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We present a computational model of basal ganglia as a key player in exploratory behavior. The model describes exploration of a virtual rat in a simulated water pool experiment. The virtual rat is trained using a reward-based or reinforcement learning paradigm which requires units with stochastic behavior for exploration of the system's state space. We model the Subthalamic Nucleus-Globus Pallidus externa (STN-GPe) segment of the basal ganglia as a pair of neuronal layers with oscillatory dynamics, exhibiting a variety of dynamic regimes such as chaos, traveling waves and clustering. Invoking the property of chaotic systems to explore state-space, we suggest that the complex exploratory dynamics of STN-GPe system in conjunction with dopamine-based reward signaling from the Substantia Nigra pars compacta (SNc) present the two key ingredients of a reinforcement learning system.

The Lie series technique is applied to solve a class of nonlinear differential equations. We show that it includes also the dynamical integration technique. In particular systems with limit cycle behavior are studied. Differential equations with chaotic behavior are also considered with this technique.

Economy is demanding new models, able to understand and predict the evolution of markets. To this respect, Econophysics offers models of markets as complex systems, that try to comprehend macro-, system-wide states of the economy from the interaction of many agents at micro-level. One of these models is the gas-like model for trading markets. This tries to predict money distributions in closed economies and quite simply, obtains the ones observed in real economies. However, it reveals technical hitches to explain the power law distribution, observed in individuals with high incomes. In this work, nonlinear dynamics is introduced in the gas-like model in an effort to overcomes these flaws. A particular chaotic dynamics is used to break the pairing symmetry of agents (i, j) ⇔ (j, i). The results demonstrate that a "chaotic gas-like model" can reproduce the Exponential and Power law distributions observed in real economies. Moreover, it controls the transition between them. This may give some insight of the micro-level causes that originate unfair distributions of money in a global society. Ultimately, the chaotic model makes obvious the inherent instability of asymmetric scenarios, where sinks of wealth appear and doom the market to extreme inequality.

In this paper, function projective synchronization of chaotic systems is investigated through nonlinear adaptive–impulsive control. To achieve synchronization, suitable nonlinear continuous and impulsive controllers are designed, according to invariant principle of impulsive dynamical systems. Sufficient conditions are given to ensure the synchronization. Numerical simulation results show the effectiveness of the proposed scheme.

In this paper, we give a general review on the application of ergodic theory to the investigation of dynamics of the Yang–Mills gauge fields and of the gravitational systems, as well as its application in the Monte Carlo method and fluid dynamics. In ergodic theory the maximally chaotic dynamical systems (MCDS) can be defined as dynamical systems that have nonzero Kolmogorov entropy. The hyperbolic dynamical systems that fulfill the Anosov C-condition belong to the MCDS insofar as they have exponential instability of their phase trajectories and positive Kolmogorov entropy. It follows that the C-condition defines a rich class of MCDS that span over an open set in the space of all dynamical systems. The large class of Anosov–Kolmogorov MCDS is realized on Riemannian manifolds of negative sectional curvatures and on high-dimensional tori. The interest in MCDS is rooted in the attempts to understand the relaxation phenomena, the foundations of the statistical mechanics, the appearance of turbulence in fluid dynamics, the nonlinear dynamics of Yang–Mills field and gravitating $N$-body systems as well as black hole thermodynamics. Our aim is to investigate classical- and quantum-mechanical properties of MCDS and their role in the theory of fundamental interactions.

The ultimate state of projective synchronization is usually considered to be hardly controllable due to its close dependence on the initial conditions of both drive and response systems. In this letter, we show that the scaling factor of projective synchronization can be controlled to be proportional to a given scalar function with respect to time even without the knowledge of the initial conditions of response system. Further, we apply it to the chaotic encryption. Comparing with some existing implementations, it is found that the present scheme owns several remarkable advantages. To our knowledge, it is for the first time that the projective synchronization is applied to chaos-based encryption.

In this paper, we propose a new lag synchronization scheme for a general class of chaotic systems with external disturbance. Based on Lyapunov theory and linear matrix inequality (LMI) approach, the lag synchronization controller is presented to not only guarantee exponential lag synchronization but also reduce the effect of external disturbance to an norm constraint. The proposed controller can be obtained by solving a convex optimization problem represented by the LMI. As an application of the proposed method, the lag synchronization problem for Genesio–Tesi system is investigated.

Unidirectional coupled synchronization of two identical or different chaotic systems has been carefully studied based on the master–slave synchronization scheme, where the synchronous state is that of the master system and cannot be changed after they realized synchronization. In this paper, a general bidirectional synchronization scheme is presented which made the master–slave scheme a special case. It is straightforward to tune the synchronous state by just changing the value of a parameter. Based on the general bidirectional synchronization scheme, active control method is used to tune the synchronous state of two pairs of different chaotic systems: the Lorenz and Chen systems; and then the Lü and Rössler systems.

In this paper, the problem of adaptive ℋ_∞ synchronization for unified chaotic systems with unknown parameter and external disturbance is studied. It is noticed that this unified chaotic system contains the noted Lorentz, Lü and Chen systems. Based on Lyapunov theory and linear matrix inequality (LMI) formulation, the novel feedback controller with adaptive law is established to not only guarantee stable synchronization of both master and slave systems but also reduce the effect of external disturbance to an ℋ_∞ norm constraint. A criterion for existence of the controller is given in terms of LMIs. Finally, a numerical example shows the effectiveness of the proposed method.

In this paper, we have proposed a new chaotic megastable oscillator which has both conservative and dissipative characters depending on the selection of parameters. Various dynamical characteristics including megastability of the new system are investigated and presented. The bifurcation plots and the corresponding Lyapunov exponents (LEs) confirm the existence of both dissipative and conservative oscillations in the system. The proposed megastable oscillator is used as a carrier generator in a differential chaos shift keying (DCSK). Another application of the new chaotic oscillator is shown by using it in developing a random number generator (RNG) and the NIST test results are presented to show the statistical complexity of the new system.

We adopt long short-term memory (LSTM) networks to model and characterize chaotic systems rather than conventional dynamical equations. We find that a well-trained LSTM system can synchronize with its learned chaotic system via transmitting a common signal. In the same fashion, we show that when learning an identical chaotic system, the trained LSTM systems can also be synchronized. Remarkably, we find that a cascading synchronization will be achieved among chaotic systems and their trained LSTM systems in the same manner. We further validate that this synchronization behavior is robust even the transmitting signal is contaminated with relatively a high level of white noise. Our work reveals that synchronization is a common behavior linking chaotic systems and their learned LSTM networks.

Random numbers are important parameters for the security of cryptographic applications. In this study, a secure and efficient generator is proposed to generate random numbers. The first part of the generator is a true random number generator that consists of chaotic systems implemented on FPGA. The second part of the generator is a post-processing algorithm used to overcome the problems that emerge from the generator or environmental factors. As the post-processing algorithm, Keccak, the latest standard of hash algorithm, was rearranged and used. Random numbers with the proposed approach meet the security requirements for cryptographic applications. Furthermore, the NIST 800-22 test suite and autocorrelation test are used to ensure the generated numbers have no statistical weakness. The successful test results demonstrate the security of the generated numbers. An important advantage of the proposed generator does not cause any data loss and perform 100% efficiency although data loss can be up to 70% in some post-processing algorithms.

By diode bridging an inductor to implement a memristor bipole, with active Wien-bridge oscillator, a simple and feasible third-order autonomous memristive chaotic oscillator is presented. The dynamical characteristics of the proposed circuit are investigated both theoretically and numerically, from which it can be found that the circuit has one unstable equilibrium point. Through the analysis of the bifurcation diagram, Lyapunov exponent spectrum and the 0–1 test chaos detection, it is shown that this system displays limit cycle orbit with different periodicity, quasi-periodic behavior, chaotic behavior and bursting behavior. The bursting behavior found in this circuit is periodic, quasi-periodic and chaotic bursting. We confirm the feasibility of the proposed theoretical model using Pspice simulations and a physical realization based on an electronic analog implementation of the model.

This paper proposes a novel jerk circuit obtained by using an alternative nonlinear bipole component of inductor and diode in parallel. The circuit is described by five differential equations and investigated by the stability analysis, equilibria points, Kaplan–Yorke dimension, phase portraits, Lyapunov characteristic exponent estimation, bifurcation diagram and the 0–1 test chaos detection. The control parameter is adopted by varying the inductor $L$ value, this system can display periodic orbit, quasi-periodic orbit and chaotic behavior. The dynamic influence of transit diode capacitance is done and this confirms the robustness of the system to noise influence. The validity of the numerical simulations is experimentally realized through the phase portraits of the circuit. Finally, the synchronization of the systems is studied and time simulation results are presented.

This paper proposes synchronization and chaotic communication for a class of Lur'e type discrete-time chaotic systems. The scalar outputs are suitably chosen in a flexible manner to be linear, nonlinear, or predictive, and along with the drive system are then written in an output injection form. Then with a suitable design of an observer-based response system, dead-beat performance is achieved for synchronization and chaotic communications. Then disturbances in the drive system are considered. Using an ℋ^∞ performance criterion, the disturbance is attenuated to a prescribed level by solving linear matrix inequalities (LMIs). Numerical simulations are carried out to verify the dead-beat performance.

In this paper, by means of case studies we discuss an observable feature of 3D continuous-time autonomous chaotic systems through scalar output and its time derivatives. We observe that the Lorenz system, the Rössler system, the Chua circuit and the Chen system are all observable based on scalar output and its derivatives. This leads to our conjecture that chaotic motion described by 3D continuous-time autonomous dynamical system is observable based on a scalar output and its first- and second-order derivatives. Finally, we present some mathematical analysis and put up some theoretical questions for future studies.

This study addresses the adaptive synchronization of Rössler and Chua circuit systems with unknown parameters. By using Lyapunov stability theory the adaptive synchronization law with a single-state variable feedback is derived, such that the trajectory of the two systems are globally stabilized to an equilibrium point of the uncontrolled system (globally stable means that the method of the solution is restricted in area of phase space i.e. globally in a subset of a phase space with bounded zero volume). We use the Lyapunov direct method to study the asymptotic stability of the solutions of error system. Numerical simulations are given to explain the effectiveness of the proposed control scheme.

We consider observer based synchronization of continuous-time chaotic systems. We present two message transmission schemes for such systems. The first one is based on chaotic masking and modulation, and the second one is based on only chaotic modulation. We show that in these schemes, the message may be recovered under certain conditions. We show that the proposed schemes are robust with respect to noise and parameter mismatch. We also present some simulation results.

We call a chaotic dynamical system pseudo-deterministic when it does not produce numerical, or pseudo-trajectories that stay close, or shadow chaotic true trajectories, even though the model equations are strictly deterministic. In this case, single chaotic trajectories may not be meaningful, and only statistical predictions, at best, could be drawn on the model, like in a stochastic system. The dynamical reason for this behavior is nonhyperbolicity characterized either by tangencies of stable and unstable manifolds or by the presence of periodic orbits embedded in a chaotic invariant set with a different number of unstable directions. We emphasize herewith the latter by studying a low-dimensional discrete-time model in which the phenomenon appears due to a saddle-repeller bifurcation. We also investigate the behavior of the finite-time Lyapunov exponents for the system, which quantifies this type of nonhyperbolicity as a system parameter evolves past a critical value. We argue that the effect of unstable dimension variability is more intense when the invariant chaotic set of the system loses transversal stability through a blowout bifurcation.

We consider a model-based approach for the anticontrol of some continuous time systems. We assume the existence of a chaotic model in an appropriate form. By using a suitable input, we match the dynamics of the controlled system and the chaotic model. We show that controllable systems can be chaotified with the proposed method. We give a procedure to generate such chaotic models. We also apply an observer-based synchronization scheme to compute the required input.