Hamilton Energy Control for the Chaotic System with Hidden Attractors

In this paper, the dynamic behavior and control of chaotic systems with hidden attractors are studied. Firstly, a class of autonomous chaotic systems without the equilibrium point is proposed. Secondly, quantitative analysis methods are applied to explore the dynamic behavior of the new chaotic systems. Then, the Hamilton energy function of the new system is calculated by the Helmholtz theorem and the energy feedback controller is designed. Finally, the effectiveness of the controller is verified by numerical simulations. Compared with the line feedback control, the control effect of Hamilton energy control is better.

A note on the Maxwell Equations of electromagnetic irradiation in a holed spatial region

We make relevant and important comments (on engineering context of Maxwell equations for the electrical and magnetic field ) on the electromagnetic irradiation in a holed spatial region with a non trivial topology ,by rewriting those Maxwell Equations on light of the Helmholtz theorem for vectorial fields .

Jefimenko’s Gravito-Electromagnetic Equations as an Implication of the Helmholtz Theorem, Adapted to Time-Varying Vector Fields

Using the Helmholtz theorem (modified to time-varying vector fields) the Jefimenko’s gravito-electromagnetic equations are deduced. Newton’s gravitational field emerges as a particular case when an action at a distance theory is considered.

Helmholtz Theorem for Nondifferentiable Hamiltonian Systems in the Framework of Cresson’s Quantum Calculus

We derive the Helmholtz theorem for nondifferentiable Hamiltonian systems in the framework of Cresson’s quantum calculus. Precisely, we give a theorem characterizing nondifferentiable equations, admitting a Hamiltonian formulation. Moreover, in the affirmative case, we give the associated Hamiltonian.

Relativistic analysis of application of Helmholtz theorem to classical electrodynamics

In this work we discuss the relationship between the instantaneous-action-at-a-distance solutions of Maxwell’s equations obtained using Helmholtz theorem and the Lorentz’s invariant solutions of the same equations obtained using Special Relativity postulates. We show that Special Relativity postulates are not consistent with Helmholtz’s theorem in the presence of charges and currents, but in the vacuum, without charges and currents, Helmholtz’s theorem and Special Relativity agree because the instantaneous-action-at-a-distance solutions can be eliminated using a gauge transformation.

An Application of the Helmholtz Theorem in Extracting the Externally Induced Deformation Field from the Total Wind Field in a Limited Domain

A new application of the Helmholtz theorem that divides the horizontal wind into purely rotational, purely divergent, and harmonic deformational flow is put forward in this study. The formulas and methods, with consideration of avoiding the nonunique problem in solving the two Poisson equations with coupled boundary conditions, are constructed and tested for applicability and accuracy in an ideal experiment. Numerical tests show that the three extracted components together with the reconstructed wind almost recover the ideal fields by using the extended Helmholtz theorem, while other methods fail. The physical meaning of the extracted externally induced deformational flow is explained in the context of its connection to frontogenesis in a real weather event. It may provide potential usage in analyzing certain types of real weather.