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Showing papers on "Bifurcation diagram published in 2018"


Journal ArticleDOI
TL;DR: A chaotic image encryption algorithm is proposed according to the extended Lu system with coexisting attractors and the performance of the algorithm is numerically analyzed.
Abstract: This paper introduces an extended Lu system with coexisting attractors. The number and stability of equilibria are determined. The coexisting attractors of the system are displayed by the bifurcation diagrams, Lyapunov exponent spectrum, phase portraits. It is shown that the system has a pair of strange attractors, a pair of limit cycles, a pair of point attractors for different initial conditions. The circuit implementation of the chaotic attractor and coexisting attractors of the system are presented. The control problem of the system is studied as well. A controller is designed to stabilize the system to the origin and realize the switching between two chaotic attractors based on the passive control method. Moreover, a chaotic image encryption algorithm is proposed according to the system. The performance of the algorithm is numerically analyzed.

140 citations


Journal ArticleDOI
TL;DR: In this article, the authors presented a multi-stable electromagnetic-induction energy harvesting (MEH) system by magnetic levitation oscillation, which has a nonlinear restoring force and a multiwell restoring force potential, offering an improvement upon their linear counterparts by broadening its frequency response.

124 citations


Journal ArticleDOI
TL;DR: The fractional-order simplest memristor-based chaotic circuit is investigated based on the novel conformable Adomian decomposition method (CADM), and it is found the minimum order of this system for generating chaos is 1.08.
Abstract: In this paper, the fractional-order simplest memristor-based chaotic circuit is investigated based on the novel conformable Adomian decomposition method (CADM). Dynamics of this circuit is analyzed by employing bifurcation diagram, Lyapunov exponent spectrum, Poincare section and other methods. The result shows that it has rich dynamical behaviors and we found the minimum order of this system for generating chaos is 1.08. To implement the system in digital circuit, the CADM iteration results with different items are compared to balance the speed and accuracy, and the suitable items are chosen for further application. Finally, DSP implementation of the system verifies the effectiveness of the solution algorithm.

98 citations


Journal ArticleDOI
TL;DR: This work presents a new chaotic hyperjerk system having two exponential nonlinearities, which can be useful in scientific fields such as Random Number Generators (RNGs), data security, data hiding, etc.
Abstract: Hyperjerk systems have received significant interest in the literature because of their simple structure and complex dynamical properties This work presents a new chaotic hyperjerk system having two exponential nonlinearities Dynamical properties of the chaotic hyperjerk system are discovered through equilibrium point analysis, bifurcation diagram, dissipativity and Lyapunov exponents Moreover, an adaptive backstepping controller is designed for the synchronization of the chaotic hyperjerk system Also, a real circuit of the chaotic hyperjerk system has been carried out to show the feasibility of the theoretical hyperjerk model The chaotic hyperjerk system can also be useful in scientific fields such as Random Number Generators (RNGs), data security, data hiding, etc In this work, three implementations of the chaotic hyperjerk system, viz RNG, image encryption and sound steganography have been performed by using complex dynamics characteristics of the system

98 citations


Journal ArticleDOI
TL;DR: In this paper, a generalized double-humped (DH) logistic map was used for pseudo-random number key generation (PRNG) and an image encryption algorithm was introduced based on the proposed generalized DH map offering secure communication transfer of medical MRI and X-ray images.

87 citations


Journal ArticleDOI
TL;DR: Experimental results and security analysis show that SHAM-IEA has strong capability to withstand statistical analysis, differential attack, chosen-plaintext and chosen-ciphertext attacks.
Abstract: Based on the one-dimensional Sine map and the two-dimensional Henon map, a new two-dimensional Sine-Henon alteration model (2D-SHAM) is hereby proposed. Basic dynamic characteristics of 2D-SHAM are studied through the following aspects: equilibria, Jacobin eigenvalues, trajectory, bifurcation diagram, Lyapunov exponents and sensitivity dependence test. The complexity of 2D-SHAM is investigated using Sample Entropy algorithm. Simulation results show that 2D-SHAM is overall hyperchaotic with the high complexity, and high sensitivity to its initial values and control parameters. To investigate its performance in terms of security, a new 2D-SHAM-based image encryption algorithm (SHAM-IEA) is also proposed. In this algorithm, the essential requirements of confusion and diffusion are accomplished, and the stochastic 2D-SHAM is used to enhance the security of encrypted image. The stochastic 2D-SHAM generates random values, hence SHAM-IEA can produce different encrypted images even with the same secret key. Experimental results and security analysis show that SHAM-IEA has strong capability to withstand statistical analysis, differential attack, chosen-plaintext and chosen-ciphertext attacks.

79 citations


Journal ArticleDOI
TL;DR: This paper proposes an adaptive robust finite-time control method based on a global sliding surface for the synchronization of a class of chaotic systems and obtains a four-dimensional system with two interesting features.
Abstract: This paper proposes an adaptive robust finite-time control method based on a global sliding surface for the synchronization of a class of chaotic systems. New chattering-free control laws are designed to guarantee the removal of the reaching mode and realize the existence of the sliding mode around the designed surface right from the first moment. The proposed adaptive-tuning controllers eliminate the requirement of knowledge about disturbance bounds. Using the suggested control technique, superior master–slave synchronization is achieved, the chattering problem is fully solved, and the amplitudes of the control signals are noticeably decreased. Demonstrative simulation results for a Lu chaotic system are presented to indicate the efficiency and usefulness of the proposed scheme. In the end, using a state-feedback controller, we obtain a four-dimensional system with two interesting features. First, some hyperchaotic solutions are proposed, and then a continuous bifurcation diagram showing chaos for a wide...

72 citations


Journal ArticleDOI
TL;DR: An interesting new oscillator with coexisting limit cycles and point attractors is obtained by modifying a known two-dimensional oscillator and changing this new system to its forced version and choosing a proper set of parameters, which introduces a chaotic system with some very interesting features.
Abstract: In this paper, by modifying a known two-dimensional oscillator, we obtain an interesting new oscillator with coexisting limit cycles and point attractors. Then by changing this new system to its fo...

68 citations


Journal ArticleDOI
TL;DR: A novel fractional-order two-gene regulatory network model with delays is proposed, which can describe the memory and hereditary properties of genetic regulatory networks more suitably and the onset of the Hopf bifurcation increases distinctly when the fractional order deceases.

68 citations


Journal ArticleDOI
Bocheng Bao, Pingye Wu, Han Bao, Quan Xu, Mo Chen 
TL;DR: In this article, a novel third-order autonomous memristive chaotic oscillator is presented, which is accomplished by parallelly coupling a simple diode bridge emulator into a Sallen-key low-pass filter.
Abstract: This paper presents a novel third-order autonomous memristive chaotic oscillator, which is accomplished by parallelly coupling a simple memristive diode bridge emulator into a Sallen–Key low-pass filter (LPF). With the modeling of this oscillator, stability analyses of the equilibrium point and numerical simulations of the phase plane orbit, time-domain sequence, bifurcation diagram, and finite-time Lyapunov exponent spectrum are performed, from which period, quasi-period, chaos, and quasi-period to chaos route are found. Particularly, two types of dynamical phenomena of quasi-periodic behavior and point-cycle chaotic bursting that are further identified by using 0–1 test are observed in such a third-order autonomous memristive oscillator, which have been rarely reported in the previous literatures. Additionally, hardware experiments are implemented and the quasi-periodic behavior and point-cycle chaotic bursting are well confirmed.

68 citations


Journal ArticleDOI
TL;DR: It is revealed that the stability and bifurcation of high-dimension fractional ring-structured neural networks with multiple time delays heavily relies on the sum of time delays for the proposed networks, and the stability performance of such networks can be markedly improved by selecting carefully the total time delays.

Journal ArticleDOI
TL;DR: Analysis results show that the variation of meshing frequency as the external excitation could transit the states of the system and the higher damping coefficient and suitable backlash could suppress the region of chaos.

Journal ArticleDOI
TL;DR: In this article, the qualitative behavior of two discrete-time glycolysis models is discussed and the parametric conditions for local asymptotic stability of positive steady-state are investigated.
Abstract: In this paper, the qualitative behavior of two discrete-time glycolysis models is discussed. The discrete-time models are obtained by implementing forward Euler’s scheme and nonstandard finite difference method. The parametric conditions for local asymptotic stability of positive steady-states are investigated. Moreover, we discuss the existence and directions of period-doubling and Neimark–Sacker bifurcations with the help of center manifold theorem and bifurcation theory. OGY feedback control and hybrid control methods are implemented in order to control chaos in discrete-time glycolysis model due to emergence of period-doubling and Neimark–Sacker bifurcations. Numerical simulations are provided to illustrate theoretical discussion.

Journal ArticleDOI
17 Jan 2018-PLOS ONE
TL;DR: Results indicate that memristor coupling can be effective to control chaotic circuits and it is also useful to reproduce dynamical behaviors for neuronal activities.
Abstract: In this paper, a new four-variable dynamical system is proposed to set chaotic circuit composed of memristor and Josephson junction, and the dependence of chaotic behaviors on nonlinearity is investigated. A magnetic flux-controlled memristor is used to couple with the RCL-shunted junction circuit, and the dynamical behaviors can be modulated by changing the coupling intensity between the memristor and the RCL-shunted junction. Bifurcation diagram and Lyapunov exponent are calculated to confirm the emergence of chaos in the improved dynamical system. The outputs and dynamical behaviors can be controlled by the initial setting and external stimulus as well. As a result, chaos can be suppressed and spiking occurs in the sampled outputs under negative feedback, while applying positive feedback type via memristor can be effective to trigger chaos. Furthermore, it is found that the number of multi-attractors in the Jerk circuit can be modulated when memristor coupling is applied on the circuit. These results indicate that memristor coupling can be effective to control chaotic circuits and it is also useful to reproduce dynamical behaviors for neuronal activities.

Journal ArticleDOI
TL;DR: It is found that appropriate parameters setting can induce distinct chaotic and periodical states by analyzing the output series and the chaos encryption based on Josephson junction circuit coupled by memristor is investigated.

Journal ArticleDOI
TL;DR: In this paper, a new memristive system is proposed, which can have no equilibrium and a line of equilibrium based on the value of its controlling parameter, changing that parameter can cause the system having both chaotic and hyperchaotic solutions.
Abstract: A new memristive system is proposed in this paper which can have no equilibrium and a line of equilibrium based on the value of its controlling parameter. Also, changing that parameter can cause the system having both chaotic and hyperchaotic solutions. This system has a multi-wing strange attractor. Dynamical properties of this system such as Lyapunov exponents and bifurcation diagram are calculated. This system belongs to the category of systems with hidden and multistable attractors. A system with all the above-mentioned properties is not common in the literature. Finally, an adaptive sliding mode control method is applied to synchronize this chaotic system.

Journal ArticleDOI
TL;DR: A novel 3D fractional-order chaotic system is proposed in this paper and the corresponding implementation circuit is designed and the Multisim simulations and the hardware experimental results are well in accordance with numerical simulations of the same system on the Matlab platform.

Journal ArticleDOI
TL;DR: In this paper, a discrete-time prey-predator model with Allee effect is proposed and parametric conditions for local asymptotic stability of steady-state are investigated.
Abstract: In this paper, a discrete-time prey-predator model with Allee effect is proposed. The parametric conditions for local asymptotic stability of steady-states are investigated. Moreover, we discuss the existence and directions of period-doubling and Neimark-Sacker bifurcations with the of help center manifold theorem and bifurcation theory. In order to control chaos due to emergence of Neimar-Sacker bifurcation, we apply OGY feedback control method and the hybrid control methodology is also implemented. Numerical simulations are provided to illustrate theoretical discussion.

Journal ArticleDOI
TL;DR: In this article, an analytical expression of a stroboscopic controlled hybrid Poincare map is presented for determining the fixed point of such map and for studying its stability, and conditions for the localization of the border-collision bifurcation are established.

Journal ArticleDOI
TL;DR: A new fractionalized logistic map is proposed based on the numerical algorithm of fractional differentiation definition that holds rich dynamical behaviors because of its memory effect and new types of behaviors of bifurcation and chaos are found.

Journal ArticleDOI
TL;DR: In this article, the authors considered an ecological model, in a predator-prey interaction with the presence of a herd behavior, and proved the existence of positive solution and also the existence Hopf bifurcation, Turing driven instability, and Turing hopf point.
Abstract: We consider in this paper an ecological model, in a predator–prey interaction with the presence of a herd behavior. For the analysis of the model, the existence of positive solution and also the existence Hopf bifurcation, Turing driven instability, and Turing–Hopf bifurcation point have bee proved. Then by calculating the normal form, on the center of the manifold associated to the Hopf bifurcation points, the stability of the periodic solution has been proved. In the last part of the paper, numerical simulations has been given to illustrate our theoretical analysis.

Journal ArticleDOI
TL;DR: This paper presents a new three-dimensional autonomous chaotic system with hyperbolic sine equilibrium, which has been discovered by using equilibrium analysis, phase portrait, Poincaré map, bifurcation diagram and Lyapunov spectrum.
Abstract: For the past 4 years, there has been a rapid rise in the study of chaotic systems with curves of equilibria which are categorized as systems with hidden attractors. There is still significant controversy surrounding the shapes of equilibrium points. This paper presents a new three-dimensional autonomous chaotic system with hyperbolic sine equilibrium. Fundamental dynamical properties and complex dynamics of the system have been discovered by using equilibrium analysis, phase portrait, Poincare map, bifurcation diagram and Lyapunov spectrum. It is crucial to note that there are bistable hidden chaotic attractors in the introduced system. Furthermore, in order to show the feasibility of the new system with hyperbolic sine equilibrium, its electronic circuit has been implemented.

Journal ArticleDOI
09 Mar 2018-Pramana
TL;DR: In this article, a chaotic jerk system with non-hyperbolic equilibrium is proposed and the dynamics of this new system is revealed through equilibrium analysis, phase portrait, bifurcation diagram and Lyapunov exponents.
Abstract: The literature on chaos has highlighted several chaotic systems with special features. In this work, a novel chaotic jerk system with non-hyperbolic equilibrium is proposed. The dynamics of this new system is revealed through equilibrium analysis, phase portrait, bifurcation diagram and Lyapunov exponents. In addition, we investigate the time-delay effects on the proposed system. Realisation of such a system is presented to verify its feasibility.

Journal ArticleDOI
TL;DR: In this article, a simple 4D chaotic system with no equilibrium point and having hidden attractors with the coexistence of attractors (i.e. multistability) is presented.
Abstract: The finding of hidden attractors in a chaotic/hyperchaotic system is more important, interesting and difficult than a self-excited attractor. This paper reports a new simple 4-D chaotic system with no equilibrium point and having hidden attractors with the coexistence of attractors (i.e. multistability). The proposed system has a total of eight terms including only one nonlinear term and hence, it is simple. It has only one bifurcation parameter. The system has complex dynamical behaviour. It exhibits 3-torus, 2-torus, chaotic and chaotic 2-torus behaviours. The coexistence of hidden attractors in the proposed system is analysed with phase portrait, Finite time Lyapunov spectrum, bifurcation diagram, Poincare map, instantaneous phase plot and 0–1 test. The system has chaotic behaviour with $$({+,0,-,-})$$ sign of distinct Lyapunov exponents although the Jacobian matrix has rank less than four. Electronic circuit realisation is shown to validate the chaotic behaviour of the proposed system.

Journal ArticleDOI
TL;DR: In this paper, deflated continuation is used to compute steady-state solitary waveforms in a one-component, two-dimensional nonlinear Schrodinger equation with a parabolic trap and repulsive interactions.

Journal ArticleDOI
TL;DR: In this article, the stick-slip dynamics of a vibration-driven locomotion system from a sliding bifurcation perspective were analyzed. And a simplified single degree-of-freedom model was established, with the rationality of simplification been explained theoretically and numerically.

Journal ArticleDOI
TL;DR: Based on closed-loop modulation coupling pattern and the model of sinusoidal cavity, a high-dimensional sinusoid cavity hyperchaotic system is proposed in this paper, where the number of cavities is controlled by the system parameters.
Abstract: Based on closed-loop modulation coupling pattern and the model of sinusoidal cavity, a high-dimensional sinusoidal cavity hyperchaotic system is proposed. The number of sinusoidal cavities is controlled by the system parameters. By designing a piecewise-linear controller, the grid sinusoidal cavity attractors are obtained. The equilibrium points are theoretically analyzed through mathematical calculation. Taking the two-dimensional grid sinusoidal cavity hyperchaotic map as an example, dynamics of the system are analyzed by phase diagram, equilibrium points, Lyapunov exponents spectrum, bifurcation diagram, complexity and distribution characteristics. The results show that it has rich dynamical behaviors, including complicated phase space trajectory, hyperchaotic behavior, large maximum Lyapunov exponent and typical bifurcations. The proposed hyperchaotic map has advantages in complexity and distribution in the whole parameter space. Therefore, it has good application prospects in secure communication.

Journal ArticleDOI
TL;DR: This paper proposes a delayed fractional-order gene regulatory network model, and it is shown that the fractional order can be effectively manipulated to control the dynamics of such network, and the stability domain can be changed with different fractional orders.
Abstract: In this paper, we propose a delayed fractional-order gene regulatory network model. Firstly, the sum of delays is chosen as the bifurcation parameter, and the conditions of the existence for Hopf bifurcations are achieved through analyzing its characteristic equation. Secondly, it is shown that the fractional order can be effectively manipulated to control the dynamics of such network, and the stability domain can be changed with different fractional orders. The fractional-order genetic network can generate a Hopf bifurcation (oscillation appears) as the sum of delays passes through some critical values. Therefore, we can achieve some desirable dynamical behaviors by choosing the appropriate fractional order. Finally, numerical simulations are carried out to illustrate the validity of our theoretical analysis.

Journal ArticleDOI
TL;DR: In this paper, the authors presented five 4D autonomous conservative chaotic systems having non-hyperbolic equilibria with various characteristics, and verified the chaotic behaviours of the proposed systems by using phase portrait plot, Poincare map, local Lyapunov spectrum, bifurcation diagram and frequency spectrum plots.
Abstract: Very little research is available in the field of 4-D autonomous conservative chaotic systems. This paper presents five new 4-D autonomous conservative chaotic systems having non-hyperbolic equilibria with various characteristics. The proposed systems have different numbers of non-hyperbolic equilibrium points. One of the new systems has four non-hyperbolic equilibria points along with lines of equilibria. Hence, this system may belong to the category of hidden attractors chaotic system. The first, second, fourth and fifth type of the systems exhibit coexistence of chaotic flow, whereas the third type of the system exhibits coexistence of chaotic flows with quasi-periodic behaviour. The chaotic behaviours of the proposed systems are verified by using phase portrait plot, Poincare map, local Lyapunov spectrum, bifurcation diagram and frequency spectrum plots. The conservative nature of the proposed systems is proved by finding the sum of finite-time local Lyapunov exponents, finite-time local Lyapunov dimensions and divergence of the vector field. The sum of the finite-time local Lyapunov exponents and divergence of the vector field are equal to zero, and local Lyapunov dimension is equal to the order of the system confirm the conservative nature of the new chaotic systems.

Journal ArticleDOI
TL;DR: An algorithm to determine which folding joints to make stiffer in order to ensure that the sheet is folded into the chosen state is proposed, enabling speed-dependent selection of different folded states.
Abstract: Disordered mechanical systems, when strongly deformed, have complex configuration spaces with multiple stable states and pathways connecting them. The topology of such pathways determines which states are smoothly accessible from any part of configuration space. Controlling this topology would allow us to limit access to undesired states and select desired behaviors in metamaterials. Here, we show that the topology of such pathways, as captured by bifurcation diagrams, can be tuned using imperfections such as stiff hinges in elastic networks and creased thin sheets. We derive Linear Programming-like equations for designing desirable pathway topologies. These ideas are applied to eliminate the exponentially many ways of misfolding self-folding sheets by making some creases stiffer than others. Our approach allows robust folding by entire classes of external folding forces. Finally, we find that the bifurcation diagram makes pathways accessible only at specific folding speeds, enabling speed-dependent selection of different folded states.