to investigate the dynamics of two spatially distributed species in (i) a competitive relation, and (ii) a predator-prey relation. In all cases, we discuss how the dynamics of the population(s) depend on the amount of suitable space that is available to them. We find a number of bifurcations, such as period-doubling sequences and Naimark-Sacker bifurcations, which we illustrate through simulations.

Bifurcations and chaos in Mira 2 map

In this paper, Mira 2 map is investigated. The conditions of the existence for fold bifurcation, flip bifurcation and Naimark-Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. And the conditions of the existence for chaos in the sense of Marroto are obtained. Numerical simulation results not only show the consistence with the theoretical analysis but also display complex dynamical behaviors, including period-n orbits, crisis, some chaotic attractors, period-doubling bifurcation to chaos, quasi-period behaviors to chaos, chaos to quasi-period behaviors, bubble and onset of chaos.

Naimark-Sacker bifurcation of second order rational difference equation with quadratic terms

We investigate the global asymptotic stability and Naimark-Sacker bifurcation of the dierence equation xn+1 = F bxnxn 1 +cx 2 1 +f ; n = 0; 1;:::

Bifurcations behavior of bending vibrations of beams with two breathing cracks

The lateral vibrations of the beams with two breathing cracks are considered. Two contact parameters are used to describe breathing of the cracks. The finite degrees of freedom nonlinear dynamical system with two constraints is derived to describe the beam vibrations. These constraints describe breathing of the cracks. The continuation technique is applied to analyze numerically the periodic vibrations. The periodic motions, their stability and bifurcations are investigated. It is analyzed the sub-harmonic motions, which originate from the period-doubling bifurcations. The sub-harmonic motions undergo the Naimark-Sacker bifurcations. Therefore, these motions are transformed into the quasiperiodic ones. The influences of the cracks lengths on the sub-harmonic motions are investigated numerically.

Bifurcation behavior of steady vibrations of cantilever plates with geometrical nonlinearities interacting with three-dimensional inviscid potential flow

Cantilever plates with geometrical nonlinearities interacting with three-dimensional inviscid potential flow are considered. The new method for analysis of stability and bifurcations of the plate self-sustained vibrations is suggested. The basis of this method is the solution of the singular integral equations with respect to aerodynamic derivatives of the plate pressure drop. The Von Karman equations with respect to transversal displacements and a stress function are used to describe the plate vibrations with geometrical nonlinearity. The combination of the shooting technique and the continuation algorithm are used to analyze bifurcations and stability of the plate self-sustained vibrations. The plate aeroelastic self-sustained vibrations lose stability due to the Naimark-Sacker bifurcation. As a result of this bifurcation, almost periodic and chaotic vibrations are observed. It is shown, when it is acceptable to use one-dimensional models of plate self-sustained vibrations and when the plate model must be two-dimensional.

From Mandelbrot-like sets to Arnold tongues

A transition from Mandelbrot-like sets to Arnold tongues is characterized via a coupling of two non-identical quadratic maps proposed by us. A two-dimensional parameter-space considering the parameters of the individual quadratic maps was used to demonstrate numerically the event. The location of the parameter sets where Naimark–Sacker bifurcations occur, which is exactly the place where Arnold tongues of arbitrary periods are born, was computed analytically.

The dynamics of a symmetric coupling of three modified quadratic maps

We investigate the dynamical behavior of a symmetric linear coupling of three quadratic maps with exponential terms, and identify various interesting features as a function of two control parameters. In particular, we investigate the emergence of quasiperiodic states arising from Naimark—Sacker bifurcations of stable period-1, period-2, and period-3 orbits. We also investigate the multistability in the same coupling. Lyapunov exponents, parameter planes, phase space portraits, and bifurcation diagrams are used to investigate transitions from periodic to quasiperiodic states, from quasiperiodic to mode-locked states and to chaotic states, and from chaotic to hyperchaotic states.

Resonant many-mode periodic and chaotic self-sustained aeroelastic vibrations of cantilever plates with geometrical non-linearities in incompressible flow

The plates interacting with inviscid, incompressible, potential gas flow are analyzed. Many modes interaction is considered to describe self-sustained vibrations of plates. The singular integral equation is solved to obtain gas pressures acting on the plate. The Von Karman equations with respect to three displacements are used to describe the plate geometrical non-linear vibrations. The Galerkin method is applied to each partial differential equation to obtain the finite-degree-of-freedom model of the plate vibrations. Self-sustained vibrations, which take place due to the Hopf bifurcation, are investigated. These vibrations undergo the Naimark–Sacker bifurcation and the periodic motions are transformed into the almost periodic ones. If the stream velocity is increased, almost periodic motions are transformed into chaotic ones. As a result of the internal resonance, the saturation of the vibration mode is observed. The non-linear dynamics of low- and high-aspect-ratio plates is analyzed.

Numerical Study of a Three-Dimensional Hénon Map

We consider a three-dimensional generalization of the two-dimensional Hénon map. We first investigate the emergence of quasiperiodic states, as a result of Naimark-Sacker bifurcations of period-1 and period-2 orbits. Secondly we investigate the disappearance of the resonance torus in the transition from quasiperiodicity to chaos.

DYNAMICS OF ASYMMETRIC COUPLINGS OF TWO HÉNON MAPS

In this paper, we present and analyze two asymmetric couplings of two Hénon maps. In the first case, we investigate numerically phenomena associated with the appearence, in the parameter-space, of periodic structures like Arnold tongues, which are windows of periodicity immersed in a quasiperiodic region resulting from a Naimark–Sacker bifurcation. We show that these structures organize themselves in period-adding bifurcation cascades, and that successive windows have monotonically decreasing width. In the second case, we show that when the individual Hénon maps in coupling are both chaotic, there are specific parameter values that may force the coupled maps into periodic orbits. Therefore, parameter-space regions of the coupled system where the chaotic dynamics is driven towards regular dynamics are found. In the two cases, Lyapunov exponents, bifurcation diagrams, parameter-space and phase-space plots are used to characterize the dynamics observed as parameters are changed.

Chaos and Hyperchaos in a Symmetric Coupling of Three Quadratic Maps

In this paper we investigate the dynamical behavior of a symmetric coupling of three quadratic maps, and identify various interesting features as a function of the nonlinearity and the coupling parameters. In particular, we study the emergence of quasiperiodic states arising from Naimark-Sacker bifurcations of stable periodic orbits pertaining to 1 × 2 n cascade, namely period-1 and period-2 orbits. Lyapunov exponent plots, parameter-space and three-dimensional phase-space portraits, and bifurcation diagrams are used to study the transition from periodic to quasiperiodic states, from quasiperiodic to mode-locked and to chaotic states, and from chaotic to hyperchaotic states. We also investigate the modifications introduced in the parameter-space, by the increase of the number of maps in the coupling.

Arnold tongues and quasiperiodicity in a prey–predator model

We consider a two-dimensional map in which one of the fixed points is destabilized via a supercritical Naimark–Sacker bifurcation. We investigate, via analytical and numerical simulations, phenomena associated with the appearance, in the parameter-space, of structures like Arnold tongues involved in the Naimark–Sacker bifurcation. We determine analytically the location of the parameter sets where Naimark–Sacker bifurcation occurs, and the location on this line where tongues of arbitrary period are born. Lyapunov exponents, bifurcation diagrams, parameter-space and phase-space diagrams are used to show the transition from quasiperiodic to chaotic states.

Complex dynamics in a two-dimensional noninvertible map

A two-dimensional noninvertible map is investigated. The conditions of existence for pitchfork bifurcation, flip bifurcation and Naimark–Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. Chaotic behavior in the sense of Marotto’s definition of chaos is proven. And numerical simulations not only show the consistence with the theoretical analysis but also exhibit the complex dynamical behaviors, including period-34, period-5 orbits, quasi-period orbits, intermittency, boundary crisis as well as chaotic transient. The computation of Lyapunov exponents conforms the dynamical behaviors.

Naimark–Sacker bifurcations in a delay quartic map

In this paper, we consider a two-dimensional map in which one of the fixed points is destabilized via a supercritical Naimark–Sacker bifurcation. We investigate, via numerical simulations, phenomena associated with the appearance, in the phase-space, of closed invariant curves involved in the Naimark–Sacker bifurcation. Lyapunov exponents, parameter-space and phase-space diagrams are used to show that the transition from quasiperiodic to chaotic states generally do not happen in this case. We determine numerically the location of the parameter sets where the Naimark–Sacker bifurcation occurs.

Generation of quasiperiodic oscillations in pairs of coupled maps

We investigate analytically the emergence of quasiperiodic and mode-locked states of arbitrary period in pairs of coupled maps. Quasiperiodic and mode-locked states arise from Naimark–Sacker bifurcations and exhibit very rich local dynamics. We determine analytically both states for pairs of maps under symmetric and asymmetric couplings. Lyapunov spectra, portraits in parameter-space and in phase-space are used to show the transition from quasiperiodic and from mode-locked states into chaotic states and to check the range of validity of the approximations implicit in the standard normal forms.

Bifurcations and chaos of a delayed ecological model

In this paper, a complicated delayed ecological model is studied in detail. First, the parameter plane is divided into three different regions, in which there exist various fixed points of the model. Second, by using the center manifold theorem, it is shown that there exists the transcritical bifurcation rather than the saddle-node bifurcation with respect to the zero fixed point of the model. For the non-zero fixed points of the model, their stabilities and bifurcations are studied when some parameters are varied. Third, the numerical simulation results not only show the consistence with the theoretical analysis but also display some new interesting dynamical behaviors and various bifurcations, including periodic invariant n -cycles circles ( n = 6, 7, 8, 16, 22), continuous closed invariant circles, transcritical bifurcations, Naimark–Sacker bifurcations and inverse Naimark–Sacker bifurcations. In particular, some false phenomena, such as a 3-cycle and many 4-cycles, obtained only from the bifurcation diagram are revised by our qualitative analysis. Finally, the computations of the maximum Lyapunov exponents confirm that there exist chaotic motions in this model.