Nonlinear Dynamics
Nonlinear dynamics describes systems in which the forces acting on those systems do not increase linearly with quantities such as position, mass or velocity. It also describes systems which exhibit a "sensitive dependence on their initial conditions." This expression was first coined in the 1960s by the meteorologist Edward Lorenz, who was working on weather prediction models. He realized that a weather parameter curve would diverge from the original input curve if the starting points of the two curves were infinitesimally different. This behavior is now referred to as the butterfly effect, i.e. small changes in initial conditions may lead to qualitatively and quantitatively different long-term behavior. It led Lorenz to conclude that it was impossible to predict the weather accurately because many parameters can cause slight changes to a weather system at one time thus causing the later behavior of the meteorological system to become completely different. This insight also led to the rapid development of chaos theory, a cornerstone of nonlinear dynamics. Mathematically, nonlinearity also implies that a small variation in an input quantity causes a considerable change in the response of the system for which linear algebraic methods are no longer adequate to find solutions to the stated problem.
Nonlinear dynamics are important because no model of a real system is truly linear and as such, nonlinear dynamics apply very concretely to a wide variety of problems in fields such as biology (complex patterns of neuronal growth), chemistry (diffusion processes in chemical reactions), economics (the irregular fluctuations of the stock market), engineering (instabilities at rigid interfaces), geology (shock wave patterns in earthquakes), medicine (properties of normal and pathological biological cells), materials science (stress propagation in crystals and glassy materials), meteorology (hurricane prediction) and mining (dynamics of fluidized flotation beds), to give but a few examples.
Aside from sensitive dependence on initial conditions, the distinctive behavior of nonlinear systems is also characterized by spatial or temporal symmetry-breaking, which means that the temporal response of the system is not the same as that of the force driving the system, or that the spatial patterns will be independent from the boundary conditions. A good example is the case of the nonlinear oscillator. A linear or simple harmonic oscillator, such as an object oscillating on a spring, is characterized by a type of motion described by Hooke's law and can be the sum of many contributing harmonic motions if the oscillator is complex. Such an oscillator will always be displaced by a force acting on the mass which increases linearly with its displacement. But the motions involved in the nonlinear oscillator are not the simple linear combinations of the harmonic oscillations and so the nonlinear oscillator will oscillate at several amplitudes unlike the linear oscillator which is characterized by a single resonant maximum amplitude. The contributing factors to the nonlinear behavior of such an oscillator are friction, damping forces, or other input elements which vary nonlinearly and which cause the oscillator to be described not by a parabolic potential curve, as is the case with the harmonic oscillator, but by a multitude of non-parabolic potentials. This translates into differential equations with solutions that can only be solved numerically, if at all, while the linear oscillator has analytical solutions.
Nonlinear dynamics describes instabilities, chaos, and pattern formation in systems driven far from thermodynamic equilibrium. Such systems are referred to as turbulent systems and they have a special importance in nonlinear dynamics. The major goal of turbulence physics, now understood to be a function of dynamical chaos in nonlinear systems, is to predict the effects of turbulence and investigate means of controlling it in applications as diverse as industrial mixers and burners, car engines, nuclear reactors, aircraft and ship turbines, and spray nozzles.
Examples of everyday nonlinear phenomena include the erosion pattern created by a stream of water as it flows along a path or a mound of sand generated by a gentle flow of sand piling up on the ground. The stream of water generates repetitive miniature mountains, ponds, and secondary streams as it winds through the soil. This is one of nature's most complex processes and it can be described by highly nonlinear fractal patterns. In the case of the sand, a complex equilibrium keeps the mound stable until it reaches the point of criticality, i.e. the point at which addition of more sand pebbles will cause collapse of the mound. After the collapse, a new mound starts building up if the sand flow is not interrupted. Each new failure cycle relieves pressure on the mound and similar patterns of alternating growth and failure are characteristic of many complex systems. But no matter how distinct such complex systems may be, they all reach their respective points of criticality via bifurcations, or dramatic changes of behavior which occur over extremely small parameter ranges. Whereas linear dynamical systems depend on quantities such as temperature or pressure which, if changed gradually, produce predictable and gradual variations in the system, nonlinear systems become unpredictable when the point of criticality is abruptly exceeded. Bifurcation theory studies the onset of turbulence in nonlinear systems as some control parameter is continuously varied.
Another important theoretical concept in nonlinear dynamics is that of phase space, which consists of the independent dynamical variables used as coordinates to define the space for the system and represents the collection of possible states for that system. An example is the coin toss which has two states in phase space: heads or tails. For a deterministic system, the future state of the system depends on the current state of the system, represented as a point in phase space. The classical pendulum also exists in two-dimensional phase space and according to Newton's laws, knowing its two states (i.e., position and velocity), we can predict its subsequent motion. One or two-dimensional phase space does not exhibit nonlinear behavior, but it becomes possible in three-dimensional phase spaces, which often consist of a classical two-phase state system to which the time dimensionality is added. As the system progresses in time, a trajectory of its state-space evolution is obtained and can be mapped to yield a function of the phase space that will yield the next state of the system. These phase maps have geometrical properties and nonlinear systems are often characterized by chaotic fractal patterns.
Nonlinear dynamics have had a crucial impact on 20th century physics. Its contributions are considered as important as those of quantum mechanics and relativity in shaping the new physics because manifestations of nonlinearity are an inescapable part of our real physical world.
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