Narrow Results

One purpose of this paper is to document the fact that, in dynamical systems described by ordinary differential equations, the trajectories can be organized not only around fixed points (steady states), but also around lines. In 2D, these lines are the nullclines themselves, in 3D, the intersections of the nullclines two by two, etc.

We precise the concepts of "partial steady states" (i.e. steady states in a subsystem that consists of sections of phase space by planes normal to one of the axes) and of "partial multistationarity" (multistationarity in such a subsystem).

Steady states, nullclines or their intersections are revisited in terms of circuits, defined from nonzero elements of the Jacobian matrix. It is shown how the mere examination of the Jacobian matrix and the sign patterns of its circuits can help interpreting (and often predicting) aspects of the dynamics of systems.

The results reinforce the idea that chaotic dynamics requires both a positive circuit, to provide (if only partial) multistationarity, and a negative circuit, to provide sustained oscillations. As shown elsewhere, a single circuit may suffice if it is ambiguous (i.e. positive or negative depending on the location in phase space).

The description in terms of circuits is by no means exclusive of the classical description. In many cases, a fruitful approach involves repeated feedback between the two viewpoints.

Using the Routh–Hurwitz stability criterion and a systematic computer search, 23 simple chaotic flows with quadratic nonlinearities were found that have the unusual feature of having a coexisting stable equilibrium point. Such systems belong to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.

Using a systematic computer search, four simple chaotic flows with cubic nonlinearities were found that have the unusual feature of having a curve of equilibria. Such systems belong to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.

Using a systematic computer search, a simple four-dimensional chaotic flow was found that has the unusual feature of having a plane of equilibria. Such a system belongs to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.

In this note, hidden attractors in chaotic maps are investigated. Although there are many new researches on hidden attractors in chaotic flows, no investigation has been done on hidden attractors in maps based on our knowledge. In addition, a new interesting chaotic map with a bifurcation diagram starting from any desired period and then continuing with period doubling is introduced in this paper.

Perpetual points represent a new interesting topic in the literature of nonlinear dynamics. This paper introduces some chaotic flows with four different structural features from the viewpoint of fixed points and perpetual points.

Designing chaotic systems with specific features is a very interesting topic in nonlinear dynamics. However most of the efforts in this area are about features in the structure of the equations, while there is less attention to features in the topology of strange attractors. In this paper, we introduce a new chaotic system with unique property. It has been designed in such a way that a specific property has been injected to it. This new system is analyzed carefully and its real circuit implementation is presented.

In this note, we define four main categories of conservative flows: (a) those in which the dissipation is identically zero, (b) those in which the dissipation depends on the state of the system and is zero on average as a consequence of the orbits being bounded, (c) those in which the dissipation depends on the state of the system and is zero on average, but for which the orbit need not be bounded and a different proof is required, and (d) those in which the dissipation depends on the initial conditions and cannot be determined from the equations alone. We introduce a new 3D conservative jerk flow to serve as an example of the first two categories and show what might be the simplest examples for each category. Also, we categorize some of the existing known systems according to these definitions.