Study of Nonlinear Dynamics Using Logistic Map

INTRODUCTION NONLINEAR DYNAMICS

Nonlinear dynamics is concerned with the nonlinear systems whose time evolution equations (differential equations) are nonlinear; i.e., the dynamical variables describing the system properties appear in the equation in a nonlinear form. Nonlinear systems have always played an important role in the study of natural phenomena, but the last few decades have seen an intensified interest and renewed vigour in nonlinear systems research (Robert C. Hilborn, 1994). The main reason for this growth is the recent availability of inexpensive computing power. Unlike linear systems, which have closed form solutions, few nonlinear systems possess closed form solutions and therefore, numerical simulations play a crucial role in the process of finding and analyzing nonlinear phenomenon. Before the advent of low cost computers, the ability to perform nonlinear simulations was restricted to researchers with access to a large computing facility; now, anyone with a personal computer may simulate a nonlinear system (Robert C. Hilborn, 1994;Paul S Addison, 1997). One of the elementary tenets of science is that deterministic systems are predictable: given the initial condition and the equations describing a system, the behaviour of the system can be predicted for all time. The discovery of chaotic systems has eliminated this point of view. Simply put, a chaotic system is a deterministic system that demonstrates random behaviour. Chaos also called strange behaviour is currently one of the most exciting topics in nonlinear systems research (Soltani et al, 2003). Chaos is an aperiodic behaviour in a deterministic system that shows sensitive dependence on initial conditions. Basically it is a phenomenon that occurs widely in nonlinear dynamical systems. Chaotic system exhibits apparently random and unpredictable behaviour. In a deterministic system starting from an exactly known initial condition we can repeat the sequence of outcomes as many times as we feel like. Whereas, in a pure random (probabilistic) system the sequence of the outcomes can not be repeated (Heinz-Otto Peitgen et al, 1992).

Generally, chaos has the following characteristics: 1. A power spectrum with a continuous part.

2. An infinite number of periodic solutions to the associated differential equation, each solution being unstable.

3. Extreme sensitivity of the trajectories with respect to the initial conditions 4. Extreme sensitivities of the trajectories with respect to parameters.

Consequently chaos may be described as a bounded, aperiodic, and noisy like oscillation: a deterministic system appears to behave randomly even though there is no random input. In unstable nonlinear systems a variety of strange effects are observed including subharmonics, quasiperiodic oscillation, intermittency and chaotic behaviour; erratic apparently random motion (Steven H. Strogatz, 1994). The term bifurcation means an abrupt variation in the behaviour of a nonlinear system when some parameters e.g., input or feedback are changed. Therefore, bifurcations are a potential source of engineering malfunction and failure. There are many types of bifurcations, e.g., period doubling or subharmonic bifurcation, flip bifurcation, fold bifurcation, pitchfork bifurcation, transcritical bifurcation, etc (Steven H. Strogatz, 1994;Dettmer, 1993).

NONLINEARITY AND CHAOS

All real systems are nonlinear at least to some extent. Some abrupt and dramatic changes in nonlinear systems may give rise to the complex behaviour called chaos. The words, chaos and chaotic are employed to describe the time behaviour of a system when that behaviour is aperiodic (it never exactly repeats) and is "apparently" random or noisy. Beneath this apparent chaotic randomness is an order determined by the system equations. Actually, most of the chaotic systems are completely deterministic (Steven H. Strogatz, 1994).

Generally, the behaviour of a system can be determined by these guidelines: 1. The time evolution equations 2. The parameter values describing the equation 3. The initial conditions A system is deterministic if the above given guidelines in principle completely determine the subsequent behaviour of the system i.e., a system in which the later states of the system follow from, or are determined by, the earlier ones. Such a system contrasts with a "stochastic or random system" in which future states are not determined from previous ones e.g., sequence of heads or tails of an unbiased coin, or radioactive decay (Robert C. Hilborn, 1994).

If a system is deterministic, this doesn't necessarily imply that later states of the system are predictable from knowledge of the earlier ones. In this way, chaos is similar to a random system. For example, chaos has been termed "deterministic chaos" since, although it is determined by simple rules, its property of sensitive dependence on initial conditions makes a chaotic system, in practice, largely unpredictable. Hence uunpredictable behaviour of deterministic system is called "Chaos". The crux of the problem is to harmonize this underlying determinism with the apparent randomness (Robert C. Hilborn, 1994;Soltani et al, 2003).

LOGISTIC EQUATION

The logistic equation also called the "Verhulst model" is given by the equation:

Where r is a parameter representing growth rate and x n is the variable at the nth iteration and n is the running variable. This is a model of population growth first published by Pierre François Verhulst (1804-1849). The discrete version of the logistic equation is known as the "logistic map". The logistic map (a map is simply a function, f, on the phase space that gives the next state, f (z) (the image), of the system given its current state, z. In the notation z' = f (z), the prime means the next point, not the derivative.) is a polynomial mapping, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple nonlinear dynamical equations. It was popularized in a seminal paper by the biologist Robert May (Robert C. Hilborn, 1994). The logistic map is a non-invertible map i.e., the map can be iterated forward in time with each x n leading to a unique subsequent value x n+1 , the reverse is not true. It is also called "iterated map function" as it maps one value of x, say x 0 , into another value of x that we call x 1 . By repeatedly iterating the logistic map, various kinds of behaviors can be observed. The sequence of iterated solutions to the map is called a "trajectory" as well as "orbit" (Paul S Addison, 1997).

BIFURCATION DIAGRAM

A bifurcation is a qualitative change in the dynamics that occurs as a system parameter is changed. A bifurcation diagram shows the possible long term values a variable of a system can obtain as function of a parameter of the system. An example is the bifurcation diagram of the logistic map. In this case, the parameter r is shown on the horizontal axis of the plot and the vertical axis shows the density of the possible long term population values of the logistic function (Paul S Addison, 1997; Robert C. Hilborn, 1994). shows that for r less than one, all the points are plotted at zero. Zero is the one point attractor for r less than 1. For r between 1 and 3, we still have one point attractors, but the 'attracted' value of x increases as r increases, at least to r = 3. Bifurcations occur at r = 3, r = 3.45, 3.54, 3.564, 3.569 etc., until just beyond 3.57, where the multiple unstable periodic oscillations, which means chaotic behaviour. We can no longer see any oscillations. Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos (Heinz-Otto Peitgen et al, 1992;Paul S Addison, 1997). The period doubling sequence occurred through the bifurcation (splitting into two) of the previous fixed points when they become unstable. It nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. This splitting is known as "Pitchfork Bifurcation'. At each period doubling bifurcation point the previously stable attracting periodic fixed point becomes unstable and two new stable fixed points emerge. This equation demonstrates how deterministic systems (the stable outcomes) can, when pushed, produce unpredictable and chaotic outcomes. At higher values of r (3.5-4) the system demonstrates sensitive dependence on initial conditions i.e., minor changes in the value of r results in markedly different outcomes (Heinz-Otto Peitgen et al, 1992). An interesting feature of this diagram is that as the periods go to infinity, r remains finite. When r is greater than approximately 3.57, the orbits become chaotic. Hence this bifurcation diagram demonstrates a nice example of the importance of chaos theory in even very simple nonlinear systems. Bifurcation diagram is the basic tool of studying the change in system behaviour in response to the variation of system parameters (Heinz-Otto Peitgen et al, 1992;Paul S Addison, 1997).

MATHEMATICAL RESULTS

Fig 2

In fig. 2, for k=0.2, the first thirty iterated solutions to the logistic map X n are plotted against n. This final solution is known as "period-1 orbit", as the iterates tend to a fixed value where x n+1 = x n for large n. fig 4, the solutions rapidly converge to two alternating attracting fixed points. These solutions to the map oscillate between two values forever, i.e., x n+2 = x n . This is termed as "period-2 orbit".

Fig 5

In fig 5 for k=3.52, the solutions to the logistic map repeat every fourth value, i.e., x n+4 = x n . This is called "period-4 orbit". In it the iterated solutions do not converge and further iterations will never produce a repeating and hence periodic sequence of solutions. This aperiodic behaviour is known as "chaotic orbit" or simply "chaos".

CONCLUSIONS

Nonlinear systems show nonlinear dynamics. Practically, all real systems are nonlinear. Almost all power electronics circuits exhibit nonlinear behavior e.g., quasi-periodicity, subharmonic oscillations, bifurcations and chaos. We have tried to observe the nonlinear behavior shown by the logistic map in this paper.