Integral Value Transformations: A Class of Discrete Dynamical Systems

IVTs have been studied on similar lines as Cellular automata, introduced by Von Neumann. Cellular automata have been studied extensively over the past forty years since its inception by computer scientists and mathematicians alike and yielded many beautiful results. The characterisation of Cellular automata by Hedlund [5] provided a new dimension of looking at these functions and thus started the endeavours of studying Cellular automata through topological dynamics. .

A semi-group acting on a space M is called a dynamical system if a mapping T: G x M → M defined as T (g, x) = such that . Further, if G = or G = Z, then the system is called a Discrete Dynamical System [DDS] [5,6].

IVTs form a discrete dynamical system when applied iteratively and this opens up a vast unexplored area. It is interesting to see how these functions evolve over time, form chaotic patterns, etc. The real motivation is to make an attempt at understanding how these IVTs evolve over time. Dynamical systems could throw some light in this aspect thereby aiding us in comprehending the time evolution of these functions. To meet this end, we first define a dynamical system of IVTs which is done in section 3 along with a few results.

Theorem-2.1: is a discrete dynamical system. The function T is defined as T: as Where

Clearly, is a semi-group acting on the space .

[Remark: by another definition,

Therefore, is a discrete dynamical system.

Here T (n, x) is the evolution function and n is the evolution parameter of the dynamical system and is the initial state/condition and is the state/phase space.

Corresponding to each p and j, we get a different dynamical system.

For a fix variable n, then = : is called the flow through x.

Further, if is an invertible map, then the dynamical system is invertible.

The orbit of is a set of points denoted by For any , x= will be a fixed point of i.e. ( ) if for each i=1,2,….,n.

A fixed point of is called a periodic point of period n of f and the set of periodic pints is denoted by Per (f).

Example: We know is a Collatz like function so we expect its orbit around any point to be finite and containing 0. Its orbit around

It’s worth noting that the orbits of the Merseene numbers are 3-point sets which consists of the number itself and 0 and 1 in case of . The orbits basically give the path/trajectory of convergence of the Collatz like functions.

, ( ) = { , }= { }

The orbit of any steady state equilibrium/fixed point will be the point itself. For , ( ) = { , }

It is possible to classify the orbits in the following categories of steady state equilibria/fixed points, periodic points and non-periodic points.

If is periodic of period n, then the orbit of

Finding the set of fixed points itself is an arduous task in the first place owing to the complexity of the functions. We will first look at the set of fixed points in a particular case and then make an attempt towards a generalisation.

The dynamical system of IVTs forms a Non-linear system and here it has been exploited the existing literature in this area to analyse the stability of fixed points. { } gives us the trajectory of the non-linear system and the iterative scheme is given by .

A steady state equilibrium of the equation is a point ̅ Є such that ̅ ̅ that is ̅ .

Stability Analysis of steady state equilibria of discrete dynamical systems is based on some propositions and/or explicit solution of the non-linear, autonomous (the parameters/coefficients a and b in the difference equation are independent of time), one-dimensional dynamical systems after reducing the non-linear system to a linear system.

A linear system is called locally stable if for a small perturbation to the system, it converges asymptotically to the original equilibrium. A linear system is called globally stable if irrespective of the extent of perturbation, it converges asymptotically to the original equilibrium. Mathematically, the definition is as follows: Through the Taylor series expansion of the non-linear system around the fixed point ̅ of (1), it can be reduced to a linear system around the steady state equilibrium ̅ to approximate the behaviour around the fixed point by a linear system. The Taylor series expansion is the following

where ̅ denotes the kth derivative of ̅. The concept of derivative in is defined in [3] The linearized system around the fixed point is ̅ ̅ ̅ neglecting the higher order terms Since the above linear system (2) is obtained by linearizing the non-linear system around ̅ (i.e; in a neighbourhood of ̅), therefore the global stability of the linear system (2)ensures only the local stability of the non-linear system (1).

Thus, the dynamical system (1) is locally stable around the steady state equilibrium/fixed point ̅ iff | ̅ | .

Thus we have established the conditi