A three-scroll chaotic attractor

This Letter introduces a new chaotic member to the three-dimensional smooth autonomous quadratic system family, which derived from the classical Lorenz system but exhibits a three-scroll chaotic attractor. Interestingly, the two other scrolls are symmetry related with respect to the z-axis as for the Lorenz attractor, but the third scroll of this three-scroll chaotic attractor is around the z-axis. Some basic dynamical properties, such as Lyapunov exponents, fractal dimension, Poincaré map and chaotic dynamical behaviors of the new chaotic system are investigated, either numerically or analytically. The obtained results clearly show this is a new chaotic system and deserves further detailed investigation.     

A novel 3D autonomous system with different multilayer chaotic attractors

This Letter proposes a novel three-dimensional autonomous system which has complex chaotic dynamics behaviors and gives analysis of novel system. More importantly, the novel system can generate three-layer chaotic attractor, four-layer chaotic attractor, five-layer chaotic attractor, multilayer chaotic attractor by choosing different parameters and initial condition. We analyze the new system by means of phase portraits, Lyapunov exponent spectrum, fractional dimension, bifurcation diagram and Poincaré maps of the system. The three-dimensional autonomous system is totally different from the well-known systems in previous work. The new multilayer chaotic attractors are also worth causing attention.     

The stability control of fractional order unified chaotic system with sliding mode control theory

This paper studies the stability of the fractional order unified chaotic system with sliding mode control theory. The sliding manifold is constructed by the definition of fractional order derivative and integral for the fractional order unified chaotic system. By the existing proof of sliding manifold, the sliding mode controller is designed. To improve the convergence rate, the equivalent controller includes two parts: the continuous part and switching part. With Gronwall’s inequality and the boundness of chaotic attractor, the finite stabilization of the fractional order unified chaotic system is proved, and the controlling parameters can be obtained. Simulation results are made to verify the effectiveness of this method.     

Chaotic dynamics of the fractional Lorenz system

In this Letter we introduce a generalization of the Lorenz dynamical system using fractional derivatives. Thus, the system can have an effective noninteger dimension Sigma defined as a sum of the orders of all involved derivatives. We found that the system with Sigma<3 can exhibit chaotic behavior. A striking finding is that there is a critical value of the effective dimension Sigma(cr), under which the system undergoes a transition from chaotic dynamics to regular one.     

Two dimensional fractional projectile motion in a resisting medium

In this paper we propose a fractional differential equation describing the behavior of a two dimensional projectile in a resisting medium. In order to maintain the dimensionality of the physical quantities in the system, an auxiliary parameter k was introduced in the derivative operator. This parameter has a dimension of inverse of seconds (sec)^?1 and characterizes the existence of fractional time components in the given system. It will be shown that the trajectories of the projectile at different values of γ and different fixed values of velocity v ₀ and angle θ, in the fractional approach, are always less than the classical one, unlike the results obtained in other studies. All the results obtained in the ordinary case may be obtained from the fractional case when γ = 1.     

A semi-dissipative crisis

This article reports a sudden chaotic attractor change in a system described by a conservative and dissipative map concatenation. When the driving parameter passes a critical value, the chaotic attractor suddenly loses stability and turns into a transient chaotic web. The iterations spend super-long random jumps in the web, finally falling into several special escaping holes. Once in the holes, they are attracted monotonically to several periodic points. Following Grebogi, Ott, and Yorke, we address such a chaotic attractor change as a crisis. We numerically demonstrate that phase space areas occupied by the web and its complementary set (a fat fractal forbidden net) become the periodic points' “riddled-like” attraction basins. The basin areas are dominated by weaker dissipation called “quasi-dissipation”. Small areas, serving as special escape holes, are dominated by classical dissipation and bound by the forbidden region, but only in each periodic point's vicinity. Thus the crisis shows an escape from a riddled-like attraction basin. This feature influences the approximation of the scaling behavior of the crisis's averaged lifetime, which is analytically and numerically determined as 〈τ〉∝(b-b₀)^γ, where b₀ denotes the control parameter's critical threshold, and γ≃-1.5.     

TS fuzzy realization of chaotic Lü system

The Lü attractor is a new chaotic attractor, which connects the Lorenz attractor and the Chen attractor and represents the transition from one to the other. The Letter presents a hybrid TS fuzzy modeling approach for the newly coined chaotic Lü system. Then the abundant and fundamental dynamical behaviors of the chaotic Lü system are completely and comprehensive investigated based on this novel hybrid TS fuzzy model.     

Parametric characterisation of a chaotic attractor using the two scale Cantor measure

A chaotic attractor is usually characterised by its multifractal spectrum which gives a geometric measure of its complexity. Here we present a characterisation using a minimal set of independent parameters which is uniquely determined by the underlying process that generates the attractor. The method maps the f(α) spectrum of a chaotic attractor on to that of a general two scale Cantor measure. We show that the mapping can be done in practice with reasonable accuracy for many of the standard chaotic attractors. In order to implement this procedure, we also propose a generalisation of the standard equations for the two scale Cantor set in one dimension to that in higher dimensions. Another interesting result we have obtained both theoretically and numerically is that, the f(α) characterisation gives information only up to two scales, even when the underlying process generating the multifractal involves more than two scales.     

Chaotic and pseudochaotic attractors of perturbed fractional oscillator

We consider a nonlinear oscillator of the Duffing type with fractional derivative of the order 1相似文献   

The proto Bhalekar–Gejji system

Recently construction of new chaotic attractors for various design demands has drawn much attention. This paper provides a complete construction of a new chaotic attractor, called proto Bhalekar–Gejji (B–G) system. This proto B–G system is a quotient of the B–G system. The covers L_n of the proto B–G system are constructed that lead to n-eared strange attractor. The design of Hamiltonian energy function of proto B–G system concludes that the energy is decreased as the multi-wing number increased. Moreover, complex dynamics of the proto B–G system are discussed in detail for a specific set of parameters.     

Projective synchronization of fractional order chaotic system based on linear separation

This Letter analyses the dynamical behavior of fractional order unified system, based on the stability criterion of linear systems, a new approach for constructing projective synchronization of fractional order unified system is proposed. Numerical simulations of fractional order Chen system, fractional order Lü system and fractional order Lorenz-like system are achieved via the linear separation method.     

Dynamical Analysis of a Fractional Order Rosenzweig-MacArthur Model Incorporating Herbivore Induced Plant Volatile

Plants send signals through releasing Volatile Organic Compounds (VOCs), to attract beneficial natural carnivorous insects as reinforcement against harmful herbivorous insects which are responsible for hampering the growth of plants. In this work, we explore the dynamical behavior of a volatile mediated plant-herbivore-carnivore system with fractional order differential equations. Basic results on the existence, uniqueness, non-negativity and boundedness of the solutions, local and global stability of coexistence equilibrium points and limit cycles emerging through Hopf bifurcation are investigated. Stability behavior around coexistence equilibrium point changes with varying fractional order (α). Also the existence of Hopf bifurcation is established by considering the fractional order α as a bifurcation parameter. Moreover, the attraction factor of plant volatile to carnivore and predation rate for plant-herbivore are responsible for changing the system dynamics. Numerical simulations using matlab software are performed to support the analytical findings.     

基于参数切换算法的混沌系统吸引子近似及其电路设计

基于参数切换算法和离散混沌系统, 设计一种新的混沌系统参数切换算法, 给出了两算法的原理. 采用混沌吸引子相图观测法, 研究了不同算法下统一混沌系统和Rössler混沌系统参数切换结果, 最后引入方波发生器, 设计了Rössler混沌系统参数切换电路. 结果表明, 采用参数切换算法可以近似出指定参数下的系统, 其吸引子与该参数下吸引子一致; 基于离散系统的参数切换结果更为复杂, 当离散序列分布均匀时, 只可近似得到指定参数下的系统; 相比传统切换混沌电路, 参数切换电路不用修改原有系统电路结构, 设计更为简单, 输出结果受方波频率影响, 通过加入合适频率的方波发生器, 数值仿真与电路仿真结果一致.     

Stabilization and Synchronization of a Complex Hidden Attractor Chaotic System by Backstepping Technique

In this paper, the stabilization and synchronization of a complex hidden chaotic attractor is shown. This article begins with the dynamic analysis of a complex Lorenz chaotic system considering the vector field properties of the analyzed system in the Cn domain. Then, considering first the original domain of attraction of the complex Lorenz chaotic system in the equilibrium point, by using the required set topology of this domain of attraction, one hidden chaotic attractor is found by finding the intersection of two sets in which two of the parameters, r and b, can be varied in order to find hidden chaotic attractors. Then, a backstepping controller is derived by selecting extra state variables and establishing the required Lyapunov functionals in a recursive methodology. For the control synchronization law, a similar procedure is implemented, but this time, taking into consideration the error variable which comprise the difference of the response system and drive system, to synchronize the response system with the original drive system which is the original complex Lorenz system.     

Crossing bifurcations and unstable dimension variability

A crisis is a global bifurcation in which a chaotic attractor has a discontinuous change in size or suddenly disappears as a scalar parameter of the system is varied. In this Letter, we describe a global bifurcation in three dimensions which can result in a crisis. This bifurcation does not involve a tangency and cannot occur in maps of dimension smaller than 3. We present evidence of unstable dimension variability as a result of the crisis. We then derive a new scaling law describing the density of the new portion of the attractor formed in the crisis. We illustrate this new type of bifurcation with a specific example of a three-dimensional chaotic attractor undergoing a crisis.     

Chaotic dynamics of the fractional-order Lü system and its synchronization

In this Letter we numerically investigate the chaotic behaviors of the fractional-order Lü system. A striking finding is that the lowest order for this system to have chaos is 0.3, which is the lowest-order chaotic system among all the found chaotic systems reported in the literature to date. Period-doubling routes to chaos in the fractional-order Lü system are also found. Master–slave synchronization of chaotic fractional-order Lü systems with linear coupling is also studied.     

The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems

In this paper a time fractional Fourier law is obtained from fractional calculus. According to the fractional Fourier law, a fractional heat conduction equation with a time fractional derivative in the general orthogonal curvilinear coordinate system is built. The fractional heat conduction equations in other orthogonal coordinate systems are readily obtainable as special cases. In addition, we obtain the solution of the fractional heat conduction equation in the cylindrical coordinate system in terms of the generalized H-function using integral transformation methods. The fractional heat conduction equation in the case 0<α≤1 interpolates the standard heat conduction equation (α=1) and the Localized heat conduction equation (α→0). Finally, numerical results are presented graphically for various values of order of fractional derivative.     

Coexistence of attractors in autonomous Van der Pol–Duffing jerk oscillator: Analysis,chaos control and synchronisation in its fractional-order form

Dynamics at infinity and a Hopf bifurcation for a Sprott E system with a very small perturbation constant are studied in this paper. By using Poincaré compactification of polynomial vector fields in \(R^3\), the dynamics near infinity of the singularities is obtained. Furthermore, in accordance with the centre manifold theorem, the subcritical Hopf bifurcation is analysed and obtained. Numerical simulations demonstrate the correctness of the dynamical and bifurcation analyses. Moreover, by choosing appropriate parameters, this perturbed system can exhibit chaotic, quasiperiodic and periodic dynamics, as well as some coexisting attractors, such as a chaotic attractor coexisting with a periodic attractor for \(a>0\), and a chaotic attractor coexisting with a quasiperiodic attractor for \(a=0\). Coexisting attractors are not associated with an unstable equilibrium and thus often go undiscovered because they may occur in a small region of parameter space, with a small basin of attraction in the space of initial conditions.     

A special type of attractor and transitions in a delayed system

We discuss strange nonchaotic attractors (SNAs) in addition to chaotic and regular attractors in a quasiperiodically driven system with time delays. A route and the associated mechanism are described for a special type of attractor called strange-nonchaotic-attractor-like (SNA-like) through T² torus bifurcation. The type of attractor can be observed in large parameter domains and it is easily mistaken for a true SNA judging merely from the phase portrait, power spectrum and the largest Lyapunov exponent. SNA-like attractor is not strange and has no phase sensitivity. Conditions for Neimark-Sacker bifurcation are obtained by theoretical analysis for the unforced system. Complicated and interesting dynamical transitions are investigated among the different tongues.     

A novel hyperchaotic system and its circuit implementation

A four-dimensional hyperchaotic system with five parameters is proposed. Its dynamical properties such as dissipativity, equilibrium points, Lyapunov exponent, Lyapunov dimension, bifurcation diagrams and Poincare maps are analyzed theoretically and numerically. Theoretical analyses and simulation tests indicate that the new system's dynamics behavior can be periodic attractor, chaotic attractor and hyperchaotic attractor as the parameter varies. Finally, the circuit of this new hyperchaotic system is designed and realized by Multisim software. The simulation results confirm that the chaotic system is different from the existing chaotic systems and is a novel hyperchaotic system. The system is recommendable for many engineering applications such as information processing, cryptology, secure communications, etc.