Dynamical system in the context of "Strange attractor"

Self-replication is any behavior of a dynamical system that yields construction of an identical or similar copy of itself. Biological cells, given suitable environments, reproduce by cell division. During cell division, DNA is replicated and can be transmitted to offspring during reproduction. Biological viruses can replicate, but only by commandeering the reproductive machinery of cells through a process of infection. Harmful prion proteins can replicate by converting normal proteins into rogue forms. Computer viruses reproduce using the hardware and software already present on computers. Self-replication in robotics has been an area of research and a subject of interest in science fiction. Any self-replicating mechanism which does not make a perfect copy (mutation) will experience genetic variation and will create variants of itself. These variants will be subject to natural selection, since some will be better at surviving in their current environment than others and will out-breed them.

In physics, specifically classical mechanics, the three-body problem is to take the initial positions and velocities (or momenta) of three point masses orbiting each other in space and then to calculate their subsequent trajectories using Newton's laws of motion and Newton's law of universal gravitation.

Unlike the two-body problem, the three-body problem has no general closed-form solution, meaning there is no explicit formula for the positions of the bodies. When three bodies orbit each other, the resulting dynamical system is chaotic for most initial conditions, and the only way to predict the motions of the bodies is to estimate them using numerical methods.

A coupled human–environment system (known also as a coupled human and natural system, or CHANS) characterizes the dynamical two-way interactions between human systems (e.g., economic, social, cultural) and natural (e.g., hydrologic, atmospheric, biological, geological) systems. This coupling expresses the idea that the evolution of humans and environmental systems may no longer be treated as individual isolated systems. The complexity that CHANS research reveals is useful to inform policy decisions regarding environmental sustainability.

CHANS research is a broad field. Some research programs draw from, and build on, the perspectives developed in trans-disciplinary fields such as human ecology, ecological anthropology, environmental geography, economics, as well as others. In contrast, other research programs, such as Critical Zone science, aim to develop a more quantitative theoretic framework focusing on the development of analytical and numerical models, by building on theoretical advances in complex adaptive systems, complexity economics, dynamical systems theory, and the earth sciences. To some extent, all CHANS programs recognize the need to move beyond traditional research methods developed in the social and natural sciences, as these are not sufficient to quantify the highly nonlinear dynamics often present in CHANS. Some research into CHANS emulates the more traditional research programs that tended to separate the social from the ecological sciences.

In control theory, a time-invariant (TI) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is a function of the time-dependent input function. If this function depends only indirectly on the time-domain (via the input function, for example), then that is a system that would be considered time-invariant. Conversely, any direct dependence on the time-domain of the system function could be considered as a "time-varying system".

Mathematically speaking, "time-invariance" of a system is the following property:

Chaos theory is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause or prevent a tornado in Texas.

Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution and is fully determined by their initial conditions, with no random elements involved. In other words, despite the deterministic nature of these systems, this does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:

Complexity characterizes the behavior of a system or model whose components interact in multiple ways and follow local rules, leading to non-linearity, randomness, collective dynamics, hierarchy, and emergence.

The term is generally used to characterize something with many parts where those parts interact with each other in multiple ways, culminating in a higher order of emergence greater than the sum of its parts. The study of these complex linkages at various scales is the main goal of complex systems theory.

In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space in which the system moves, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored into smaller components. Ergodic theory is the study of systems possessing ergodicity.

Ergodic systems occur in a broad range of systems in physics and in geometry. This can be roughly understood to be due to a common phenomenon: the motions of particles, that is, geodesics, on a hyperbolic manifold are divergent; when that manifold is compact, that is, of finite size, those orbits return to the same general area, eventually filling the entire space.

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.

A macroeconomic model is an analytical tool designed to describe the operation of the problems of economy of a country or a region. These models are usually designed to examine the comparative statics and dynamics of aggregate quantities such as the total amount of goods and services produced, total income earned, the level of employment of productive resources, and the level of prices.

Macroeconomic models may be logical, mathematical, and/or computational; the different types of macroeconomic models serve different purposes and have different advantages and disadvantages. Macroeconomic models may be used to clarify and illustrate basic theoretical principles; they may be used to test, compare, and quantify different macroeconomic theories; they may be used to produce "what if" scenarios (usually to predict the effects of changes in monetary, fiscal, or other macroeconomic policies); and they may be used to generate economic forecasts. Thus, macroeconomic models are widely used in academia in teaching and research, and are also widely used by international organizations, national governments and larger corporations, as well as by economic consultants and think tanks.