Search

This paper proposes a new four-dimensional hyper-chaotic system capable of generating multi-wing chaotic attractors by introducing active magnetron memristors, multi-segmented square functions and trigonometric functions. The dynamical properties of this new hyper-chaotic system, such as equilibrium point, dissipation, Lyapunov exponential spectrum, bifurcation diagram and Poincaré cross-section and attraction basin, are analyzed theoretically and simulated numerically, and the complexity of this system with different parameters is analyzed. It is observed that this hyper-chaotic system has periodic, chaotic and hyper-chaotic variations with an infinite number of equilibria and coexisting attractors under different parameter conditions. The circuit simulation was performed using Multisim and the results obtained were consistent with the numerical analysis of the dynamics, and the chaotic circuit system is designed by FPGA to verify the realizability of the system. Finally, an image encryption algorithm is designed in conjunction with the DNA algorithm to enable a new system chaotic sequence for image encryption. The results show that the hyper-chaotic system has rich dynamical behavior and has high-security performance when applied to image encryption with strong chaotic key and plaintext sensitivity and large key space in image encryption. [ABSTRACT FROM AUTHOR]

Memristor, as a typical nonlinear element, is able to produce chaotic signals in chaotic systems easily. Chaotic systems have potential applications in secure communications, information encryption, and other fields. Therefore, it is of importance to generate abundant dynamic behaviors in a single chaotic system. In this paper, a novel memristor-based chaotic system without equilibrium points is proposed. One of the essential features is the absence of symmetry in this system, which increases the complexity of the new system. Then, the nonlinear dynamic behaviors of the system are analyzed in terms of chaos diagrams, bifurcation diagrams, Poincaré maps, Lyapunov exponent spectra, the sum of Lyapunov exponents, phase portraits, 0–1 test, recurrence analysis and instantaneous phase. The results of the sum of Lyapunov exponents show that the given system is a quasi-Hamiltonian system with certain initial conditions (IC) and parameters. Next, other critical phenomena, such as hidden multi-scroll attractors, abundant coexistence characteristics, are found characterized through basins of attraction and others. Especially, it reveals some rare phenomena in other systems that multiple hidden hyperchaotic attractors coexist. Finally, the circuit implementation based on Micro Control Unit (MCU) confirms theoretical analysis and the numerical simulation. [ABSTRACT FROM AUTHOR]

This paper presents the Chen system as a controlled weather model. Mathematically, the Chen system is dual to the Lorenz system via time reversal. Physically, the Chen system can be viewed as a controlled weather model from the anti-control perspective. This paper illustrates the physical principle of this controlled weather model, and develops an engineering design of the model for real indoor climate (temperature-humidity) regulation, with a perspective on outdoor weather control application. [ABSTRACT FROM AUTHOR]

An epidemic model for pest management with impulsive control over a patchy environment is proposed in this paper. We investigate the dynamical behaviors on extinction and permanence and obtain the threshold value R 0 of dynamics for the impulsive system by utilizing a small amplitude perturbation method, matrix spectral analysis and persistence theory. We prove that the periodic pest-eradication solution of the system is globally asymptotically stable if R 0 < 1 , while the system is persistent if R 0 > 1. Furthermore, by discussion on the two-patch case, we analyze the effects of the dispersal and impulsive control on dynamical behaviors of the system. Some numerical examples are given to illustrate the effectiveness of the obtained results and to demonstrate the complexity such as chaotic characteristic of the system. [ABSTRACT FROM AUTHOR]

The paper discusses the fractional Rössler system and the dependence of its dynamics on some entry parameters. An explicit algorithm for a priori determination of fractional Hopf bifurcations is derived and scenarios documenting a route of the system from stability to chaos are performed with respect to a varying system’s fractional order as well as to a varying system’s coefficient. Contrary to the existing results, the searched values of the fractional Hopf bifurcations follow directly from a revealed analytical dependence between these two systems’ entries. Their various critical values are established and confirmed by numerical experiments demonstrating not only the loss of stability of an equilibrium point, but also other phenomena of transition to chaotic behavior. In addition, we suggest an active control method for synchronization of two chaotic fractional-order Rössler systems. Our theoretical analysis enables to synchronize them for any value of a free parameter under which the master system displays a chaotic behavior. [ABSTRACT FROM AUTHOR]

In this paper, we present a mechanism of generation of a class of switched dynamical system without equilibrium points that generates a chaotic attractor. The switched dynamical systems are based on piecewise linear (PWL) systems. The theoretical results are formally given through a theorem and corollary which give necessary and sufficient conditions to guarantee that a linear affine dynamical system has no equilibria. Numerical results are in accordance with the theory. [ABSTRACT FROM AUTHOR]

Lyapunov exponent is an important index for describing chaotic systems behavior, and the largest Lyapunov exponent can be used to determine whether a system is chaotic or not. For discrete-time dynamical systems, the Lyapunov exponents are calculated by an eigenvalue method. In theory, according to eigenvalue method, the more accurate calculations of Lyapunov exponent can be obtained with the increment of iterations, and the limits also exist. However, due to the finite precision of computer and other reasons, the results will be numeric overflow, unrecognized, or inaccurate, which can be stated as follows: (1) The iterations cannot be too large, otherwise, the simulation result will appear as an error message of NaN or Inf; (2) If the error message of NaN or Inf does not appear, then with the increment of iterations, all Lyapunov exponents will get close to the largest Lyapunov exponent, which leads to inaccurate calculation results; (3) From the viewpoint of numerical calculation, obviously, if the iterations are too small, then the results are also inaccurate. Based on the analysis of Lyapunov-exponent calculation in discrete-time systems, this paper investigates two improved algorithms via QR orthogonal decomposition and SVD orthogonal decomposition approaches so as to solve the above-mentioned problems. Finally, some examples are given to illustrate the feasibility and effectiveness of the improved algorithms. [ABSTRACT FROM AUTHOR]

Zero Average Dynamics (ZAD) strategy has been reported in the last decade as an alternative control technique for power converters, and a lot of work has been devoted to analyze it. From a theoretical point of view, this technique has the advantage that it guarantees fixed switching frequency, low output error and robustness, however, no high correspondence between numerical and experimental results has been obtained. These differences are basically due to model assumptions; in particular, all elements in the circuit were modeled as ideal elements and simulations and conclusions about steady state stability and transitions to chaos have been carried out with this ideal model. Regarding the practical point of view and the digital implementation, we include in this paper internal resistances, quantization effects and 1-period delay to the model. This paper shows in an experimental and numerical way the effects of these elements to the model and their incidence in the results. Now, experimental and numerical analyses fully agree. [ABSTRACT FROM AUTHOR]

In this paper, the analytical solutions for period-m flows and chaos in nonlinear dynamical systems are presented through the generalized harmonic balance method. The nonlinear damping, periodically forced, Duffing oscillator was investigated as an example to demonstrate the analytical solutions of periodic motions and chaos. Through this investigation, the mechanism for a period-m motion jumping to another period-n motion in numerical computation is found. In this problem, the Hopf bifurcation of periodic motions is equivalent to the period-doubling bifurcation via Poincare mappings of dynamical systems. The stable and unstable period-m motions can be obtained analytically. Even more, the stable and unstable chaotic motions can be achieved analytically. The methodology presented in this paper can be applied to other nonlinear vibration systems, which is independent of small parameters. [ABSTRACT FROM AUTHOR]

In systems theory and science, emergence is the way complex systems and patterns arise out of a multiplicity of relatively simple interactions. Emergence is central to the theories of integrative levels and of complex systems [Aziz-Alaoui & Bertelle, 2009]. In this paper, we use the emergent property of the ultra weak multidimensional coupling of p 1-dimensional dynamical chaotic systems which leads from chaos to randomness. Generation of random or pseudorandom numbers, nowadays, is a key feature of industrial mathematics. Pseudorandom or chaotic numbers are used in many areas of contemporary technology such as modern communication systems and engineering applications. More and more European or US patents using discrete mappings for this purpose are obtained by researchers of discrete dynamical systems [Petersen & Sorensen, 2007; Ruggiero et al., 2006]. Efficient Chaotic Pseudo Random Number Generators (CPRNG) have been recently introduced. They use the ultra weak multidimensional coupling of p 1-dimensional dynamical systems which preserve the chaotic properties of the continuous models in numerical experiments. Together with chaotic sampling and mixing processes, ultra weak coupling leads to families of (CPRNG) which are noteworthy [Hénaff et al., 2009a, 2009b, 2009c, 2010]. In this paper we improve again these families using a double threshold chaotic sampling instead of a single one. We analyze numerically the properties of these new families and underline their very high qualities and usefulness as CPRNG when very long series are computed. Moreover, a determining property of such improved CPRNG is the high number of parameters used and the high sensitivity to the parameters value which allows choosing it as cipher-keys. It is why we call these families multiparameter chaotic pseudo-random number generators (M-p CPRNG). [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behaviors of the Tinkerbell map are investigated in detail. Conditions for the existence of fold bifurcation, flip bifurcation and Hopf bifurcation are derived, and chaos in the sense of Marotto is verified by both analytical and numerical methods. Numerical simulations include bifurcation diagrams in two- and three-dimensional spaces, phase portraits, and the maximum Lyapunov exponent and fractal dimension, as well as the distribution of dynamics in the parameter plane, which exhibit new and interesting dynamical behaviors. More specifically, this paper reports the findings of chaos in the sense of Marotto, a route from an invariant circle to transient chaos with a great abundance of periodic windows, including period-2, 7, 8, 9, 10, 13, 17, 19, 23, 26 and so on, and suddenly appearing or disappearing chaos, convergence of an invariant circle to a period-one orbit, symmetry-breaking of periodic orbits, interlocking period-doubling bifurcations in chaotic regions, interior crisis, chaotic attractors, coexisting (2, 10, 13) chaotic sets, two coexisting invariant circles, two attracting chaotic sets coexisting with a non-attracting chaotic set, and so on, all in the Tinkerbell map. In particular, it is found that there is no obvious road from period-doubling bifurcations to chaos, but there is a route from a period-one orbit to an invariant circle and then to transient chaos as the parameters are varied. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the Tinkerbell map is obtained. [ABSTRACT FROM AUTHOR]

This paper studies chaos synchronization via unidirectional coupling. The self-synchronization of Lorenz systems, modified coupled dynamos systems and hyperchaotic Chen systems is studied by three methods: the Lyapunov function method, the global synchronization method and the numerical calculation of the largest Lyapunov exponent method. In regard to application to communication, we show that via transmitting single signal the synchronization of the drive system and the response system can be achieved. An example of applying self-synchronization of hyperchaotic Chen systems to chaotic masking secure communication is presented in this paper. Simulation results show the effectiveness of the method. [ABSTRACT FROM AUTHOR]

Illuminative deformation patterns of a honeycomb structure are presented. A representative volume element of a honeycomb structure consisting of 2 × 2 hexagonal cells is modeled to be a ${\rm D}_6\dot{+}({\rm C}_2\times\tilde{\rm C}_2)$-equivariant system. The bifurcation mechanism and an exhaustive list of possible bifurcated patterns are obtained by group-theoretic bifurcation theory. A flower mode of the honeycomb is shown to have the same symmetry as the so-called anti-hexagon in the Rayleigh–Bénard convection. A numerical bifurcation analysis is conducted on an elastic in-plane honeycomb structure consisting of 2×2 cells to produce beautiful wallpapers of bifurcating deformation patterns and, in turn, to highlight the achievement of the paper. New deformation patterns of a honeycomb structure have been found and classified in a systematic manner. Knowledge of the symmetries of the bifurcating solutions has turned out to be vital in the successful numerical tracing of the bifurcated paths. This paper paves the way for the introduction of the results hitherto obtained for flow patterns in fluid dynamics into the study of patterns on materials. [ABSTRACT FROM AUTHOR]

In this paper, we carefully reexamine the chaotic RF model, first studied by Rabinovich and Fabrikant [1979], and have found many new and rich complex dynamics of the model that were mostly not reported before. The chaotic RF model has proved to be a great challenge to classical numerical methods, in the sense that most classical numerical methods have not been very successful in the study of complex dynamics of this special RF model. Therefore, in this paper, we present and apply a special numerical method, the Local Iterative Linearization (LIL) method, along with a special Turbo Pascal code based on this accurate LIL algorithm, for a careful numerical study of this complex RF model. Many interesting new findings are summarized and reported in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system governing the motion of two point vortices in a flow excited by an external acoustic forcing. It is known that the system of two vortices is integrable in the absence of acoustic forcing. However, the addition of the acoustic forcing makes the system much more complex, and the system becomes nonintegrable and loses the phase volume preservation property. The objective of our research is to study chaotic dynamics and typical bifurcations. Numerical analysis has shown that the reversible pitchfork bifurcation is typical. Also, we show that the existence of strange attractors is not characteristic for the system under consideration. [ABSTRACT FROM AUTHOR]

This paper describes a new reliable numerical method for computing chaotic solutions of dynamical systems and, in special cases, is applied to Chen strange attractor. The numerical precision of the computation is finely mastered. We introduce a modification of the method of power series for the construction of approximate solutions of the Chen system together with forward/backward control of the precision. As a test for the method, we obtained the region of convergence of series and researched the behavior of the trajectories on this attractor. The results of a numerical experiment are presented. [ABSTRACT FROM AUTHOR]

In this paper, we analyze chaotic dynamics for two-dimensional nonautonomous maps through the use of a nonautonomous version of the Conley-Moser conditions given previously. With this approach we are able to give a precise definition of what is meant by a chaotic invariant set for nonautonomous maps. We extend the nonautonomous Conley-Moser conditions by deriving a new sufficient condition for the nonautonomous chaotic invariant set to be hyperbolic. We consider the specific example of a nonautonomous Hénon map and give sufficient conditions, in terms of the parameters defining the map, for the nonautonomous Hénon map to have a hyperbolic chaotic invariant set. [ABSTRACT FROM AUTHOR]

In previous papers [Isagi et al., 1997; Satake & Iwasa, 2000], a forest model was proposed. The authors demonstrated numerically that the mature forest could possibly exhibit annual reproduction (fixed point synchronization), periodic and chaotic synchronization as the energy depletion constant is gradually increased. To understand such rich synchronization phenomena, we are led to study global dynamics of a piecewise smooth map containing two parameters and . Here is the energy depletion quantity and is the coupling strength. In particular, we obtain the following results. First, we prove that has a chaotic dynamic in the sense of Devaney on an invariant set whenever , which improves a result of [Chang & Chen, 2011]. Second, we prove, via the Schwarzian derivative and a generalized result of [Singer, 1978], that exhibits the period adding bifurcation. Specifically, we show that for any , has a unique global attracting fixed point whenever () and that for any , has a unique attracting period point whenever is less than and near any positive integer . Furthermore, the corresponding period point instantly becomes unstable as moves pass the integer . Finally, we demonstrate numerically that there are chaotic dynamics whenever is in between and away from two consecutive positive integers. We also observe the route to chaos as increases from one positive integer to the next through finite period doubling. [ABSTRACT FROM AUTHOR]

In this paper, a general autonomous van der Pol-Duffing oscillator is studied. Several issues, such as periodic bifurcations and the dynamical structures of the system are investigated either analytically or numerically. Especially, a phenomenon of hidden attractors is noticed and an algorithm for the location of hidden attractors is given. The obtained results show that hidden attractors exist around chaotic attractors. [ABSTRACT FROM AUTHOR]

In this paper, on the basis of the Laypunov stability and the cascade synchronization approach, an effectual scroll controller and improving cascade synchronization approach are derived. The chaotic attractor can be displayed to be lopsided, till to be 1-scroll and periodic via the different scroll controller. And the proposed synchronization technique is applied to the new system, which the synchronization controller obtained by improving cascade method and adaptive functions. Numerical results are used to verify the effectiveness of the scheme. [ABSTRACT FROM AUTHOR]

In this paper, chaos and adaptive synchronization for a fractional-order Genesio-Tesi system with fifth order nonlinearity are investigated. The minimum effective dimension for the system to remain chaotic is 2.784 in the commensurate-order case and 2.793 in the incommensurate-order case. A period-doubling bifurcation and an interior crisis from single-scroll to double-scroll attractors are also found with the variation of derivative order. Furthermore, based on the stability theory of fractional-order systems, the adaptive synchronization of the system with unknown parameters is realized by designing appropriated controllers. Numerical simulations are carried out to demonstrate the effectiveness and flexibility of the controllers. [ABSTRACT FROM AUTHOR]

In this paper, subharmonic and chaotic behavior in current controlled DC drive has been investigated. The effects of variation of some chosen parameters on the qualitative behavior of the system have been studied. To avoid occurrence of chaos we present a method for controlling chaos applicable to DC drives. The results of numerical investigation are presented. [ABSTRACT FROM AUTHOR]

This paper studies the problem of chaos synchronization between two different hyperchaotic systems with uncertain parameters. Based on the Lyapunov stability theory, we obtain the sufficient condition of synchronization between two different hyperchaotic systems with uncertain parameters. A new adaptive controller with parameter update laws is designed to synchronize these chaotic systems. We proved it in theory with an uncertain hyperchaotic Lorenz system and an uncertain hyperchaotic Rössler system. Numerical results verified the validation of the proposed scheme. [ABSTRACT FROM AUTHOR]

A new approach to generate chaotic phenomenon, called chaos entanglement, is proposed in this paper. The basic principle is to entangle two or multiple stable linear subsystems by entanglement functions to form an artificial chaotic system such that each of them evolves in a chaotic manner. Firstly, a new attractor, entangling a two-dimensional linear subsystem and a one-dimensional one by sine function, is presented as an example. Dynamical analysis shows that both entangled subsystems are bounded and all equilibra are unstable saddle points when chaos entanglement is achieved. Also, numerical computation shows that this system has one positive Lyapunov exponent, which implies chaos. Furthermore, two conditions are given to achieve chaos entanglement. Along this way, by different linear subsystems and different entanglement functions, a variety of novel chaotic attractors have been created and abundant complex dynamics are exhibited. Our discovery indicates that it is not difficult any more to construct new artificial chaotic systems/networks for engineering applications such as chaos-based secure communication. Finally, a possible circuit is given to realize these new chaotic attractors. [ABSTRACT FROM AUTHOR]

In this paper, asymmetric periodic motions in a periodically forced, softening Duffing oscillator are presented analytically through the generalized harmonic balance method. For the softening Duffing oscillator, the symmetric periodic motions with jumping phenomena were understood very well. However, asymmetric periodic motions in the softening Duffing oscillators are not investigated analytically yet, and such asymmetric periodic motions possess much richer dynamics than the symmetric motions in the softening Duffing oscillator. For asymmetric motions, the bifurcation tree from asymmetric period-1 motions to chaos is discussed comprehensively. The corresponding, unstable and stable, asymmetric and symmetric, periodic motions in the softening Duffing oscillator are presented, and numerical illustrations of stable and unstable periodic motions are completed. This investigation provides a better picture of complex motion in the softening Duffing oscillator. [ABSTRACT FROM AUTHOR]

This paper studied the dynamic behavior of the fractional order hyper-chaotic Lorenz system and the fractional order hyper-chaotic Rössler system, then numerical analysis of the different fractional orders hyper-chaotic systems are carried out under the predictor-corrector method. We proved the two systems are in hyper-chaos when the maximum and the second largest Lyapunov exponential are calculated. Also the smallest orders of the systems are proved when they are in hyper-chaos. The diverse structure synchronization of the fractional order hyper-chaotic Lorenz system and the fractional order hyper-chaotic Rössler system is realized using active control method. Numerical simulations indicated that the scheme was always effective and efficient. [ABSTRACT FROM AUTHOR]

This paper investigates the global synchronization problem of master-slave coupled multi-scroll saturated chaotic systems via single-state linear feedback control. An algebraic criterion for global synchronization is rigorously proven by means of the absolute stability theory. The obtained algebraic criterion is then optimized thereby deriving a less-conservative result, which is finally verified by numerical examples. [ABSTRACT FROM AUTHOR]

Chaos generation is an interesting research topic in the study of coupled complex dynamical networks. In this paper, based on mathematical analysis of Lyapunov exponent and boundedness of networks, the emergence of chaos for a class of nonlinear complex networks is investigated and some new criteria of chaos generation are derived. The effectiveness of theoretical results is verified by a numerical example. [ABSTRACT FROM AUTHOR]

This paper presents a simple scheme for synchronization of fractional-order chaotic systems. The scheme utilizes a recently developed LMI (Linear matrix inequality) stabilization theorem for fractional-order linear interval systems to design a linear controller. In contrast to existing schemes in the literature, the present scheme is straightforward and does not require that nonlinear parts of synchronization error dynamics are cancelled by the controller. The fractional-order Rössler, Lorenz, and hyperchaotic Chen systems are used as demonstrative examples. Numerical results illustrate the effectiveness of the present scheme. [ABSTRACT FROM AUTHOR]

In this paper, an image encryption algorithm based on couple multiple chaotic systems is presented. It made the one-dimensional Coupled Map Lattice (CML) formed by Skew Tent map as spatiotemporal chaotic system and made its output sequence as the initial value of logistic and meanwhile did iterative of specific times to get the final key sequence, and then did XOR operations with corresponding pixels to finish the encryption. Numerical analysis expresses that this algorithm has large enough space and high security. [ABSTRACT FROM AUTHOR]

Based on the stability theory of fractional order systems and the pole placement technique, this paper designs a synchronization scheme with the state observer method and achieves the projective synchronization of a class of fractional order chaotic systems. Taking an example for the fractional order unified system by using this observer controller, and numerical simulations of fractional order Lorenz-like system, fractional order Lü system and fractional order Chen system are provided to demonstrate the effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we present a simple and accessible way to enhance the stable behaviors of a chaotic dynamical system which models a cancer growth, as presented in [Itik & Banks, 2010]. The algorithm presented in [Danca et al., 2012], approximates numerically any attractor of a system belonging to a defined class of dynamical systems, by alternating the control parameter in relatively short periods of time. When switching the control parameter within a set of values corresponding to some chaotic behaviors, the result may be a stable evolution or, reversely a chaotic behavior may be obtained by switching the parameter within a set of values corresponding to stable evolutions. This apparently surprising phenomenon is, in fact, a generalization of the known Parrondo's paradox. [ABSTRACT FROM AUTHOR]

By utilizing the fractional calculus techniques and spatiotemporal chaos theory, this paper brings Lorenz system to fractional-order spatiotemporal coupled differential equation for the first time, and proposes the fractional-order spatiotemporal coupled Lorenz system. Based on that, we study the problem of chaotic synchronization of fractional-order spatiotemporal coupled Lorenz systems, design the linear controller and nonlinear controller by utilizing the Lyapunov stability theory and prove the correctness in theory. The numerical simulation results demonstrate the validity of controllers in high-dimension fractional-order spatiotemporal coupled Lorenz system. [ABSTRACT FROM AUTHOR]

As originally formulated, the Generalized Alignment Index (GALI) method of chaos detection has so far been applied to distinguish quasiperiodic from chaotic motion in conservative nonlinear dynamical systems. In this paper, we extend its realm of applicability by using it to investigate the local dynamics of periodic orbits. We show theoretically and verify numerically that for stable periodic orbits, the GALIs tend to zero following particular power laws for Hamiltonian flows, while they fluctuate around nonzero values for symplectic maps. By comparison, the GALIs of unstable periodic orbits tend exponentially to zero, both for flows and maps. We also apply the GALIs for investigating the dynamics in the neighborhood of periodic orbits, and show that for chaotic solutions influenced by the homoclinic tangle of unstable periodic orbits, the GALIs can exhibit a remarkable oscillatory behavior during which their amplitudes change by many orders of magnitude. Finally, we use the GALI method to elucidate further the connection between the dynamics of Hamiltonian flows and symplectic maps. In particular, we show that, using the components of deviation vectors orthogonal to the direction of motion for the computation of GALIs, the indices of stable periodic orbits behave for flows as they do for maps. [ABSTRACT FROM AUTHOR]

Based on the principle of chaotification for continuous-time autonomous systems, which relies on two basic properties of chaos, i.e. being globally bounded with necessary positive-zero-negative Lyapunov exponents, this paper derives a feasible and unified chaotification method for designing a general chaotic continuous-time autonomous nonlinear system. For a system consisting of a linear and a nonlinear subsystems, chaotification is achieved using separation of state variables, which decomposes the system into two open-loop subsystems interacting through mutual feedback resulting in an overall closed-loop nonlinear feedback system. Under the condition that the nonlinear feedback control output is uniformly bounded where the nonlinear function is of bounded-input/bounded-output, it is proved that the resulting system is chaotic in the sense of being globally bounded with a required placement of Lyapunov exponents. Several numerical examples are given to verify the effectiveness of the theoretical design. Since linear systems are special cases of nonlinear systems, the new method is also applicable to linear systems in general. [ABSTRACT FROM AUTHOR]

In this paper, we study the chaotic dynamics of a voltage-mode controlled buck converter, which is typically a switched piecewise linear system. For the two-dimensional hybrid system, we consider a properly chosen cross-section and the corresponding Poincaré map, and show that the dynamics of the system is semi-conjugate to a 2-shift map, which implies the chaotic behavior of this system. The essential tool is a topological horseshoe theory and numerical method. [ABSTRACT FROM AUTHOR]

In this paper, we have studied an important class of first order differential equations with nonlinear delayed feedback. The nonlocal dynamics of such equations was investigated by analytical and numeric-analytical methods. An approach based on asymptotic analysis made it possible to study attractors consisting of impulse type solutions. We obtained some numerical characteristics of irregular oscillations and studied their dependence on the delay. [ABSTRACT FROM AUTHOR]

In this paper, we perform a global analysis of the dynamics of the Chen system x = a(y -- x), y = (c -- a)x -- xz + cy, z = xy -- bz, where (x,y,z) ∊ R3 and (a,b,c) ∊ R3. We give the complete description of its dynamics on the sphere at infinity. For six sets of the parameter values, the system has invariant algebraic surfaces. In these cases, we provide the global phase portrait of the Chen system and give a complete description of the α- and w-limit sets of its orbits in the Poincaré ball, including its boundary S2, i.e. in the compactification of R3 with the sphere S2 of infinity. Moreover, combining the analytical results obtained with an accurate numerical analysis, we prove the existence of a family with infinitely many heteroclinic orbits contained on invariant cylinders when the Chen system has a line of singularities and a first integral, which indicates the complicated dynamical behavior of the Chen system solutions even in the absence of chaotic dynamics. [ABSTRACT FROM AUTHOR]

In this paper we discuss a novel route to chaos in cam-follower systems, a class of mechanical devices of relevance in applications. We show that a sudden transition to chaos is observed when a complete chattering motion (characterized theoretically by infinitely many impacts) is interrupted. Using numerical computations and semi-analytical methods we show that a cascade of grazing bifurcations is then originated, leading the system into chaos. This novel phenomenon is described using an appropriate model that is shown to predict correctly the numerical observations. The results are used to explain the transition to chaos observed in an experimental cam-follower system recently presented in the literature. [ABSTRACT FROM AUTHOR]

This paper presents a fractional-order population model which consists of the two-predators and one-prey scheme. For this new model, the numerical solution is derived and the simulations are performed for various sets of model parameters together with stability analysis for commensurate and incomensurate orders of the fractional-order population model. The results obtained via the simulations show that chaos can be observed in such population model. [ABSTRACT FROM AUTHOR]

We show analytically that almost all three-dimensional dissipative quadratic systems of ordinary differential equations with a total of five terms on the right-hand side and one nonlinear term (namely 5-1 cases) are not chaotic except twenty one of them. Indeed we find nine systems that exhibit chaos, which were discovered by Sprott and Malasoma earlier. They are the simplest dissipative chaotic systems found so far. In this paper, we also extend Heidel-Zhang's theorem which provides sufficient conditions for solutions in the three-dimensional autonomous systems with polynomials and rational expressions on the right-hand side being nonchaotic. We then investigate the twenty one systems analytically and numerically. We show the portraits of some typical chaotic and nonchaotic solutions in phase space. For two of the systems that exhibit chaos we found stable period 1, 2, 4, 8 and 12 orbits numerically. [ABSTRACT FROM AUTHOR]

In this paper, we have proposed a novel three-dimensional Lorenz-like chaotic system. Some basic properties of the system, such as dynamical behaviors, bifurcation diagram. Lyapunov exponents and Poincare mapping are investigated either analytically or numerically. Furthermore, the control problem of the new chaotic system was studied via nonlinear backstepping method. The single backstepping control input was designed according to Lyapunov stability criterion. Numerical simulations are carried out in order to demonstrate the effectiveness of the proposed control design. [ABSTRACT FROM AUTHOR]

In this two-part paper the problem of evaluating robustness and strength of chaos in thermomechanically-based Shape Memory Oscillators (SMOs) is addressed. In Part I, several tools for the theoretical prediction of the main features of the pseudoelastic loops of Shape Memory Devices (SMDs) have been proposed. In this Part II, the Method of Wandering Trajectories (MWT), that has already been validated as a systematic, yet affordable, numerical tool for the evaluation of the chaotic response of SMOs, is enhanced by complementing it with a quantitative indicator of chaoticity: the maximum value of the displacement normalized separation over a fixed time interval. The method is used to compute numerical 3D behavior charts for several model parameters sets, properly selected in order to highlight the influence of various model parameters on the nonlinear dynamical response of SMOs. From the numerical analyses it turns out that two main aspects of the SMD behavior do influence the robustness and strength of chaos, namely the hardening of the pseudoelastic plateaus and the area of the hysteresis loop, both of them being meaningfully affected by the model mechanical and thermal parameters. The numerical results are also interpreted by means of the theoretical indicators discussed in Part I, which provide a reliable framework for the prediction of the main features of dynamic response before actual computation of the trajectories. [ABSTRACT FROM AUTHOR]

In this two-part paper the problem of evaluating robustness and strength of chaos in thermomechanically-based Shape Memory Oscillators (SMO) is addressed. In the first part, a theoretical analysis of the main features of the pseudoelastic loops exhibited by the underlying Shape Memory Devices (SMD) is accomplished with the aim to establish a predictive framework for accompanying numerical investigations. The analysis is based on the evaluation of suitable synthetic indicators of the SMD behavior that can be computed from the model parameters before the computation of SMO actual trajectories, and provide information about the hysteresis loops and their dependence on temperature variations. By means of such indicators, a detailed analysis of the influence of thermomechanical coupling on the rate-dependent mechanical response is presented. It is shown that a careful interpretation of the synthetic indicators permits to obtain a reasonable estimation of the influence of various model parameters on the hysteresis loop area and slopes of the pseudoelastic plateaus, that are the main global aspects influencing the occurrence of chaotic responses. In the second part, the theoretical predictions based on the synthetic indicators will be exploited to interpret the results of a systematic numerical investigation based on an enhanced version of the Method of Wandering Trajectories. [ABSTRACT FROM AUTHOR]

The Šil'nikov homoclinic theorem provides one analytic criterion for proving the existence of chaos in three-dimensional autonomous nonlinear systems. In applications of the theorem, however, the existence of a homoclinic orbit that usually determines the geometric structure of the chaotic attractor is not easily verified mainly because there are no available efficient methods. In this paper, based on the undetermined coefficient approach we present a framework of how to find homoclinic orbits in two classes of three-dimensional autonomous nonlinear systems of normal forms, including how to set a reasonable form of expanding series of the homoclinic orbit, how to determine all coefficients in the expansion, and how to find a numerical homoclinic orbit. Numerical examples show that the proposed framework in combination with computer simulation is very efficient. [ABSTRACT FROM AUTHOR]

Synchronization of multiple identical chaotic systems, coupled in a ring structure, was studied in this paper. Ueda attractor was used as an example to perform the synchronization of several chaotic systems. The bound of the Ueda attractor was also estimated by numerical simulation. The above results were also adopted to design a controller to synchronize chaotic systems. The stability of the synchronization of multiple identical chaotic systems was investigated using Lyapunov's direct method. Some sufficient conditions of asymptotical synchronization were attained from rigorous mathematical theory. Further, numerical results were also demonstrated in order to validate the proposed synchronization approach. [ABSTRACT FROM AUTHOR]

In this paper autonomous and nonautonomous modified hyperchaotic complex Lü systems are proposed. Our systems have been generated by using state feedback and complex periodic forcing. The basic properties of these systems are studied. Parameters range for hyperchaotic attractors to exist are calculated. These systems have very rich dynamics in a wide range of parameters. The analytical results are tested numerically and excellent agreement is found. A circuit diagram is designed for one of these hyperchaotic complex systems and simulated using Matlab/Simulink to verify the hyperchaotic behavior. [ABSTRACT FROM AUTHOR]

In this paper, the analytical conditions for the controlled pendulum synchronizing with periodic motions in the Duffing oscillator are developed using the theory of discontinuous dynamical systems. From the analytical conditions, the synchronization invariant domain is obtained. The partial and full synchronizations of the controlled pendulum with periodic motions in the Duffing oscillator are discussed. The control parameter map for the synchronization is achieved from the analytical conditions, and numerical illustrations of the partial and full synchronizations are carried out to illustrate the analytical conditions. This synchronization is different from the controlled Duffing oscillator synchronizing with chaotic motion in the periodically excited pendulum. Because the periodically forced, damped Duffing oscillator possesses periodic and chaotic motions, further investigation on the controlled pendulum synchronizing with complicated periodic and chaotic motions in the Duffing oscillator will be accomplished in sequel. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a master-slave synchronization scheme of two n-dimensional nonautonomous chaotic systems coupled by sinusoidal state error feedback control, where parameter mismatch exists between the external harmonic excitation of master system and that of slave one. A concept of synchronization with error bound is introduced due to parameter mismatch, and then the bounds of synchronization error are estimated analytically. Some synchronization criteria are firstly obtained in the form of matrix inequalities by the Lyapunov direct method, and then simplified into some algebraic inequalities by the Gerschgorin disc theorem. The relationship between the estimated synchronization error bound and system parameters reveals that the synchronization error can be controlled as small as possible by increasing the coupling strength or decreasing the magnitude of mismatch. A three-dimensional gyrostat system is chosen as an example to verify the effectiveness of these criteria, and the estimated synchronization error bounds are compared with the numerical error bounds. Both the theoretical and numerical results show that the present sinusoidal state error feedback control is effective for the synchronization. Numerical examples verify that the present control is robust against amplitude or phase mismatch. [ABSTRACT FROM AUTHOR]

In this paper, a novel approach for generating multi-wing chaotic attractors via switching control is proposed. By using a switching controller, multi-wing chaotic attractors can be generated from a double-wing system. The presented method is different from that of classical multi-scroll chaotic attractors generated by odd-symmetric multi-segment linear functions from Chua system. Furthermore, the basic dynamical behaviors, including equilibrium points, maximum Lyapunov exponents and bifurcations, are further investigated. An improved module-based unified circuit is designed for realizing 4, 6, 8 and 10-wing chaotic attractors, and the experimental result is also demonstrated, which is consistent with the numerical simulation. [ABSTRACT FROM AUTHOR]