Chaos theory has wide applications and its importance can be seen by the voluminous publications on various applications in several branches of science, commerce and engineering. Control, tracking or regulation and synchronization of different types of chaotic systems are importance areas of research in the control literature and various methods have been adopted over the past few decades for tackling these research problems. Also, the discovery of novel chaotic and hyperchaotic systems in various applications, their qualitative properties and the control of such systems are also important research areas in chaos theory. This paper announces a novel 4-D hyperchaotic Rikitake dynamo system, which is derived by adding a state feedback control to the famous 3-D Rikitake two-disk dynamo system (1958). The frequent and irregular reversals of the Earth’s magnetic field inspired a number of early studies involving electrical currents within the Earth’s molten core. One of the first such models to exhibit reversals was Rikitake’s two-disk dynamo system (Rikitake, 1958). This paper discusses the qualitative properties of the novel hyperchaotic Rikitake dynamo system. We note that the novel hyperchaotic Rikitake dynamo system has no equilibrium points. The Lyapunov exponents of the hyperchaotic Rikitake dynamo system are found as L! = 0.09136, L! = 0.02198, L! = 0 and L! = −2.11190.  The Kaplan-Yorke fractional dimension of the novel hyperchaotic Rikitake dynamo system is found as D!\" = 3.05367.   Next, this paper discusses control and synchronization of the novel hyperchaotic Rikitake dynamo system with unknown parameters using adaptive control method. The main results are established using Lyapunov stability theory and numerically illustrated using MATLAB. Finally, for the 4-D novel hyperchaotic system, an electronic circuit realization in SPICE has been described to confirm the feasibility of the theoretical hyperchaotic Rikitake dynamo model.

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Analysis, Control, Synchronization and SPICE Implementation of a Novel 4 - D Hyperchaotic Rikitake Dynamo System without Equilibrium

Bifurcation analysis of two disc dynamos with viscous friction and multiple time delays

(2 < p < 4) [200]. (Uq(∫u(1, 1)), oq1/2(2n)) [92]. 1 [273, 79, 304, 119]. 1 + 1 [252]. 2 [352, 318, 226, 40, 233, 157, 299, 60]. 2× 2 [185]. 3 [456, 363, 58, 18, 351]. ∗ [238]. 2 [277]. 3 [350]. p [282]. B−L [427]. α [216, 483]. α− z [322]. N = 2 [507]. D [222]. ẍ+ f(x)ẋ + g(x) = 0 [112, 111, 8, 5, 6]. Eτ,ηgl3 [148]. g [300]. κ [244]. L [205, 117]. L [164]. L∞ [368]. M [539]. P [27]. R [147]. Z2 [565]. Z n 2 [131]. Z2 × Z2 [25]. D(X) [166]. S(N) [110]. ∫l2 [154]. SU(2) [210]. N [196, 242]. O [386]. osp(1|2) [565]. p [113, 468]. p(x) [17]. q [437, 220, 92, 183]. R, d = 1, 2, 3 [279]. SDiff(S) [32]. σ [526]. SLq(2) [185]. SU(N) [490]. τ [440]. U(1) N [507]. Uq(sl 2) [185]. φ 2k [283]. φ [553]. φ4 [365]. ∨ [466]. VOA[M4] [33]. Z [550].

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A Complete Bibliography of Publications in the Journal of Mathematical Physics: 2005{2009

Characterizing testing methods for context-aware software systems: Results from a quasi-systematic literature review

For competitive Lotka–Volterra systems, Ahmad and Lazer’s work [S. Ahmad, A.C. Lazer, Average growth and total permanence in a competitive Lotka–Volterra system, Annali di Matematica 185 (2006) S47–S67] on total permanence of systems without delays has been extended to delayed systems [Z. Hou, On permanence of all subsystems of competitive Lotka–Volterra systems with delays, Nonlinear Analysis: Real World Applications 11 (2010) 4285–4301]. In this paper, existence and boundedness of nonnegative solutions and permanence are considered for general Lotka–Volterra systems with delays including competitive, cooperative, predator–prey and mixed type systems. First, a condition is established for the existence and boundedness of solutions on a half line. Second, a necessary condition on the limits of the average growth rates is provided for permanence of all subsystems. Then the result for competitive systems is also proved for the general systems by using the same techniques. Just as for competitive systems, the eminent finding is that permanence of the system and all of its subsystems is completely irrelevant to the size and distribution of the delays.

On permanence of Lotka–Volterra systems with delays and variable intrinsic growth rates

In this paper we present some relevant dynamical properties of a three-dimensional Lotka–Volterra system from the Poisson dynamics point of view.

On a Hamiltonian version of a three-dimensional Lotka–Volterra system

In this paper we analyze the quadratic and homogeneous Hamiltonian systems on (&#x1d530;&#x1d52c;(3))* from the Poisson dynamics and geometry point of view.

New Aspects on the Geometry and Dynamics of Quadratic Hamiltonian Systems on ((3))

In this paper, we present some relevant dynamical properties of the Rossler system from the Poisson dynamics point of view.

On the completely integrable case of the Rössler system

In this paper we will describe some relevant dynamical properties of the Rikitake dynamo system from the Poisson geometry point of view.

A Hamiltonian look at the Rikitake two-disk dynamo system

In this paper, we describe some relevant dynamical and geometrical properties of the Rabinovich system from the Poisson geometry and the dynamics point of view. Starting with a Lie-Poisson realization of the Rabinovich system, we determine and then completely analyze the Lyapunov stability of all the equilibrium states of the system, study the existence of periodic orbits, and then using a new geometrical approach, we provide the complete dynamical behavior of the Rabinovich system, in terms of the geometric semialgebraic properties of a two-dimensional geometric figure, associated with the problem. Moreover, in tight connection with the dynamical behavior, by using this approach, we also recover all the dynamical objects of the system (e.g. equilibrium states, periodic orbits, homoclinic and heteroclinic connections). Next, we integrate the Rabinovich system by Jacoby elliptic functions, and give some Lax formulations of the system. The last part of the article discusses some numerics associated with the Poisson geometrical structure of the Rabinovich system.

Anania Gîrban

Papers

On a Hamiltonian version of a three-dimensional Lotka–Volterra system

New Aspects on the Geometry and Dynamics of Quadratic Hamiltonian Systems on ((3))

On the completely integrable case of the Rössler system

A Hamiltonian look at the Rikitake two-disk dynamo system

On the Hamiltonian dynamics and geometry of the Rabinovich system