Marotto Theorem Research Articles

Generating chaos via decentralized linear state feedback and a class of nonlinear functions

Abstract This paper proposes a general chaotification algorithm for a class of nonlinear discrete-time dynamical systems. Based on a given nonlinear discrete-time system, the new chaotification algorithm uses the decentralized linear state feedback control and a class of nonlinear functions that are only required to satisfy some mild assumptions to construct a chaotic nonlinear system. The sine and sawtooth function used in the existing literature on chaotification only are the special cases of the proposed nonlinear function. Based on the corrected version of the Marotto theorem, we mathematically prove that the constructed nonlinear system is indeed chaotic in the sense of Li and Yorke. In particular, an explicit formula for the computation of chaotification parameters is also obtained. The theoretical results in this paper can be used to supervise one to construct different chaotification algorithms.

Chaotifying a stable Takagi–Sugeno fuzzy system by a sawtooth nonlinearity

The problem of making a stable Takagi–Sugeno (TS) fuzzy system chaotic by using the state feedback control of arbitrarily small magnitude is studied. The feedback controller used is a simple sawtooth function of the system state. The improved Marotto theorem has been obtained and applied to mathematically prove that the controlled system is chaotic in the sense of Li and Yorke. In particular, an explicit formula for the computation of chaotification parameters is obtained. A numerical example is used to visualize and illustrate the theoretical results.

A simple time-delay feedback anticontrol method made rigorous.

An effective method of chaotification via time-delay feedback for a simple finite-dimensional continuous-time autonomous system is made rigorous in this paper. Some mathematical conditions are derived under which a nonchaotic system can be controlled to become chaotic, where the chaos so generated is in a rigorous mathematical sense of Li-Yorke in terms of the Marotto theorem. Numerical simulations are given to verify the theoretical analysis.

Chaotifying a linear time-invariant system by the state feedback controller and sawtooth function

Another algorithm for chaotification of any given linear time-invariant discrete-time systems is presented. The new chaotification algorithm uses the decentralized control and the continuous sawtooth function, which can generate discrete chaos with an arbitrarily desired amplitude bound. Based on the Marotto theorem, we mathematically prove that the controlled system is chaotic in the sense of Li and Yorke. Finally, a simple example is used to illustrate the effectiveness of the proposed theory and method.

A new method of determining chaos-parameter-region for the tent map

Abstract In this paper, we apply the modified Marotto Theorem developed by Li and Chen to determination of the chaos-parameter-region of the tent map, which provides a new approach to detection of the chaos-parameter-region for the nonlinear maps.

Dynamical Behaviors in a Two-dimensional Map

We consider the dynamics of a two-dimensional map proposed by Maynard Smith as a population model. The existence of chaos in the sense of Marotto's theorem is first proved, and the bifurcations of periodic points are studied by analytic methods. The numerical simulations not only show the consistence with the theoretical analysis but also exhibi the complex dynamical behaviors.

The Marotto Theorem on planar monotone or competitive maps

In 1978, Marotto generalized Li–Yorke’s results on the criterion for chaos from one-dimensional discrete dynamical systems to n-dimensional discrete dynamical systems, showing that the existence of a non-degenerate snap-back repeller implies chaos in the sense of Li–Yorke. This theorem is very useful in predicting and analyzing discrete chaos in multi-dimensional dynamical systems. Yet, besides it is well known that there exists an error in the conditions of the original Marotto Theorem, and several authors had tried to correct it in different way, Chen, Hsu and Zhou pointed out that the verification of “non-degeneracy” of a snap-back repeller is the most difficult in general and expected, “almost beyond reasonable doubt,” that the existence of only degenerate snap-back repeller still implies chaotic, which was posed as a conjecture by them. In this paper, we shall give necessary and sufficient conditions of chaos in the sense of Li–Yorke for planar monotone or competitive discrete dynamical systems and solve Chen–Hsu–Zhou Conjecture for such kinds of systems.

ON SPATIAL LYAPUNOV EXPONENTS AND SPATIAL CHAOS

In this paper, we investigate the periodic orbits, spatial Lyapunov exponents, and stability of spatially periodic orbits of the general 2D Logistic system [Formula: see text] where a is a real constant and μ is a parameter. The existence of spatial chaos in the sense of Li and Yorke is proved using the Marotto theorem. These results extend the corresponding results in the 1D Logistic system [Formula: see text] where n0is a fixed integer. These results also improve some existing results of the 2D coupled map lattice (CML) model [Formula: see text] where ε > 0 is the coupling constant.

Chaotic dynamics of an integrate-and-fire circuit with periodic pulse-train input

In this brief, we introduce the working principle of a pacemaker neuron type integrate-and-fire circuit having two states with a periodic pulse-train input, first proposed by Nakano et al. (1999). The dynamics of this circuit can be described by a standard impulsive differential equation and applied to simulate the evolution of the pulse-coupled neural networks. By applying the Marotto theorem (1978), we theoretically prove that the circuit becomes chaotic as the parameters enter some regions. We find that the circuit does not exhibit chaotic dynamics with a small period of the pulse-train input but that a chaotic phenomenon appears with increase of the period. The relations between this circuit and that of symbolic dynamics are further investigated. Numerical simulations and corresponding calculation, as illustrative examples, reinforce our theoretical proof and theory.

On the Marotto–Li–Chen theorem and its application to chaotification of multi-dimensional discrete dynamical systems

This paper further discusses the modified Marotto Theorem developed recently by Li and Chen (called “Marotto–Li–Chen Theorem” here for distinction). A simple yet rigorous chaotification (i.e., “anticontrol of chaos”) schemes is proposed for multi-dimensional dynamical systems based on the Marotto–Li–Chen Theorem. An illustrative example is included to show that the constructed chaotification algorithm is effective.

An improved version of the Marotto Theorem

In 1975, Li and Yorke introduced the first precise definition of discrete chaos and established a very simple criterion for chaos in one-dimensional difference equations, “period three implies chaos” for brevity. After three years. Marotto generalized this result to n-dimensional difference equations, showing that the existence of a snap-back repeller implies chaos in the sense of Li–Yorke. This theorem is up to now the best one in predicting and analyzing discrete chaos in multidimensional difference equations. Yet, it is well known that there exists an error in the condition of the original Marotto Theorem, and several authors had tried to correct it in different ways. In this paper, we further clarify the issue, with an improved version of the Marotto Theorem derived.

Chaos and asymptotical stability in discrete-time neural networks

This paper aims to theoretically prove by applying Marotto's Theorem that both transiently chaotic neural networks (TCNN) and discrete-time recurrent neural networks (DRNN) have chaotic structure. A significant property of TCNN and DRNN is that they have only one fixed point, when absolute values of the self-feedback connection weights in TCNN and the difference time in DRNN are sufficiently large. We show that this unique fixed point can actually evolve into a snap-back repeller which generates chaotic structure, if several conditions are satisfied. On the other hand, by using the Lyapunov functions, we also derive sufficient conditions on asymptotical stability for symmetrical versions of both TCNN and DRNN, under which TCNN and DRNN asymptotically converge to a fixed point. Furthermore, generic bifurcations are also considered in this paper. Since both of TCNN and DRNN are not special but simple and general, the obtained theoretical results hold for a wide class of discrete-time neural networks. To demonstrate the theoretical results of this paper better, several numerical simulations are provided as illustrating examples.

Chaos from PWM Electrohydraulic Servo

The phenomenon of CHAOS for discret models control systems is discussed. A PWM electrohidraulic servo Is studied. This nonlinear servo control has Its fixed point determined, it has its stability regions delimited as a function of its parameters. We use Marotto's theorem to show the sufficient conditions to exist CHAOS. We make use of a digital simulation with nonlinear programming techniques to show these conditions and to observe the dynamic behavior of the system. The main results are the possibilité of application of CHAOS theory to to discret models control systems, description of the behavior of such systems in unstable regions and the identification and prevision of CHAOS.

The dynamics of a non‐linear discrete oscillator

AbstractThe dynamic behaviour of a simple second‐order discrete system with a single non‐linear factor, specified by (x, y)‐ (Ax + B(y + sin(2π(x +y))), x), (A, B) ϵ D = {(A, B): A, B ≥ 0, A + B < 1} ‐ is studied. the parameter subspace of D is analysed in detail. the bifurcation sets of the system are found. the existence of chaos in the system is proved by applying Marotto's theorem. the result of computer simulation agrees well with its analytical counterpart, and confirms that the single non‐linear factor ϕ = sin(2π(x + y)) is indeed the crucial point of a very complicated dynamic behaviour, including the alternative emergence of periodic, quasi‐periodic and chaotic phenomena, of this simple second‐order oscillator.