A new autonomous 4D nonlinear model with two nonlinearities describing the dynamics of change of voltage and current in the contact railway electric network is offered. This model is a connection of two 2D oscillatory circuits for current and voltage in the contact electric network. In the found system for the defined values of parameters an existence of limit cycles is proved. By introduction of new variables this system can be reduced to 5D system only with one quadratic nonlinearity. The constructed model may be used for the control by voltage stability in a direct current power supply system.
Let [Formula: see text] be a chaotic attractor generated by a quadratic system of ordinary differential equations [Formula: see text]. A method for constructing new chaotic attractors based on the attractor [Formula: see text] is proposed. The idea of the method is to replace the state vector [Formula: see text] located on the right side of the original system with new vector [Formula: see text]; where [Formula: see text], [Formula: see text], and [Formula: see text] are odd power functions; [Formula: see text]. (In other words, a state feedback [Formula: see text] is introduced into the right side of the system under study: [Formula: see text].) As a result, the newly obtained system generates new chaotic attractors, which are topologically not equivalent (generally speaking) to the attractor [Formula: see text]. In addition, for an antisymmetric neural ODE system with a homoclinic orbit connected at a saddle point, the conditions for the occurrence of chaotic dynamics are found.
The normal system of ordinary differential equations, whose right-hand sides are the ratios of linear and nonlinear positive functions, is considered. A feature of these ratios is that some of their denominators can take on arbitrarily small nonzero values. (Thus, the modules of the corresponding derivatives can take arbitrarily large value.) In the sequel, the constructed system of differential equations is used to model strongly oscillating processes (for example, processes determined by the rhythms of electroencephalograms measured at certain points in the cerebral cortex). The obtained results can be used to diagnose human brain diseases.
Robust chaos is determined by the absence of periodic windows in bifurcation diagrams and coexisting attractors with parameter values taken from some regions of the parameter space of a dynamical system. Reliable chaos is an important characteristic of a dynamic system when it comes to its practical application. This property ensures that the chaotic behavior of the system will not deteriorate or be adversely affected by various factors. There are many methods for creating chaotic systems that are generated by adjusting the corresponding system parameters. However, most of the proposed systems are functions of well-known discrete mappings. In view of this, in this paper we consider a continuous system that illustrates some robust chaos properties.
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