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1.
In this paper, a new simple 4D smooth autonomous system is proposed, which illustrates two interesting rare phenomena: first, this system can generate a four-wing hyperchaotic and a four-wing chaotic attractor and second, this generation occurs under condition that the system has only one equilibrium point at the origin. The dynamic analysis approach in the paper involves time series, phase portraits, Lyapunov exponents, bifurcation diagram, and Poincaré maps, to investigate some basic dynamical behaviors of the proposed 4D system. The physical existence of the four-wing hyperchaotic attractor is verified by an electronic circuit. Finally, it is shown that the fractional-order form of the system can also generate a chaotic four-wing attractor.  相似文献   

2.
This paper presents a new four-dimensional (4-D) smooth quadratic autonomous chaotic system, which can present periodic orbit, chaos, and hyper-chaos under the conditions on different parameters. Importantly, the system can generate a four-wing hyper-chaotic attractor and a pair of coexistent double-wing hyper-chaotic attractors with two symmetrical initial conditions. Furthermore, a four-wing transient chaos occurs in the system. The dynamic analysis approach- in the paper involves time series, phase portraits, Poincaré maps, bifurcation diagrams, and Lyapunov exponents, to investigate some basic dynamical behaviors of the proposed 4-D system.  相似文献   

3.
In this paper we numerically investigate the fractional-order sliding-mode control for a novel fractional-order hyperchaotic system. Firstly, the dynamic analysis approaches of the hyperchaotic system involving phase portraits, Lyapunov exponents, bifurcation diagram, Lyapunov dimension, and Poincaré maps are investigated. Then the fractional-order generalizations of the chaotic and hyperchaotic systems are studied briefly. The minimum orders we found for chaos and hyperchaos to exist in such systems are 2.89 and 3.66, respectively. Finally, the fractional-order sliding-mode controller is designed to control the fractional-order hyperchaotic system. Numerical experimental examples are shown to verify the theoretical results.  相似文献   

4.
This paper first formulates a Hamiltonian system with hyperchaotic phenomena and investigates the equilibrium point and double Hopf bifurcation of the system. We obtain the result that the Hamiltonian system has hyperchaotic behaviors when any system parameter varies. The influences of holonomic constraint and nonholonomic constraint on the equilibrium points, invariance and the hyperchaotic state of the Hamiltonian system are then studied. Finally, we achieve the hyperchaotic control of the Hamiltonian system by introducing the constraint method. The studies indicate that the constraint can not only change the Hamiltonian system from hyperchaotic state to periodic state or chaotic state, but also make the Hamiltonian system become globally asymptotically stable. Numerical simulations, including Lyapunov exponents, bifurcation diagrams, Poincaré maps and phase portraits for systems, exhibit the complex dynamical behaviors.  相似文献   

5.
In this paper, a chaotic fractional-order modified hybrid optical system is presented. Some basic dynamical properties are further investigated by means of Poincaré mapping, parameter phase portraits, and the largest Lyapunov exponents. Fractional Hopf bifurcation conditions are proposed; it is found that Hopf bifurcation occurs on the proposed system when the fractional-order varies and passes a sequence of critical values. The chaotic motion is validated by the positive Lyapunov exponent. Finally, some numerical simulations are also carried out to illustrate our results.  相似文献   

6.
In this article, the stretch-twist-fold (STF) flow is numerically studied using phase portraits, sensitive dependence on initial conditions, Lyapunov exponents, power spectrum, and the Poincaré map. The stretch-twist-fold flow is a two-parameter family of Stokes flows defined in a unit sphere that is associated with the fluid particle motion that naturally arises in the dynamo theory, which proposes a mechanism by which celestial bodies, such as earth and sun can maintain and amplify the magnetic field continuously. For this continuous growth of magnetic field, scientists are interested to invent new tools for the nonfuel consumption magnetism propulsion for the low earth orbit of spacecrafts or satellites. General properties of a chaotic dynamical system reference to the stretch-twist-fold flow model are addressed and numerical solutions are generated to explain some of these properties. Analytically, we studied the local behavior at equilibrium points. The predictability of chaos in the STF flow with the numerical calculation of Lyapunov exponents and Poincaré map is presented in this paper.  相似文献   

7.
This paper presents a new class of chaotic systems with infinite number of equilibrium points like a three-leaved clover. They signify an exciting class of dynamical systems which represent many major characteristics of regular and chaotic motions. These chaotic systems belong to the general class of chaotic systems with hidden attractors. By using a systematic computer search, three chaotic systems with three-leaved-clover-shaped equilibria were found which are classified into dissipative systems. Dynamics of the chaotic system with the three-leaved-clover-equilibria has been investigated by using phase portraits, bifurcation diagram, Lyapunov exponents, Kaplan–Yorke dimension and Poincaré map. Moreover, an electronic circuit implementation of the theoretical system is designed to check its effectiveness. Random number generator design has been realized with newly developed chaotic systems. The obtained random bit sequences are used for image encryption. Security analysis of image encryption processes has been performed.  相似文献   

8.
In the present part of the paper various problems of non-linear dynamics of nano-beams within the modified couple stress theory as well as the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Reddy-Levinson models are studied taking into account the geometric non-linearity. Different characteristics of the vibrational process, including Fourier spectra, wavelet spectra, phase portraits, Poincaré maps as well as the largest Lyapunov exponents, are studied for the same physical-geometric parameter with and without consideration of the size-dependent behaviour. Vibration graphs are constructed and analysed, and scenarios of transition from regular to chaotic vibrations are illustrated and discussed.  相似文献   

9.
This article presents a new hyperchaotic system of four-dimensional quadratic autonomous ordinary differential equations, which has one equilibrium point and two quadratic nonlinearities. Some basic dynamical properties are further investigated by means of Poincaré mapping, parameter phase portraits, and calculated Lyapunov exponents and power spectra. The existence of the hyperchaotic system is verified not only by theoretical analysis but also by conducting a novel fourth-order electronic circuit experiment. Various attractors of experimental results show that this 4D hyperchaotic system is different from the historically proposed system and has good engineering application prospects.  相似文献   

10.
This paper analyzes the hyperchaotic behaviors of the newly presented simplified Lorenz system by using a sinusoidal parameter variation and hyperchaos control of the forced system via feedback. Through dynamic simulations which include phase portraits, Lyapunov exponents, bifurcation diagrams, and Poincaré sections, we find the sinusoidal forcing not only suppresses chaotic behaviors, but also generates hyperchaos. The forced system also exhibits some typical bifurcations such as the pitchfork, period-doubling, and tangent bifurcations. Interestingly, three-attractor coexisting phenomenon happens at some specific parameter values. Furthermore, a feedback controller is designed for stabilizing the hyperchaos to periodic orbits, which is useful for engineering applications.  相似文献   

11.
Based on Rikitake system, a new chaotic system is discussed. Some basic dynamical properties, such as equilibrium points, Lyapunov exponents, fractal dimension, Poincaré map, bifurcation diagrams and chaotic dynamical behaviors of the new chaotic system are studied, either numerically or analytically. The obtained results show clearly that the system discussed is a new chaotic system. By utilizing the fractional calculus theory and computer simulations, it is found that chaos exists in the new fractional-order three-dimensional system with order less than 3. The lowest order to yield chaos in this system is 2.733. The results are validated by the existence of one positive Lyapunov exponent and some phase diagrams. Further, based on the stability theory of the fractional-order system, projective synchronization of the new fractional-order chaotic system through designing the suitable nonlinear controller is investigated. The proposed method is rather simple and need not compute the conditional Lyapunov exponents. Numerical results are performed to verify the effectiveness of the presented synchronization scheme.  相似文献   

12.
We study networks of coupled oscillators governed by ODEs and yielded by physically validated sets of a few PDEs governing dynamics of structural members (plate and beams), chaos and phase synchronization and contact/no-contact non-linear dynamics of structural members coupled via boundary conditions. We have detected, illustrated and discussed a few novel kinds of hybrid states of the studied plate-beam(s) contact/no-contact interactions as well as novel scenarios of transition into chaos exhibited by the interplay of continuous objects. Classical (time histories, phase portraits, Poincaré maps, FFT, Lyapunov exponents) and non-classical (2D Morlet wavelets) approaches are used while monitoring non-linear dynamics of the interacting spatial structural members. Our results include examples from structural mechanics and the studied objects are modelled by validated mechanical hypotheses and assumptions. Novel non-linear phenomena including switching to different vibration regimes and phase chaotic synchronization are illustrated and discussed.  相似文献   

13.
This paper reports a new five-dimensional (5D) hyperchaotic system with three positive Lyapunov exponents, which is generated by adding a linear controller to the second equation of a 4D system that is obtained by coupling of a 1D linear system and a 3D modified generalized Lorenz system. This hyperchaotic system has very simple algebraic structure but can exhibit complex dynamical behaviors. Of particular interest are the observations that the hyperchaotic system has a hyperchaotic attractor with three positive Lyapunov exponents under a unique equilibrium, three or infinite equilibria, and there are three types of coexisting attractors of this new 5D hyperchaotic system. Numerical analysis of phase trajectories, Lyapunov exponents, bifurcation, Poincaré projections and power spectrum verifies the existence of the hyperchaotic and chaotic attractors. Moreover, stability of hyperbolic or non-hyperbolic equilibria and two complete mathematical characterization for 5D Hopf bifurcation are rigorously studied. Finally, some electronic circuits are designed to implement the 5D hyperchaotic system.  相似文献   

14.
A novel memristive chaotic circuit is proposed by replacing the Chua’s diode in modified Chua’s circuit with a smooth active memristor, and the corresponding memristive model is analyzed and validated. The equilibrium point set, dissipativity and stability of this new chaotic circuit are developed theoretically. The dynamic characteristics for the new system are presented by means of phase diagrams, Lyapunov exponents, bifurcation diagrams and Poincaré maps. The coexistence of the memristive system is carried out from the perspective of asymmetric coexistence and symmetry coexistence. In addition, the coexistence of multiple states is studied by a more direct method with initial value as the system variable to gain a more intuitive observation. The circuit model of the memristive chaotic system is designed through Multisim simulation software. Finally, the memristive chaotic sequence is used to encrypt the image, and the influence of multistability on the encryption is investigated by the histogram, correlation and key sensitivity. The results show that the proposed new memristive chaotic system has high security.  相似文献   

15.
This paper presents a three-dimensional autonomous Lorenz-like system formed by only five terms with a butterfly chaotic attractor. The dynamics of this new system is completely different from that in the Lorenz system family. This new chaotic system can display different dynamic behaviors such as periodic orbits, intermittency and chaos, which are numerically verified through investigating phase trajectories, Lyapunov exponents, bifurcation diagrams and Poincaré sections. Furthermore, this new system with compound structures is also proved by the presence of Hopf bifurcation at the equilibria and the crisis-induced intermittency.  相似文献   

16.
Nonlinear dynamical behaviors of an axially accelerating viscoelastic sandwich beam subjected to three-to-one internal resonance and parametric excitations resulting from simultaneous velocity and tension fluctuations are investigated. The direct method of multiple scales is adopted to obtain a set of first-order ordinary differential equations and associated boundary conditions. The frequency and amplitude response curves along with their stability and bifurcation are numerically studied. A great number of dynamic behaviors are presented in the form of phase portraits, time traces, Poincaré sections, and FFT power spectra. Due to modal interaction, various periodic, quasiperiodic, and chaotic behaviors are displayed, depending on the initial conditions. The largest Lyapunov exponent is carried out to determine the midly chaotic response by the convergent form of exponents. Numerical results show various oscillatory behaviors indicating the influence of internal resonance and coupled effects of fluctuating axial velocity and tension.  相似文献   

17.
Chaos exists in biological systems. Through investigating synthetic genetic relaxation oscillators coupled by quorum sensing, this paper reports a chaotic system. The detailed dynamical behaviors of this chaotic biological system are investigated, including Lyapunov exponents spectrum, bifurcation, and Poincaré mapping.  相似文献   

18.
The chaos of nonholonomic systems with two external nonlinear nonholonomic constraints where the magnitude of velocity is a constant and the magnitude of the velocity is a constant with a periodic disturbance, respectively, is completely identified for the first time. The scope of the chaos study is extended to nonlinear nonholonomic systems. By applying the nonlinear nonholonomic form of Lagrange’s equations, the dynamic equation is expressed. The existence of chaos in these two nonlinear nonholonomic systems is first wholly proved by all numerical criteria of chaos, i.e., the most reliable Lyapunov exponents, phase portraits, Poincaré maps, and bifurcation diagrams. Furthermore, it is found that the Feigenbaum number still holds for nonlinear nonholonomic systems.  相似文献   

19.
Accuracy of numerical modeling of any discontinuous dynamical system plays an important role in the proper use of the tools applied for its analysis. No less important in this matter is the numerical estimation of the phase trajectories, bifurcation diagrams, and Lyapunov exponents. This paper meets these expectations presenting application of Hénon’s method to obtain good numerical estimations of the stick–slip transitions existing in the Filippov-type discontinuous dynamical systems with dry friction. Subsequent sections are focused on the problem definition, block diagrams of the numerical procedure coded in Python, application of the method that was originally proposed for Poincaré maps, its use in estimation of phase trajectories, bifurcation diagrams of tangent points, and on estimation of the Lyapunov exponents for a selected two-dimensional system with stick–slip effect.  相似文献   

20.
This letter proposes a new 3D quadratic autonomous chaotic system which displays an extremely complicated dynamical behavior over a large range of parameters. The new chaotic system has five real equilibrium points. Interestingly, this system can generate one-wing, two-wing, three-wing and four-wing chaotic attractors and periodic motion with variation of only one parameter. Besides, this new system can generate two coexisting one-wing and two coexisting two-wing attractors with different initial conditions. Furthermore, the transient chaos phenomenon happens in the system. Some basic dynamical behaviors of the proposed chaotic system are studied. Furthermore, the bifurcation diagram, Lyapunov exponents and Poincaré mapping are investigated. Numerical simulations are carried out in order to demonstrate the obtained analytical results. The interesting findings clearly show that this is a special strange new chaotic system, which deserves further detailed investigation.  相似文献   

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