First, the neural synergetics are discussed. Then based on the basic equations of synergetics, we derive quantitatively the Lorenz equations and Lorenz mode of brain, whose two wings correspond to two hemispheres of brain, and two hemispheres jump about, which describes thinking. It shows that life lies in cooperation in chaos. The neural synergetics possesses three levels: structure, activity ...

We present an algorithm for computing rigorous solutions to a large class of ordinary differential equations. The main algorithm is based on a partitioning process and the use of interval arithmetic with directed rounding. As an application, we prove that the Lorenz equations support a strange attractor, as conjectured by Edward Lorenz in 1963. This conjecture was recently listed by Steven Smal...

Lorenz like attractors are known to appear in unfoldings from certain codimension two homoclinic bifurcations for differential equations in R that possess a reflectional symmetry. This includes homoclinic loops under a resonance condition and the inclination flip homoclinic loops. We show that Lorenz like attractors also appear in the third possible codimension two homoclinic bifurcation (for h...

Article history: Received 15 August 2015 Accepted 28 September 2015 Available online 3 October 2015 Communicated by F. Porcelli

1 Problem Give expressions for the potentials of a Hertzian (point) oscillating dipole in various gauges. The form of Maxwell's equations for the fields E and B permit these fields to be related to potentials V and A according to, E = −∇V − 1 c ∂A ∂t , B ∇ × A, (1) in Gaussian units, where c is the speed of light in vacuum, and the potentials obeys the wave equations, ∇ 2 V + 1 c ∂ ∂t ∇ · A = −...

We announce and outline a proof of the existence of a homoclinic orbit of the Lorenz equations. In addition, we develop a shooting technique and two key conditions, which lead to the existence of a one-to-one correspondence between a set of solutions and the set of all infinite sequences of l's and 3's.

Recent progress of symbolic dynamics of oneand especially two-dimensional maps has enabled us to construct symbolic dynamics for systems of ordinary differential equations (ODEs). Numerical study under the guidance of symbolic dynamics is capable to yield global results on chaotic and periodic regimes in systems of dissipative ODEs which cannot be obtained neither by purely analytical means nor...

This paper investigates the problem of designing feedback controllers for regulating the output of the simplified Lorenz system, which is a novel chaotic system recently discovered by K.H. Sun and J.C. Sprott (2009). The simplified Lorenz chaotic system is a three-dimensional chaotic system with one bifurcation parameter and it includes the Lorenz attractor (1963) as a special case. In this pap...