Showing papers on "Bifurcation diagram published in 2017"

Hidden hyperchaos and electronic circuit application in a 5D self-exciting homopolar disc dynamo.

Abstract: We report on the finding of hidden hyperchaos in a 5D extension to a known 3D self-exciting homopolar disc dynamo. The hidden hyperchaos is identified through three positive Lyapunov exponents under the condition that the proposed model has just two stable equilibrium states in certain regions of parameter space. The new 5D hyperchaotic self-exciting homopolar disc dynamo has multiple attractors including point attractors, limit cycles, quasi-periodic dynamics, hidden chaos or hyperchaos, as well as coexisting attractors. We use numerical integrations to create the phase plane trajectories, produce bifurcation diagram, and compute Lyapunov exponents to verify the hidden attractors. Because no unstable equilibria exist in two parameter regions, the system has a multistability and six kinds of complex dynamic behaviors. To the best of our knowledge, this feature has not been previously reported in any other high-dimensional system. Moreover, the 5D hyperchaotic system has been simulated using a specially de...

Numerical analysis of a fractional-order chaotic system based on conformable fractional-order derivative

Abstract: In this paper, the numerical solutions of conformable fractional-order linear and nonlinear equations are obtained by employing the constructed conformable Adomian decomposition method (CADM). We found that CADM is an effective method for numerical solution of conformable fractional-order differential equations. Taking the conformable fractional-order simplified Lorenz system as an example, the numerical solution and chaotic behaviors of the conformable fractional-order simplified Lorenz system are investigated. It is found that rich dynamics exist in the conformable fractional-order simplified Lorenz system, and the minimum order for chaos is even less than 2. The results are validated by means of bifurcation diagram, Lyapunov characteristic exponents and phase portraits.

Nonlinear dynamics of the quasi-zero-stiffness SD oscillator based upon the local and global bifurcation analyses

Abstract: In this paper, the complicated nonlinear dynamics of the harmonically forced quasi-zero-stiffness SD (smooth and discontinuous) oscillator is investigated via direct numerical simulations. This oscillator considered that the gravity is composed of a lumped mass connected with a vertical spring of positive stiffness and a pair of horizontally compressed springs providing negative stiffness, which can achieve the quasi-zero stiffness widely used in vibration isolation. The local and global bifurcation analyses are implemented to reveal the complex dynamic phenomena of this system. The double-parameter bifurcation diagrams are constructed to demonstrate the overall topological structures for the distribution of various responses in parameter spaces. Using the Floquet theory and parameter continuation method, the local bifurcation patterns of periodic solutions are obtained. Moreover, the global bifurcation mechanisms for the crises of chaos and metamorphoses of basin boundaries are examined by analysing the attractors and attraction basins, exploring the evolutions of invariant manifolds and constructing the basin cells. Meanwhile, additional nonlinear dynamic phenomena and characteristics closely related to the bifurcations are discussed including the resonant tongues, jump phenomena, amplitude–frequency responses, chaotic seas, transient chaos, chaotic saddles, and also their generation mechanisms are presented.

Chaos and Hopf bifurcation control in a fractional-order memristor-based chaotic system with time delay

Abstract: In this paper, a time-delayed feedback controller is proposed in order to control chaos and Hopf bifurcation in a fractional-order memristor-based chaotic system with time delay. The associated characteristic equation is established by regarding the time delay as a bifurcation parameter. A set of conditions which ensure the existence of the Hopf bifurcation are gained by analyzing the corresponding characteristic equation. Then, we discuss the influence of feedback gain on the critical value of fractional order and time delay in the controlled system. Theoretical analysis shows that the controller is effective in delaying the Hopf bifurcation critical value via decreasing the feedback gain. Finally, some numerical simulations are presented to prove the validity of our theoretical analysis and confirm that the time-delayed feedback controller is valid in controlling chaos and Hopf bifurcation in the fractional-order memristor-based system.

Parameter shifts for nonautonomous systems in low dimension: Bifurcation- and Rate-induced tipping

Abstract: We discuss the nonlinear phenomena of irreversible tipping for non-autonomous systems where time-varying inputs correspond to a smooth 'parameter shift' from one asymptotic value to another. We express tipping in terms of properties of local pullback attractors and present some results on how nontrivial dynamics for non-autonomous systems can be deduced from analysis of the bifurcation diagram for an associated autonomous system where parameters are fixed. In particular, we show that there is a unique local pullback point attractor associated with each linearly stable equilibrium for the past limit. If there is a smooth stable branch of equilibria over the range of values of the parameter shift, the pullback attractor will remain close to (track) this branch for small enough rates, though larger rates may lead to rate-induced tipping. More generally, we show that one can track certain stable paths that go along several stable branches by pseudo-orbits of the system, for small enough rates. For these local pullback point attractors, we define notions of bifurcation-induced and irreversible rate-induced tipping of the non-autonomous system. In one-dimension, we introduce the notion of forward basin stability and use this to give a number of sufficient conditions for the presence or absence of rate-induced tipping. We apply our results to give criteria for irreversible rate-induced tipping in a conceptual climate model.

Bifurcation analysis for a singular differential system with two parameters via to topological degree theory

Abstract: Based on the relation between Leray–Schauder degree and a pair of strict lower and upper solutions, we focus on the bifurcation analysis for a singular differential system with two parameters, explicit bifurcation points for relative parameters are obtained by using the property of solution for the akin systems and topological degree theory.

Dynamics, FPGA realization and application of a chaotic system with an infinite number of equilibrium points

Abstract: Recently, systems with infinite equilibria have attracted interest because they can be considered as systems with hidden attractors. In this work, we introduce a new elegant system with an open curve of equilibrium points. It has just one parameter (a) and an exponential function. We show the dynamics of such a new chaotic oscillator by computing the Lyapunov exponents’ spectrum and bifurcation diagram, and it is implemented by using a field-programmable gate array (FPGA). The exponential function is approached by power series and then implemented with adders and multipliers within the FPGA. Experimental results are provided for the attractor as well as for the synchronization of two chaotic oscillators to transmit an image, thus demonstrating the usefulness of the new oscillator in a chaotic secure communication system.

The simplest 4-D chaotic system with line of equilibria, chaotic 2-torus and 3-torus behaviour

Abstract: The paper reports the simplest 4-D dissipative autonomous chaotic system with line of equilibria and many unique properties. The dynamics of the new system contains a total of eight terms with one nonlinear term. It has one bifurcation parameter. Therefore, the proposed chaotic system is the simplest compared with the other similar 4-D systems. The Jacobian matrix of the new system has rank less than four. However, the proposed system exhibits four distinct Lyapunov exponents with $$(+, 0, -, -)$$ sign for some values of parameter and thus confirms the presence of chaos. Further, the system shows chaotic 2-torus $$(+,0,0,-)$$ , quasi-periodic $$[(0,0,-,-), (0,0,0,-)]$$ and multistability behaviour. Bifurcation diagram, Lyapunov spectrum, phase portrait, instantaneous phase plot, Poincare map, frequency spectrum, recurrence analysis, 0–1 test, sensitivity to initial conditions and circuit simulation are used to analyse and describe the complex and rich dynamic behaviour of the proposed system. The hardware circuit realisation of the new system validates the MATLAB simulation results. The new system is developed from the well-known Rossler type-IV 3-D chaotic system.

Fractional-order PD control at Hopf bifurcations in delayed fractional-order small-world networks

Abstract: Bifurcation and control of fractional-order systems are still an outstanding problem. In this paper, a novel delayed fractional-order model of small-world networks is introduced and several topics related to the dynamics and control of such a network are investigated, such as the stability, Hopf bifurcations, and bifurcation control. The nonlinear interactive strength is chosen as the bifurcation parameter to analyze the impact of the interactive strength parameter on the dynamics of the fractional-order small-world network model. Firstly, the stability domain of the equilibrium is completely characterized with respect to network parameters, delays and orders, and some explicit conditions for the existence of Hopf bifurcations are established for the delayed fractional-order model. Then, a fractional-order Proportional-Derivative (PD) feedback controller is first put forward to successfully control the Hopf bifurcation which inherently happens due to the change of the interactive parameter. It is demonstrated that the onset of Hopf bifurcations can be delayed or advanced via the proposed fractional-order PD controller by setting proper control parameters. Meanwhile, the conditions of the stability and Hopf bifurcations are obtained for the controlled fractional-order small-world network model. Finally, illustrative examples are provided to justify the validity of the control strategy in controlling the Hopf bifurcation generated from the delayed fractional-order small-world network model.

Dynamic behaviors of a modified SIR model in epidemic diseases using nonlinear incidence and recovery rates.

Abstract: The transmission of infectious diseases has been studied by mathematical methods since 1760s, among which SIR model shows its advantage in its epidemiological description of spread mechanisms. Here we established a modified SIR model with nonlinear incidence and recovery rates, to understand the influence by any government intervention and hospitalization condition variation in the spread of diseases. By analyzing the existence and stability of the equilibria, we found that the basic reproduction number [Formula: see text] is not a threshold parameter, and our model undergoes backward bifurcation when there is limited number of hospital beds. When the saturated coefficient a is set to zero, it is discovered that the model undergoes the Saddle-Node bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation of codimension 2. The bifurcation diagram can further be drawn near the cusp type of the Bogdanov-Takens bifurcation of codimension 3 by numerical simulation. We also found a critical value of the hospital beds bc at [Formula: see text] and sufficiently small a, which suggests that the disease can be eliminated at the hospitals where the number of beds is larger than bc. The same dynamic behaviors exist even when a ≠ 0. Therefore, it can be concluded that a sufficient number of the beds is critical to control the epidemic.

Fast-Slow Bursters in the Unfolding of a High Codimension Singularity and the Ultra-slow Transitions of Classes.

Abstract: Bursting is a phenomenon found in a variety of physical and biological systems. For example, in neuroscience, bursting is believed to play a key role in the way information is transferred in the nervous system. In this work, we propose a model that, appropriately tuned, can display several types of bursting behaviors. The model contains two subsystems acting at different time scales. For the fast subsystem we use the planar unfolding of a high codimension singularity. In its bifurcation diagram, we locate paths that underlie the right sequence of bifurcations necessary for bursting. The slow subsystem steers the fast one back and forth along these paths leading to bursting behavior. The model is able to produce almost all the classes of bursting predicted for systems with a planar fast subsystem. Transitions between classes can be obtained through an ultra-slow modulation of the model’s parameters. A detailed exploration of the parameter space allows predicting possible transitions. This provides a single framework to understand the coexistence of diverse bursting patterns in physical and biological systems or in models.

Solution and dynamics of a fractional-order 5-D hyperchaotic system with four wings

Abstract: Based on the Adomian decomposition method (ADM), the numerical solution of a fractional-order 5-D hyperchaotic system with four wings is investigated. Dynamics of the system are analyzed by means of phase diagram, bifurcation diagram, Lyapunov exponents spectrum and chaos diagram. The method of one-dimensional linear path through the multidimensional parameter space is proposed to observe the evolution law of the system dynamics with parameters varying. The results illustrate that the system has abundant dynamical behaviors. Both the system order and parameters can be taken as bifurcation parameters. The phenomenon of multiple attractors is found, which means that some attractors are generated simultaneously from different initial values. The spectral entropy (SE) algorithm is applied to estimate the fractional-order system complexity, and we found that the complexity decreases with the increasing of system order. In order to verify the reliability of numerical solution, the fractional-order 5-D system with four wings is implemented on a DSP platform. The phase portraits of fractional-order system generated on DSP agree well with those obtained by computer simulations. It is shown that the fractional-order hyperchaotic system is a potential model for application in the field of chaotic secure communication.

Chaos-based application of a novel no-equilibrium chaotic system with coexisting attractors

Abstract: Novel chaotic system designs and their engineering applications have received considerable critical attention. In this paper, a new three-dimensional chaotic system and its application are introduced. The interesting aspects of this chaotic system are the absence of equilibrium points and the coexisting of limit cycle and torus. Basic dynamics of the no-equilibrium system have been executed by means of phase portraits, bifurcation diagram, continuation, and Lyapunov exponents. Experimental results of the electronic circuit realizing the no-equilibrium system have been reported to show system’s feasibility. By using the chaoticity of the new system without equilibrium, we have developed a random bit generator for practical signal encryption application. Numerical results illustrate the usefulness of the random bit generator.

A chaotic system with an infinite number of equilibrium points located on a line and on a hyperbola and its fractional-order form

Abstract: A three-dimensional autonomous chaotic system with an infinite number of equilibrium points located on a line and a hyperbola is proposed in this paper. To analyze the dynamical behaviors of the proposed system, mathematical tools such as Routh-Hurwitz criteria, Lyapunov exponents and bifurcation diagram are exploited. For a suitable choice of the parameters, the proposed system can generate periodic oscillations and chaotic attractors of different shapes such as bistable and monostable chaotic attractors. In addition, an electronic circuit is designed and implemented to verify the feasibility of the proposed system. A good qualitative agreement is shown between the numerical simulations and the Orcard-PSpice results. Moreover, the fractional-order form of the proposed system is studied using analog and numerical simulations. It is found that chaos, periodic oscillations and periodic spiking exist in this proposed system with order less than three. Then an electronic circuit is designed for the commensurate fractional order α = 0.98, from which we can observe that a chaotic attractor exists in the fractional-order form of the proposed system. Finally, the problem of drive-response generalized projective synchronization of the fractional-order form of the chaotic proposed autonomous system is considered.

A New Chaotic System With Stable Equilibrium: From Theoretical Model to Circuit Implementation

Abstract: Recent evidences suggest that complex behavior such as chaos can be observed in a nonlinear system with stable equilibria. However, few studies have investigated chaotic systems with only one stable equilibrium. This paper introduces a new 3-D chaotic system having only one stable equilibrium. Dynamics of the new system are discovered by using phase portraits, basin of attraction, bifurcation diagram, and maximal Lyapunov exponents. It is interesting that the system has a state variable related with the freedom of offset boosting. In addition, we have investigated the anti-synchronization of the system via an adaptive control. Furthermore, the feasibility of the system is also discussed through presenting its electronic circuit implementation.

Bifurcation, periodic and chaotic motions of the modified equal width-Burgers (MEW-Burgers) equation with external periodic perturbation

Abstract: The modified equal width-Burgers (MEW-Burgers) equation is introduced for the first time. The bifurcation behavior of the MEW-Burgers equation is studied. Considering an external periodic perturbation, the periodic and chaotic motions of the perturbed MEW-Burgers equation are investigated by using phase projection analysis, time series analysis, Poincare section and bifurcation diagram. The strength ( $$f_0$$ ) of the external periodic perturbation plays a crucial role in the periodic and chaotic motions of the perturbed MEW-Burgers equation.

A Novel Four-Dimensional Hyperchaotic Four-Wing System With a Saddle–Focus Equilibrium

Abstract: A novel 4-D hyperchaotic four-wing system with a saddle–focus equilibrium is introduced in this brief. The qualitative analysis of the proposed system confirms its complex dynamic behavior, which is studied by using well-known numerical tools of nonlinear theory, such as the bifurcation diagram, Lyapunov exponents, Poincare maps, and phase portraits. Furthermore, the novel hyperchaotic system is experimentally emulated by an electronic circuit, and its dynamic behavior is studied to confirm the feasibility of the theoretical model.

The effect of blade vibration on the nonlinear characteristics of rotor–bearing system supported by nonlinear suspension

Abstract: The influence of blade vibration on the nonlinear characteristics of rotor–bearing system is non-ignorable in estimating system performance. The extensive studies simplify the rotor system as lumped mass points. The influence of shaft’s bending and shear and the flexibility are usually ignored. The present paper is aim to analyze the nonlinear dynamic behavior of a continuum model. The continuum model of flexible blade–rotor–bearing coupling system is established, simplifying the shaft as Timoshenko beam. The Lagrange method is utilized to derive the differential equation of motion of system. Then, the nonlinear equations of coupling system are numerically solved using the Newmark- $$\upbeta $$ method. The results obtained through the proposed model are compared with the rotor–bearing system without the blades. The effect of several parameters such as rotational speed, the damping coefficient and the length of blade on the nonlinear dynamics of rotor system have been investigated. Inclusive of the analysis methods of bifurcation diagram, three-dimensional spectral plots, time-base analysis, Poincare maps and spectral plots are used to analyze the behavior of the coupling system under different operating conditions, which exhibits rich dynamic behavior of the system.

Stability and bifurcation analysis of an SIR epidemic model with logistic growth and saturated treatment

Abstract: In this paper, we introduce the saturated treatment and logistic growth rate into an SIR epidemic model with bilinear incidence. The treatment function is assumed to be a continuously differential function which describes the effect of delayed treatment when the medical condition is limited and the number of infected individuals is large enough. Sufficient conditions for the existence and local stability of the disease-free and positive equilibria are established. And the existence of the stable limit cycles also is obtained. Moreover, by using the theory of bifurcations, it is shown that the model exhibits backward bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcations. Finally, the numerical examples are given to illustrate the theoretical results and obtain some additional interesting phenomena, involving double stable periodic solutions and stable limit cycles.

Subcritical bifurcation in a self-excited single-degree-of-freedom system with velocity weakening-strengthening friction law: analytical results and comparison with experiments

Abstract: The dynamical behavior of a single-degree-of-freedom system that experiences friction-induced vibrations is studied with particular interest on the possibility of the so-called hard effect of a subcritical Hopf bifurcation, using a velocity weakening–strengthening friction law The bifurcation diagram of the system is numerically evaluated using as bifurcation parameter the velocity of the belt Analytical results are provided using standard linear stability analysis and nonlinear stability analysis to large perturbations The former permits to identify the lowest belt velocity $$({v_\mathrm{lw}})$$ at which the full sliding solution is stable, the latter allows to estimate a priori the highest belt velocity at which large amplitude stick–slip vibrations exist Together the two boundaries $$[v_\mathrm{lw}, v_\mathrm{up}] $$ define the range where two equilibrium solutions coexist, ie, a stable full sliding solution and a stable stick–slip limit cycle The model is used to fit recent experimental observations

Dynamics and circuit of a chaotic system with a curve of equilibrium points

Abstract: Although chaotic systems have been intensively studied since the 1960s, new systems with mysterious features are still of interest. A novel chaotic system including hyperbolic functions is proposed in this work. Especially, the system has an infinite number of equilibrium points. Dynamics of the system are investigated by using non-linear tools such as phase portrait, bifurcation diagram, and Lyapunov exponent. It is interesting that the system can display coexisting chaotic attractors. An electronic circuit for realising the chaotic system has been implemented. Experimental results show a good agreement with theoretical ones.

Four-dimensional autonomous dynamical systems with conservative flows: two-case study

Abstract: Conservative chaotic systems are rare, especially autonomous smooth dynamical systems. This paper reports two four-dimensional (4D) autonomous conservative systems. The conservation of these two systems has been verified using the trace of Jacobian matrix, perpetual point theory and Hamiltonian energy theory. Numerical analyses, including phase portrait, Poincare section, Lyapunov exponent spectrum and bifurcation diagram, verify the existence of the chaotic and quasiperiodic flows. Moreover, a electronic circuit in Multisim is built to demonstrate their chaotic dynamics, whose circuit experimental results agree well with the numerical results.

Coexistence of single- and multi-scroll chaotic orbits in a single-link flexible joint robot manipulator with stable spiral and index-4 spiral repellor types of equilibria

Abstract: This paper reports various chaotic phenomena that occur in a single-link flexible joint (SLFJ) robot manipulator. Four different cases along with subcases are considered here to show different types of chaotic behaviour in a flexible manipulator dynamics. In the first three cases, a partial state feedback as joint velocity and motor rotor velocity feedback is considered, and the resultant autonomous dynamics is considered for analyses. In the fourth case, the manipulator dynamics is considered as a non-autonomous system. The system has (1) one stable spiral and one saddle-node foci, (2) two saddle-node foci and (3) one marginally stable nature of equilibrium points. We found single- and multi-scroll chaotic orbits in these cases. However, with the motor rotor velocity feedback, the system has two unstable equilibria. One of them has an index-4 spiral repellor. In the non-autonomous case, the SLFJ robot manipulator system has an inverse crisis route to chaos and exhibits (1) transient chaos with a stable limit cycle and (2) chaotic behaviour. In all the four cases, the SLFJ manipulator dynamics exhibits coexistence of chaotic orbits, i.e. multi-stability. The various dynamical behaviours of the system are analysed using available methods like phase portrait, Lyapunov spectrum, instantaneous phase plot, Poincare map, parameter space, bifurcation diagram, 0–1 test and frequency spectrum plot. The MATLAB simulation results support various claims made about the system. These claims are further confirmed and validated by circuit implementation using NI Multisim.

The pseudo-Hopf bifurcation for planar discontinuous piecewise linear differential systems

Abstract: The creation or destruction of a crossing limit cycle when a sliding segment changes its stability, is known as pseudo-Hopf bifurcation. In this paper, under generic conditions, we find an unfolding for such bifurcation, and we prove the existence and uniqueness of a crossing limit cycle for this family.