Some open problems in chaos theory and dynamics

Here we propose six open problems in dynamical systems and chaos theory. The first open problem is concern with 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 demension on one of the Poincaré sections. The sixth problem is on rigorous proof of existence of chaotic solutions of some sysyems which exhibit chaos in numerical solutions. 1 First Open Problem Several years ago Zhang and Heidel ( [10] (1999) and [23] (1997)) showed that (almost) all dissipative and conservative three dimensional autonomous quadratic systems of ordinary differential equations with at most four terms on the right hand sides of the equations are non-chaotic. The sole exception is the system    x′ = y2 − z2 y′ = x z′ = y (1.1) which does however appear numerically to have a single unstable periodic solution and is therefore conjectured to also be non-chaotic. The above system is equivalent to the third order or jerk equation z′′′ = z′2 − z2. Very recently Malasoma [14] has shown that every jerk equation z′′′ = j(z, z′, z′′) where j is a quadratic polynomial with at most two terms, with ∗Department of Mathematics, University of Nebraska at Omaha, Omaha NE, 68182, E-mail: [email protected] †Department of Mathematics, Cheyney University of PA, Cheyney, PA 19319, E-mail: [email protected], [email protected]