Nonlinear Dynamical Systems with Self-Excited and Hidden Attractors

In this chapter, singular system theory and fractional calculus are utilized to model the biological systems in the real world, some fractional-order singular (FOS) biological systems are established, and some qualitative analyses of proposed models are performed. Through the fractional calculus and economic theory, a new and more realistic model of biological systems predator-prey, logistic map and SEIR epidemic system have been extended, and besides some mathematical analysis, the numerical simulations are considered to illustrate the effectiveness of the numerical method to explore the impacts of fractional-order and economic interest on the presented systems in biological contexts. It will be demonstrated that the presence of fractional-order changes the stability of the solutions and enrich the dynamics of system. In addition, singular models exhibit more complicated dynamics rather than standard models, especially the bifurcation phenomena and chaotic behaviors, which can reveal the instability mechanism of systems. Toward this aim, some materials including several definitions and existence theorems of uniqueness of solution, stability conditions and bifurcation phenomena in FOS systems and detailed introductions to fundamental tools for discussing complex dynamical behavior, such as chaotic behavior have been added.

In the recent decades, utilization of chaotic systems has flourished in various engineering applications. Hence, there is an increasing demand on generalized, modified and novel chaotic systems. This chapter combines the general equation of jerk-based chaotic systems with simple scaled discrete chaotic maps. Two continuous chaotic systems based on jerk-equation and discrete maps with scaling parameters are presented. The first system employs the scaled tent map, while the other employs the scaled logistic map. The effects of different parameters on the type of the response of each system are investigated through numerical simulations of time series, phase portraits, bifurcations and Maximum Lyapunov Exponent (MLE) values against all system parameters. Numerical simulations show interesting behaviors and dependencies among these parameters. Analogy between the effects of the scaling parameters is presented for simple one-dimensional discrete chaotic systems and the continuous jerk-based chaotic systems with more complicated dynamics. The impacts of these scaling parameters appear on the effective ranges of other main system parameters and the ranges of the obtained solution. The dependence of equilibrium points on the sign of one of the scaling parameters results in coexisting attractors according to the signs of the parameter and the initial point. In addition, switching can be used to generate double-scroll attractors. Moreover, bifurcation and chaos are studied for fractional-order of the derivative.

Chaos theory has attracted the interest of the scientific community because of its broad range of applications, such as in secure communications, cryptography or modeling multi-disciplinary phenomena. Continuous flows, which are expressed in terms of ordinary differential equations, can have numerous types of post transient solutions. Reporting when these systems of differential equations exhibit chaos represents a rich research field. A self-excited chaotic attractor can be detected through a numerical method in which a trajectory starting from a point on the unstable manifold in the neighborhood of an unstable equilibrium reaches an attractor and identifies it. Several simple systems based on jerk-equations and different types of nonlinearities were proposed in the literature. Mathematical analyses of equilibrium points and their stability were provided, as well as electrical circuit implementations of the proposed systems. The purpose of this chapter is double-fold. First, a survey of several self-excited dissipative chaotic attractors based on jerk-equations is provided. The main categories of the included systems are explained from the viewpoint of nonlinearity type and their properties are summarized. Second, maximum Lyapunov exponent values are explored versus the different parameters to identify the presence of chaos in some ranges of the parameters.

Self-excited vibrations in friction oscillators are known as stick-slip phenomenon. The non-linearity in the friction force characteristics introduces instability to the steady frictional sliding. The self-excited friction oscillator consists of the mass pushed horizontally on the surface, elastic element (spring) and a drive (convey or belt). Described system serves as a classic toy model for representation of stick-slip motion. Synchronization is an interdisciplinary phenomenon and can be defined as correlation in time of at least two different processes. This chapter focuses on synchronization thresholds in networks of oscillators with dry friction oscillators coupled by linear springs. Oscillators are connected in the nearest neighbour fashion into topologies of open and closed ring. In course of the numerical modelling we are interested in identification of complete and cluster synchronization regions. The thresholds for complete synchronization are determined numerically using brute force numerical integration and by means of the master stability function (MSF). Estimation of the MSF is conducted using approach called two-oscillator probe. Moreover, we perform a parameter study in two-dimensional space, where different cluster synchronization configurations are explored. The results indicate that the MSF can be applied to non-smooth system such as stick-slip oscillator. Synchronization thresholds determined using MSF occur to be in line with the one obtained numerically.

In the present chapter the combined function projective synchronization among fractional order chaotic systems in the presence of uncertain parameters and external disturbances using backstepping control method is investigated. The chaotic attractors of the systems are found for fractional-order time derivative, which is described in Caputo sense. A new lemma of Caputo derivatives is used to design the controller based on Lyapunov stability theory. During the combined function projective synchronization among the non-identical fractional order systems, the Lorenz, Rossler and Chen systems are taken to illustrate the effectiveness of the considered method. Numerical simulation and graphical results for different particular cases clearly exhibit that the method with this new procedure is easy to implement and reliable for synchronization of non-identical fractional order chaotic systems.

This chapter, after providing some background on business cycles, Kaldor’s original model and related literature, presents an original specification Orlando (Math Comput Simul 125:83–98, 2016) which adds to the cyclical behaviour some peculiar characteristics such as an asymmetric investment and consumption function, lagged investments and integration of economic shocks. A further section proves the chaotic behaviour of the model and adds some insights derived from recurrence quantification analysis. The final part draws some concluding remarks and makes some suggestions for future research. This work investigates chaotic behaviours within a Kaldor-Kalecki framework. This can be achieved by an original specification of the functions describing the investments and consumption as variants of the hyperbolic tangent function rather than the usual arctangent. Therefore fluctuations of economic systems (i.e. business cycles) can be explained by the shape of the investment and saving functions which, in turn, are determined by the behaviour of economic agents. In addition it is explained how the model can accommodate those cumulative effects mentioned by Kaldor which may have the effect of translating the saving and investment functions. This causes the so-called shocks which may be disruptive to the economy or that may have the effect of helping the system to recover from a crisis.

This chapter proposes a three-dimensional autonomous Van der Pol-Duffing (VdPD) type oscillator which is designed from a nonautonomous VdPD two-dimensional chaotic oscillator driven by an external periodic source through replacing the external periodic drive source with a direct positive feedback loop. The dynamical behavior of the proposed autonomous VdPD type oscillator is investigated in terms of equilibria and stability, bifurcation diagrams, Lyapunov exponent plots, phase portraits and basin of attraction plots. Some interesting phenomena are found including for instance, period-doubling bifurcation, symmetry recovering and breaking bifurcation, double scroll chaos, bistable one scroll chaos and coexisting attractors. Basin of attraction of coexisting attractors is computed showing that is associated with an unstable equilibrium. So the proposed autonomous VdPD type oscillator belongs to chaotic systems with self-excited attractors. A suitable electronic circuit of the proposed autonomous VdPD type oscillator is designed and its investigations are performed using ORCAD-PSpice software. Orcard-PSpice results show a good agreement with the numerical simulations. Finally, synchronization of identical coupled proposed autonomous VdPD type oscillators in bistable regime is studied using the unidirectional linear feedback methods. It is found from the numerical simulations that the quality of synchronization depends on the coupling coefficient as well as the selection of coupling variables.

This chapter deals with dynamic analysis, electronic circuit realization and adaptive function projective synchronization (AFPS) of two identical coupled Mathieu-Duffing oscillators with unknown parameters and external disturbances. The dynamics of the Mathieu-Duffing oscillator is investigated with the help of some classical nonlinear analysis techniques such as bifurcation diagrams, Lyapunov exponent plots, phase portraits as well as frequency spectrum. It is found that the oscillator experiences very rich and striking behaviors including periodicity, quasi-periodicity and chaos. An appropriate electronic circuit capable to mimic the dynamics of the Mathieu-Duffing oscillator is designed. The correspondences are established between the parameters of the system model and electronic components of the proposed circuit. A good agreement is obtained between the experimental measurements and numerical results. Furthermore, based on Lyapunov stability theory, adaptive controllers and sufficient parameter updating laws are designed to achieve the function projective synchronization between two identical drive-response structures of Mathieu-Duffing oscillators. The external disturbances are taken into account in the drive and response systems in order to verify the robustness of the proposed strategy. Analytical calculations and numerical simulations are performed to show the effectiveness and feasibility of the method.

This chapter introduces an autonomous self-exited three-dimensional Helmholtz like oscillator which is built by converting the well know autonomous Helmholtz two-dimensional oscillator to a jerk oscillator. Basic properties of the proposed Helmholtz like-jerk oscillator such as dissipativity, equilibrium points and stability are examined. The dynamics of the proposed jerk oscillator is investigated by using bifurcation diagrams, Lyapunov exponent plots, phase portraits, frequency spectra and cross-sections of the basin of attraction. It is found that the proposed jerk oscillator exhibits some interesting phenomena including Hopf bifurcation, period-doubling bifurcation, reverse period-doubling bifurcation and hysteretic behaviors (responsible of the phenomenon of coexistence of multiple attractors). Moreover, the physical existence of the chaotic behavior and the coexistence of multiple attractors found in the proposed autonomous Helmholtz like-jerk oscillator are verified by some laboratory experimental measurements. A good qualitative agreement is shown between the numerical simulations and the experimental results. In addition, the synchronization of two identical coupled Helmholtz like-jerk oscillators is carried out using an extended backstepping control method. Based on the considered approach, generalized weighted controllers are designed to achieve synchronization in chaotic Helmholtz like-jerk oscillators. Numerical simulations are performed to verify the feasibility of the synchronization method. The approach followed in this chapter shows that by combining both numerical and experimental techniques, one can gain deep insight about the dynamics of chaotic systems exhibiting hysteretic behavior.

In this chapter we study the influence of a single strongly attractively coupled external oscillator on a system of coupled Kuramoto oscillators. First we go through the original method used by Kuramoto to solve this system of coupled oscillators. Then we use a later approach developed by Ott and Antonsen. We will use this approach first to solve the original system and show that the results match. Next we will solve a variations of the this system using Ott-Antonsen method, after which we will use it to solve our particular system. We consider a variation of the Kuramoto system which shows multiple regions of synchronization. First we observe the effects of attractive and repulsive couplings. Next we qualitatively study the effect of the initial frequency distribution of the internal oscillators, both the mean and the standard deviation of different distributions like the Gaussian and Lorentzian distributions, on these synchronization regions.

Hyperchaos has important applications in many branches of science and engineering. In this work, we propose a new 4-D hyperchaotic system with three quadratic nonlinearities by modifying the dynamics of hyperchaotic Wang system (Wang et al. 2010). The proposed new hyperchaotic system has a unique equilibrium at the origin, which is a saddle point and unstable. Thus, the new hyperchaotic system exhibits self-excited hyperchaotic attractor. We describe qualitative properties of the new hyperchaotic system such as symmetry, Lyapunov exponents, Kaplan-Yorke dimension, etc. Furthermore, an active control method is derived for the synchronization of two identical new hyperchaotic systems. The circuit experimental results of the new hyperchaotic system show agreement with the numerical simulations.

This chapter announces a new chaotic finance system and show that it is a self-excited chaotic attractor. The phase portraits and qualitative properties of the new chaotic system are described in detail. An electronic circuit realization of the new chaotic finance system is carried out to verify the feasibility of the theoretical model. Next, this chapter examines the control and synchronization of the new chaotic financial system with uncertain parameters as well as known parameters using adaptive control and backstepping control techniques. The designed adaptive controller control and globally synchronizes two identical chaotic financial systems evolving from different initial conditions. The designed controller is capable of stabilizing the financial system at any position as well as controlling it to track any trajectory that is a smooth function of time. Numerical simulations are presented to demonstrate the feasibility of the proposed schemes.

In (Jafari et al, Phys Lett A 377(9):699-702, 2013) the authors gave the expressions of seventeen classes of quadratic differential systems defined in \(\mathbb {R}^3\), depending on one real parameter a, which present chaotic behavior even without having any equilibrium point, for suitable choices of the parameter \(a>0\). In that paper, such systems are denoted by NE\(_1\) to NE\(_{17}\). As these systems have no equilibrium points, a natural question arises: how chaotic motion is generated in their nonequilibrium phase spaces? In this note we combine analytical and numerical results in order to study the integrability and dynamics of systems NE\(_1\), NE\(_6\), NE\(_8\) and NE\(_9\) among those listed in Jafari et al. (2013). We show that they exhibit a quite similar dynamical behavior and, consequently, the mechanisms for birth of chaos in these systems are similar. In this way, we intend to give at least a partial answer to the above question and contribute to better understand the complicated dynamics of the considered systems, in particular concerning the existence of periodic orbits and invariant tori and the emergence of chaotic behavior. The periodic orbits are studied using the Averaging Theory while the invariant tori are proved to exist via KAM Theorem. The chaotic dynamics arises from the broken of some of these invariant tori.

Studying the memristor based chaotic circuit and their dynamical analysis has been an increasing interest in recent years because of its nonvolatile memory. It is very important in dynamic memory elements and neural synapses. In this chapter, the recent and emerging phenomenon such as hidden oscillation is studied by the new implemented memristor based autonomous Duffing oscillator. The stability of the proposed system is studied thoroughly using basin plots and eigenvalues. We have observed a different type of hidden attractors in a wide range of the system parameters. We have shown that hidden oscillations can exist not only in piecewise linear but also in smooth nonlinear circuits and systems. In addition, to control the hidden oscillation, the linear augmentation technique is used by stabilizing a steady state of augmented system.

Hyperchaos has important applications in physics, chemistry, biology, ecology, secure communications, cryptosystems and many scientific branches. In this work, we propose a novel 4-D hyperchaotic Rikitake dynamo system without any equilibrium point by adding a state feedback control to the famous 3-D Rikitake two-disk dynamo system (1958). Thus, the proposed novel hyperchaotic Rikitake dynamo system exhibits hidden attractors. We describe qualitative properties of the hyperchaotic Rikitake dynamo system such as symmetry, Lyapunov exponents, Kaplan-Yorke dimension, etc. Furthermore, an adaptive integral sliding mode control scheme is proposed for the global hyperchaos synchronization of identical hyperchaotic Rikitake dynamo systems. The adaptive control mechanism helps the control design by estimating the unknown parameters. Numerical simulations using MATLAB are shown to illustrate all the main results derived in this work. Finally, the circuit experimental results of the hyperchaotic Rikitake dynamo system show agreement with the numerical simulations.

In this work, we propose a six-term novel 3D chaotic system with hidden attractor. The novel 3D chaotic system consists of six terms and two quadratic nonlinearities. We show that the novel chaotic system has no equilibrium point and hence it exhibits hidden attractor. A detailed qualitative analysis of the 3D chaotic system is presented such as phase portrait analysis, Lyapunov exponents, bifurcation diagram and Poincaré map. The mathematical model of the novel chaotic system is accompanied by an electrical circuit implementation, demonstrating chaotic behavior of the strange attractor. Finally, the circuit experimental results of the chaotic attractors show agreement with numerical simulations.

Recently, Leonov and Kuznetsov introduced a new class of nonlinear dynamical systems, which is called systems with hidden attractors, in contrary to the well-known class of systems with self-excited attractors. In this class, dynamical systems with infinite number of equilibrium points, with stable equilibria, or without equilibrium are classified. Since then, the study of chaotic systems with hidden attractors has become an attractive research topic because this new class of dynamical systems could play an important role not only in theoretical problems but also in engineering applications. In this direction, the proposed chapter presents the bidirectional and unidirectional coupling schemes between two identical dynamical chaotic systems with no-equilibrium points. As it is observed, when the value of the coupling coefficient is increased in both coupling schemes, the coupled systems undergo a transition from desynchronization mode to complete synchronization. Also, the simulation results reveal the richness of the coupled system’s dynamical behavior, especially in the bidirectional case, showing interesting nonlinear dynamics, with a transition between periodic, quasiperiodic and chaotic behavior as the coupling coefficient increases, as well as synchronization phenomena, such as complete and anti-phase synchronization. Various tools of nonlinear theory for the study of the proposed coupling method, such as bifurcation diagrams, phase portraits and Lyapunov exponents have been used.

The design of systems without equilibrium or with line of equilibrium points is a subject which has started to attract the interest of the research community the last decade. In this direction, various chaotic systems with hidden attractors, which are based on memristors or memristive systems, have been proposed. In this chapter a new 4-D memristive system is presented. The peculiarity of the model is that it displays a line of equilibrium points for a range of the parameters as well as no-equilibrium for another range of the parameters. System in both occasions presents a chaotic behavior with hidden attractors. The behavior of the proposed system is investigated through numerical simulations, by using phase portraits, Lyapunov exponents and bifurcation diagrams. The adaptive control scheme of the system is presented in order to prove that the memristive system’s dynamical behavior can be controlled. Also, we have designed an electronic circuit to confirm the feasibility of the system in both cases.

Robotics is an emerging and interesting area in many fields of technical science. In general, a robot manipulator (rigid/flexible) is a more focused research direction in comparison with other areas of robotics. Specifically, flexible manipulators are more applicable in many fields when compared with its rigid counterparts because of many advantages like lightweight, more workspace, lower energy consumption, smaller in size, mobility, etc. These advantages give rise to many control challenges like underactuation, nonminimum phase, noncollocation, control spillover, uncertainties, nonlinearities, complex dynamical behaviours, etc. Path planning or trajectory tracking problem is considered as an interesting and challenging control problem for a flexible manipulator in comparison with the regulation problem. In recent decades, the theory of chaos is used in various technical fields. Aperiodic long time, highly sensitive to initial conditions, unpredictable behaviours, etc. are the fundamental properties of a chaotic signal arising out of a deterministic nonlinear system. Many continuous/discrete/fractional order autonomous and non-autonomous chaotic dynamical systems are available in the literature. In the recent past, more attention has been given to the design and applications of hidden chaotic dynamical systems. The path planning problem of a flexible manipulator requires a reference signal. Various reference signals are used in the literature. Recently, a chaotic signal is used as a reference signal for path planning. However, we have not found any paper wherein a reference signal using a hidden chaotic system is used for path planning. The use of a signal from a hidden chaotic attractor for path planning of a flexible manipulator can provide a new domain of research. Hidden chaotic path planning/trajectory tracking of a two-link flexible manipulator is the aim of this chapter. Use of hidden chaotic attractors as a path/trajectory reference creates extra challenges and complexity in controlling the flexible manipulator. Thus, controlling a flexible manipulator in such a scenario is a challenging task. The dynamics of a two-link flexible manipulator is first modelled using assumed modes method and divided into two parts using two-time scale separation principle (singular perturbation). One subsystem is called as the slow subsystem involving with the rigid parts and another subsystem is called as the fast subsystem which incorporates the flexible dynamics. Separate control techniques are applied to each subsystem. An adaptive sliding mode control technique is designed for the slow subsystem which tackles the uncertainties and helps in fast tracking of the desired hidden chaotic trajectory. A backstepping controller is designed for the fast subsystem system for quick suppression of tip deflections and vibration suppressions. The proposed control techniques are validated using a reference chaotic signal generated from a 3-D hidden attractors chaotic system in MATLAB simulation environment and results are demonstrated. The results reveal that the objective of the chapter is achieved successfully by the proposed control techniques.

In the present decade, chaotic systems are used and appeared in many fields like in information security, communication systems, economics, bioengineering, mathematics, etc. Thus, developing of chaotic dynamical systems is most interesting and desirable in comparison with dynamical systems with regular behaviour. The chaotic systems are categorised into two groups. These are (i) system with self-excited attractors and (ii) systems with hidden attractors. A self-excited attractor is generated depending on the location of its unstable equilibrium point and in such case, the basin of attraction touches the equilibria. But, in the case of hidden attractors, the basin of attraction does not touch the equilibria and also finding of such attractors is a difficult task. The systems with (i) no equilibrium point and (ii) stable equilibrium points belong to the category of hidden attractors. Recently chaotic systems with infinitely many equilibria/a line of equilibria are also considered under the cattegory of hidden attractors. Higher dimensional chaotic systems have more complexity and disorders compared with lower dimensional chaotic systems. Recently, more attention is given to the development of higher dimensional chaotic systems with hidden attractors. But, the development of higher dimensional chaotic systems having both hidden attractors and self-excited attractors is more demanding. This chapter reports three hyperchaotic and two chaotic, 5-D new systems having the nature of both the self-excited and hidden attractors. The systems have non-hyperbolic equilibria, hence, belong to the category of self-excited attractors. Also, the systems have many equilibria, and hence, may be considered under the category of a chaotic system with hidden attractors. A systematic procedure is used to develop the new systems from the well-known 3-D Lorenz chaotic system. All the five systems exhibit multistability with the change of initial conditions. Various theoretical and numerical tools like phase portrait, Lyapunov spectrum, bifurcation diagram, Poincaré map, and frequency spectrum are used to confirm the chaotic nature of the new systems. The MATLAB simulation results of the new systems are validated by designing their circuits and realising the same.