Frontiers in the Study of Chaotic Dynamical Systems with Open Problems

The following sections are included:

• Preface

• Contents

In this paper, we first discuss the problems that exist in the modeling used in Lorenz's chaos theory by employing formal mathematical logic and the underlying physical meanings. Then we analyze in detail the problems found in computing Lorenz's model by employing the commonly employed schemes of computational mathematics. Our results indicate that the resultant chaos doctrine is not the chaos that appears in the physical events as Lorenz described; instead it involves the error values of the mathematical differences of quasi-equal quantities, producing the apparent chaos. Therefore, the problem of how to comprehend indeterminacy emerges.

We discuss some important issues arising from computational efforts in dynamical systems and fluid dynamics. Various individuals have misunderstood these issues since the onset of these problem areas; indeed, they have been routinely misinterpreted, and even viewed as "laws" by some. This paper hopes to stimulate appropriate corrections and to realign thinking, with the overall goal being sound future progress in dynamical systems and fluid dynamics.

Here we propose six open problems in dynamical systems and chaos theory. The first open problem is concerned with a rigorous proof of a collection of quadratic ODE systems being non-chaotic. The second problem is for a universal definition of non-chaotic solutions. The third problem is about the number of systems that can have chaotic solutions when the right hand sides are polynomials. The fourth problem is topologically how complicated a 2D invariant manifold has to be to contain and/or attract chaotic solutions. The fifth open problem is to show that a specific system has a solution with a fractal dimension on one of the Poincaré sections. The sixth problem is on rigorous proof of existence of chaotic solutions of some systems which exhibit chaos in numerical solutions.

Dynamical systems described by real and complex variables are currently one of the most popular areas of scientific research. These systems play an important role in several fields of physics, engineering, and computer sciences, for example, laser systems, control (or chaos suppression), secure communications, and information science. Dynamical basic properties, chaos (hyperchaos) synchronization, chaos control, and generating hyperchaotic behavior of these systems are briefly summarized. The main advantage of introducing complex variables is the reduction of phase space dimensions by a half. They are also used to describe and simulate the physics of detuned laser and thermal convection of liquid flows, where the electric field and the atomic polarization amplitudes are both complex. Clearly, if the variables of the system are complex the equations involve twice as many variables and control parameters, thus making it that much harder for a hostile agent to intercept and decipher the coded message. Chaotic and hyperchaotic complex systems are stated as examples. Finally there are many open problems in the study of chaotic and hyperchaotic complex nonlinear dynamical systems, which need further investigations. Some of these open problems are given.

Relevance of electronic circuits in the study of nonlinear dynamical systems is now a proven fact. After the advent of Chua's Circuit, there has been tremendous development, and subsequent research have seen to be highly fruitful. In this paper, we have developed a new chaotic system with the help of an electronic circuit which was made from usual electronic components, and analyzed its subsequent output. The corresponding time series and attractor are studied from the topological point of view. It is observed that the Poincaré map leads to the binary symbolic representation of the kneading sequence and the attractor is populated by quite a high number of period 2, 3 and 4 orbits. Later the linking number and the torsion of the system are computed, and the template is designed leading to a global understanding of chaotic scenario.

The main purpose of this paper is to introduce some open problems related to some nonlinear differential equations of fourth and fifth order with constant delay.

We study the network synchronizability in terms of its local dynamics and in terms of its global dynamics. In previous work, we obtained some results about the synchronization interval, when we fix the graph underlying the network, i.e., the network connection topology and vary the dynamic in the nodes and about the effect of some graph parameters on the synchronization interval, supposing that the local dynamics in the nodes is fixed. We present numerical simulations for several network types and some conjectures suggested by these simulations. In particular, we are interested in the effect of the clustering coefficient, the conductance and the clustering performance, on the network synchronizability.

The recent development of partial differential equation theory shows that this approach is a powerful tool for the wavelet analysis of non-stationary time series including those generated by chaotic systems. The present paper reviews main directions, where it is applicable, formulates current open problems and possible ways to their solution.

We introduce several problems involving non-autonomous discrete systems which are generated by periodic sequences of continuous maps.

Attractor of pair of complex linear maps together with the Mandelbrot set associated with this iterated function system were introduced by Barnsley and Harrington in 1985. Since that time many interesting properties of these sets have been found. We continue the investigation of this attractor by computing its code structure, the notion introduced by Barnsley in 2005.

Much recent interest has focused on "open" dynamical systems, in which a classical map or flow is considered only until the trajectory reaches a "hole", at which the dynamics is no longer considered. Here we consider questions pertaining to the survival probability as a function of time, given an initial measure on phase space. We focus on the case of billiard dynamics, namely that of a point particle moving with constant velocity except for mirror-like reflections at the boundary, and give a number of recent results, physical applications and open problems.

Genes influence the expression of each other through a complex, nonlinear, dynamical network of interactions. There are a number of interesting open questions about what kind of information can be determined about the structure and dynamics of this network from limited experimental data.

At the present time, strange attractors can be classified into three principal classes: hyperbolic, Lorenz-type, and quasi-attractors. The hyperbolic attractors are the limit sets for which Smale's Axiom A is satisfied and are structurally stable. Periodic and homoclinic orbits are dense and are of the same saddle type. The Lorenz-type attractors are not quite structurally stable, although their homoclinic and heteroclinic orbits are structurally stable (hyperbolic), and no stable periodic orbits appear under small parameter variations. The quasi-attractors are the limit sets enclosing periodic orbits of different topological types and structurally unstable orbits.

In this paper, some new open problems are proposed. These problems are related to transforming non-chaotic dynamical systems to a specific defined type of chaos (one of the three types defined above) and transforming a defined type of chaos to another defined one (also one of the three types defined above). These questions have not been previously addressed in the literature. Solving such a problem opens an interesting field in chaos studies concerned with the classification and determination of the type of chaos observed experimentally, proved analytically, or tested numerically in theory and practice.

The following sections are included:

• Author Index

• Subject Index