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It is far well-known that energy function of a two-dimensional autonomous dynamical system can be simply obtained by multiplying its corresponding second-order ordinary differential equation, i.e., its equation of motion by the first time derivative of its state variable. In the nineties, one of us (J.C.S.) stated that a three-dimensional autonomous dynamical system can be also transformed into a third-order ordinary differential equation of motion todays known as jerk equation. Although a method has been developed during these last decades to provide the energy function of such three-dimensional autonomous dynamical systems, the question arose to determine by which type of term, i.e., by the first or second time derivative of their state variable , the corresponding jerk equation of these systems should be multiplied to deduce their energy function. We prove in this work that the jerk equation of such systems must be multiplied by the second time derivative of the state variable and not by the first like in dimension two. We then provide an interpretation of the new term appearing in the energy function and called jerk energy. We thus established that it is possible to obtain the energy function of a three-dimensional dynamical system directly from its corresponding jerk equation. Two and three-dimensional Van der Pol models are then used to exemplify these main results. Applications to Lorenz and Chua's models confirms their validity. • Energy function of dissipative systems. • Helmholtz theorem. • Gibbs–Appell function. • Chaotic oscillators. • Lorenz and Chua's models. [ABSTRACT FROM AUTHOR]

There are many recent investigations on chaotic systems with self-excited and hidden attractors. In this paper we introduce a chaotic system in which the attractor can change between hidden and self-excited attractor depending on the value of parameters. Dynamical properties of the proposed attractor are investigated. The attractor is then realized using off the shelf components. Also, for this new system electronic circuit is implemented and FPGA-based chaotic oscillator is designed. In the end, the fractional- order form is examined through bifurcation and stability analysis. [ABSTRACT FROM AUTHOR]

We explore the behavior of coupled chaotic oscillators where one unit has intrinsically dissimilar dynamics. We find that the presence of a single dissimilar chaotic system in the network manages to drive all the chaotic oscillators to regular limit cycles. Additionally the regular cycles that emerge are significantly smaller in size than the uncoupled chaotic attractors. Counterintuitively the more geometrically dissimilar the single distinct system is from the other chaotic oscillators, the stronger is the emergent control. Furthermore, the position of the dissimilar system in the network does not affect the control when the dissimilar element is markedly different. So surprisingly, enhanced heterogeneity in coupled systems leads to more pronounced and robust controllability. Our results can then potentially lead to the design of new control strategies in engineered systems, and also suggests mechanisms whereby naturally occurring complex systems can evolve to regular dynamics through coupling to heterogeneous sub-systems. • We explore a network of chaotic oscillators, with one dissimilar chaotic system. • Surprisingly, the dissimilar unit controlled the chaos in the network to limit cycles. • The emergent cycles were significantly smaller than the intrinsic chaotic attractors. • Presence of more geometrically distinct systems yielded more pronounced control. • So enhanced heterogeneity in nodal dynamics leads to more robust chaos suppression. [ABSTRACT FROM AUTHOR]

The collective dynamics of chaotic oscillators has drawn considerable attention in numerous fields, including agriculture and forestry. The alternate bearing of tree crops is a phenomenon in which a year of heavy yield is followed by a year of light yield. This phenomenon has been conventionally investigated using a tent map known as a resource budget model. Alternate fruiting is caused by strong synchronisation among trees in an orchard and is a major problem in fruit growing. To develop control methods for alternate fruiting, it is essential to understand the strength of synchronisation at the individual and population levels of trees in orchards and the mechanism of alternate fruiting. In this study, in-phase/out-of-phase analysis was applied to the yield data of a 9562 pistachio (Pistacia vera L.) population, and the phase transitions and mode-locking in the orchard were revealed. Using a developed network model consisting of diffusively coupled chaotic oscillators on which common noise was imposed identically, the phase transitions, mode-locking, and 1/3 power-law scaling spatial correlation were confirmed mathematically. Furthermore, the manner in which three essential factors, i.e. common noise, direct coupling, and the cropping coefficient gradient, explain the spatial synchrony of the orchard was elucidated. The proposed methodology based on nonlinear dynamics would be useful for pomology, forestry, and ecosystem management. • The spatial distribution of phase synchronization is visualized in a 9,562 pistachio plant population. • The phase transitions and mode locking in the orchard were revealed. • Common noise imposed coupled map lattices confirmed 1/3 power-law scaling in the yield data. • This work will benefit food production, forest production, and ecosystem management. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analytical formula for zero Lyapunov exponent describing the dynamics of interacting chaotic systems with noise. The deduced analytical prediction is in a good agreement with the value of zero Lyapunov exponent obtained numerically for two unidirectionally coupled Rössler oscillators. We have shown that this good agreement is observed for a wide diapason of the values of the control parameters. [ABSTRACT FROM AUTHOR]