History of science in the Renaissance

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Renaissance
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Leonardo da Vinci's Vitruvian Man, an example of the blend of art and science during the Renaissance

During the Renaissance, great advances occurred in geography, astronomy, chemistry, physics, mathematics, manufacturing, anatomy and engineering. The collection of ancient scientific texts began in earnest at the start of the 15th century and continued up to the Fall of Constantinople in 1453, and the invention of printing democratized learning and allowed a faster propagation of new ideas. Nevertheless, some have seen the Renaissance, at least in its initial period, as one of scientific backwardness. Historians like George Sarton and Lynn Thorndike criticized how the Renaissance affected science, arguing that progress was slowed for some amount of time. Humanists favored human-centered subjects like politics and history over study of natural philosophy or applied mathematics. More recently, however, scholars have acknowledged the positive influence of the Renaissance on mathematics and science, pointing to factors like the rediscovery of lost or obscure texts and the increased emphasis on the study of language and the correct reading of texts.

Marie Boas Hall coined the term Scientific Renaissance to designate the early phase of the Scientific Revolution, 1450–1630. More recently, Peter Dear has argued for a two-phase model of early modern science: a Scientific Renaissance of the 15th and 16th centuries, focused on the restoration of the natural knowledge of the ancients; and a Scientific Revolution of the 17th century, when scientists shifted from recovery to innovation.

Context

During and after the Renaissance of the 12th century, Europe experienced an intellectual revitalization, especially with regard to the investigation of the natural world. In the 14th century, however, a series of events that would come to be known as the Crisis of the Late Middle Ages was underway. When the Black Death came, it wiped out so many lives it affected the entire system. It brought a sudden end to the previous period of massive scientific change. The plague killed 25–50% of the people in Europe, especially in the crowded conditions of the towns, where the heart of innovations lay. Recurrences of the plague and other disasters caused a continuing decline of population for a century.

The Renaissance

The 14th century saw the beginning of the cultural movement of the Renaissance. By the early 15th century, an international search for ancient manuscripts was underway and would continue unabated until the Fall of Constantinople in 1453, when many Byzantine scholars had to seek refuge in the West, particularly Italy. Likewise, the invention of the printing press was to have great effect on European society: the facilitated dissemination of the printed word democratized learning and allowed a faster propagation of new ideas.

Initially, there were no new developments in physics or astronomy, and the reverence for classical sources further enshrined the Aristotelian and Ptolemaic views of the universe. Renaissance philosophy lost much of its rigor as the rules of logic and deduction were seen as secondary to intuition and emotion. At the same time, Renaissance humanism stressed that nature came to be viewed as an animate spiritual creation that was not governed by laws or mathematics. Only later, when no more manuscripts could be found, did humanists turn from collecting to editing and translating them, and new scientific work began with the work of such figures as Copernicus, Cardano, and Vesalius.

Important developments

Alchemy

Alchemy is the study of the transmutation of materials through obscure processes. It is sometimes described as an early form of chemistry. One of the main aims of alchemists was to find a method of creating gold from other substances. A common belief of alchemists was that there is an essential substance from which all other substances formed, and that if you could reduce a substance to this original material, you could then construct it into another substance, like lead to gold. Medieval alchemists worked with two main elements or principles, sulphur and mercury.

Paracelsus was an alchemist and physician of the Renaissance. The Paracelsians added a third principle, salt, to make a trinity of alchemical elements.

Astronomy

Pages from 1550 Annotazione on Sacrobosco's De sphaera mundi, showing the Ptolemaic system

The astronomy of the late Middle Ages was based on the geocentric model described by Claudius Ptolemy in antiquity. Probably very few practicing astronomers or astrologers actually read Ptolemy's Almagest, which had been translated into Latin by Gerard of Cremona in the 12th century. Instead they relied on introductions to the Ptolemaic system such as the De sphaera mundi of Johannes de Sacrobosco and the genre of textbooks known as Theorica planetarum. For the task of predicting planetary motions they turned to the Alfonsine tables, a set of astronomical tables based on the Almagest models but incorporating some later modifications, mainly the trepidation model attributed to Thabit ibn Qurra. Contrary to popular belief, astronomers of the Middle Ages and Renaissance did not resort to "epicycles on epicycles" in order to correct the original Ptolemaic models—until one comes to Copernicus himself.

Sometime around 1450, mathematician Georg Purbach (1423–1461) began a series of lectures on astronomy at the University of Vienna. Regiomontanus (1436–1476), who was then one of his students, collected his notes on the lecture and later published them as Theoricae novae planetarum in the 1470s. This "New Theorica" replaced the older theorica as the textbook of advanced astronomy. Purbach also began to prepare a summary and commentary on the Almagest. He died after completing only six books, however, and Regiomontanus continued the task, consulting a Greek manuscript brought from Constantinople by Cardinal Bessarion. When it was published in 1496, the Epitome of the Almagest made the highest levels of Ptolemaic astronomy widely accessible to many European astronomers for the first time.

Nicolaus Copernicus

The last major event in Renaissance astronomy is the work of Nicolaus Copernicus (1473–1543). He was among the first generation of astronomers to be trained with the Theoricae novae and the Epitome. Shortly before 1514 he began to revive Aristarchus's idea that the Earth revolves around the Sun. He spent the rest of his life attempting a mathematical proof of heliocentrism. When De revolutionibus orbium coelestium was finally published in 1543, Copernicus was on his deathbed. A comparison of his work with the Almagest shows that Copernicus was in many ways a Renaissance scientist rather than a revolutionary, because he followed Ptolemy's methods and even his order of presentation. Not until the works of Johannes Kepler (1571–1630) and Galileo Galilei (1564–1642) was Ptolemy's manner of doing astronomy superseded.

Mathematics

The accomplishments of Greek mathematicians survived throughout Late Antiquity and the Middle Ages by a long and indirect history. Much of the work of Euclid, Archimedes, and Apollonius, along with later authors such as Hero and Pappus, were copied and studied in both Byzantine culture and in Islamic centers of learning. Translations of these works began already in the 12th century, by the work of translators in Spain and Sicily working mostly from Arabic and Greek sources into Latin. Two of the most prolific were Gerard of Cremona and William of Moerbeke.

The greatest of all recoveries, however, took place in the 15th and 16th centuries in Italy, as attested by the numerous manuscripts dating from this period currently found in European libraries. Virtually all leading mathematicians of the era were obsessed with the need for restoring the mathematical works of the ancients. Not only did humanists assist mathematicians with the retrieval of Greek manuscripts, they also took an active role in translating these work into Latin, often commissioned by religious leaders such as Nicholas V and Cardinal Bessarion.

Some of the leading figures in this effort include Regiomontanus, who made a copy of the Latin Archimedes and had a program for printing mathematical works; Commandino (1509–1575), who likewise produced an edition of Archimedes, as well as editions of works by Euclid, Hero, and Pappus; and Maurolyco (1494–1575), who not only translated the work of ancient mathematicians but added much of his own work to these. Their translations ensured that the next generation of mathematicians would be in possession of techniques far in advance of what it was generally available during the Middle Ages.

It must be borne in mind that the mathematical output of the 15th and 16th centuries was not exclusively limited to the works of the ancient Greeks. Some mathematicians, such as Tartaglia and Luca Paccioli, welcomed and expanded on the medieval traditions of both Islamic scholars and people like Jordanus and Fibonnacci.

Medicine

With the Renaissance came an increase in experimental investigation, principally in the field of dissection and body examination, thus advancing our knowledge of human anatomy. The development of modern neurology began in the 16th century with Andreas Vesalius, who described the anatomy of the brain and other organs; he had little knowledge of the brain's function, thinking that it resided mainly in the ventricles. Understanding of medical sciences and diagnosis improved, but with little direct benefit to health care. Few effective drugs existed, beyond opium and quinine. William Harvey provided a refined and complete description of the circulatory system. The most useful tomes in medicine, used both by students and expert physicians, were materiae medicae and pharmacopoeiae.

Geography and the New World

In the history of geography, the key classical text was the Geographia of Claudius Ptolemy (2nd century). It was translated into Latin in the 15th century by Jacopo d'Angelo. It was widely read in manuscript and went through many print editions after it was first printed in 1475. Regiomontanus worked on preparing an edition for print prior to his death; his manuscripts were consulted by later mathematicians in Nuremberg.

The information provided by Ptolemy, as well as Pliny the Elder and other classical sources, was soon seen to be in contradiction to the lands explored in the Age of Discovery. The new discoveries revealed shortcomings in classical knowledge; they also opened European imagination to new possibilities. Thomas More's Utopia was inspired partly by the discovery of the New World.

Judicial activism

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Judicial activism is a judicial philosophy holding that the courts can and should go beyond the applicable law to consider broader societal implications of its decisions. It is sometimes used as an antonym of judicial restraint. It is usually a pejorative term, implying that judges make rulings based on their own political agenda rather than precedent and take advantage of judicial discretion. The definition of judicial activism and the specific decisions that are activist are controversial political issues. The question of judicial activism is closely related to judicial interpretation, statutory interpretation, and separation of powers.

Etymology

Arthur Schlesinger Jr. introduced the term "judicial activism" in a January 1947 Fortune magazine article titled "The Supreme Court: 1947".

The phrase has been controversial since its beginning. An article by Craig Green, "An Intellectual History of Judicial Activism," is critical of Schlesinger's use of the term; "Schlesinger's original introduction of judicial activism was doubly blurred: not only did he fail to explain what counts as activism, he also declined to say whether activism is good or bad."

Even before this phrase was first used, the general concept already existed. For example, Thomas Jefferson referred to the "despotic behaviour" of Federalist federal judges, in particular Chief Justice John Marshall.

Definitions

A survey of judicial review in practice during the last three decades shows that 'Judicial Activism' has characterised the decisions of the Supreme Court at different times.

Black's Law Dictionary defines judicial activism as a "philosophy of judicial decision-making whereby judges allow their personal views about public policy, among other factors, to guide their decisions."

Political science professor Bradley Canon has posited six dimensions along which judge courts may be perceived as activist: majoritarianism, interpretive stability, interpretive fidelity, substance/democratic process, specificity of policy, and availability of an alternate policymaker. David A. Strauss has argued that judicial activism can be narrowly defined as one or more of three possible actions: overturning laws as unconstitutional, overturning judicial precedent, and ruling against a preferred interpretation of the constitution.

Others have been less confident of the term's meaning, finding it instead to be little more than a rhetorical shorthand. Kermit Roosevelt III has argued that "in practice 'activist' turns out to be little more than a rhetorically charged shorthand for decisions the speaker disagrees with"; likewise, the solicitor general under George W. Bush, Theodore Olson, said in an interview on Fox News Sunday, in regards to a case for same-sex marriage he had successfully litigated, that "most people use the term 'judicial activism' to explain decisions that they don't like." Supreme Court Justice Anthony Kennedy has said that, "An activist court is a court that makes a decision you don't like."

Debate

Detractors of judicial activism charge that it usurps the power of the elected branches of government or appointed agencies, damaging the rule of law and democracy. Defenders of judicial activism say that in many cases it is a legitimate form of judicial review, and that the interpretation of the law must change with changing times.

A third view is that so-called "objective" interpretation of the law does not exist. According to law professor Brian Z. Tamanaha, "Throughout the so-called formalist age, it turns out, many prominent judges and jurists acknowledged that there were gaps and uncertainties in the law and that judges must sometimes make choices." Under this view, any judge's use of judicial discretion will necessarily be shaped by that judge's personal and professional experience and his or her views on a wide range of matters, from legal and juridical philosophy to morals and ethics. This implies a tension between granting flexibility (to enable the dispensing of justice) and placing bounds on that flexibility (to hold judges to ruling from legal grounds rather than extralegal ones).

Some proponents of a stronger judiciary argue that the judiciary helps provide checks and balances and should grant itself an expanded role to counterbalance the effects of transient majoritarianism, i.e., there should be an increase in the powers of a branch of government which is not directly subject to the electorate, so that the majority cannot dominate or oppress any particular minority through its elective powers. Other scholars have proposed that judicial activism is most appropriate when it restrains the tendency of democratic majorities to act out of passion and prejudice rather than after reasoned deliberation.

Moreover, they argue that the judiciary strikes down both elected and unelected official action, in some instances acts of legislative bodies reflecting the view the transient majority may have had at the moment of passage and not necessarily the view the same legislative body may have at the time the legislation is struck down. Also, the judges that are appointed are usually appointed by previously elected executive officials so that their philosophy should reflect that of those who nominated them, that an independent judiciary is a great asset to civil society since special interests are unable to dictate their version of constitutional interpretation with threat of stopping political donations.

United States examples

The following rulings have been characterized^{[by whom?]} as judicial activism.

• Brown v. Board of Education – 1954 Supreme Court ruling ordering the desegregation of public schools.
• Roe v. Wade – 1973 Supreme Court ruling creating the constitutional right to an abortion.
• Bush v. Gore – The United States Supreme Court case between the major-party candidates in the 2000 presidential election, George W. Bush and Al Gore. The justices voted 5—4 to halt the recount of ballots in Florida and, as a result, George Bush was chosen as president.
• Citizens United v. Federal Election Commission – 2010 Supreme Court decision declaring Congressionally enacted limitations on corporate political spending and transparency as unconstitutional restrictions on free speech.
• Hollingsworth v. Perry – 2010 decision by Vaughn R. Walker for the United States District Court for the Northern District of California overturning California's constitutional amendment to ban same-sex marriage.
• Obergefell v. Hodges – 2015 Supreme Court decision declaring same-sex marriage as a right guaranteed under the Due Process Clause and the Fourteenth Amendment.
• Janus v. AFSCME – a 2018 Supreme Court decision addressing whether unions can require dues from all workers who benefit from collective bargaining agreements. The decision overturned the 41-year old precedent of Abood v. Detroit Board of Education.
• Department of Homeland Security v. Regents of the University of California – a 2020 Supreme Court decision addressing whether the Department of Homeland Security under U.S. President Donald Trump had the authority to dismantle the Deferred Action for Childhood Arrivals program initiated by Executive Order under Former U.S. President Barack Obama.

Outside the United States

While the term was first coined and is often used in the United States, it has also been applied in other countries, particularly common law jurisdictions.

India

India has a recent history of judicial activism, originating after the Emergency in India which saw attempts by the Government to control the judiciary. Public Interest Litigation was thus an instrument devised by the courts to reach out directly to the public, and take cognizance though the litigant may not be the victim. "Suo motu" cognizance allows the courts to take up such cases on its own. The trend has been supported as well criticized. New York Times author Gardiner Harris sums this up as

India’s judges have sweeping powers and a long history of judicial activism that would be all but unimaginable in the United States. In recent years, judges required Delhi’s auto-rickshaws to convert to natural gas to help cut down on pollution, closed much of the country’s iron-ore-mining industry to cut down on corruption and ruled that politicians facing criminal charges could not seek re-election. Indeed, India’s Supreme Court and Parliament have openly battled for decades, with Parliament passing multiple constitutional amendments to respond to various Supreme Court rulings.

All such rulings carry the force of Article 39A of the Constitution of India, although before and during the Emergency the judiciary desisted from "wide and elastic" interpretations, termed Austinian, because Directive Principles of State Policy are non-justiciable. This despite the constitutional provisions for judicial review and B R Ambedkar arguing in the Constituent Assembly Debates that "judicial review, particularly writ jurisdiction, could provide quick relief against abridgment of Fundamental Rights and ought to be at the heart of the Constitution."

Fundamental Rights as enshrined in the Constitution have been subjected to wide review, and have now been said to encompass a right to privacy, right to livelihood and right to education, among others. The 'basic structure' of the Constitution has been mandated by the Supreme Court not to be alterable, notwithstanding the powers of the Legislature under Article 368. This was recognized, and deemed not applicable by the High Court of Singapore in Teo Soh Lung v. Minister for Home Affairs.

Recent examples quoted include the order to Delhi Government to convert the Auto rickshaw to CNG, a move believed to have reduced Delhi's erstwhile acute smog problem (it is now argued to be back) and contrasted with that of Beijing.

Israel

The Israeli approach to judicial activism has transformed significantly in the last three decades, and currently presents an especially broad version of robust judicial review and intervention. Additionally, taking into consideration the intensity of public life in Israel and the challenges that the country faces (including security threats), the case law of the Israeli Supreme Court touches on diverse and controversial public matters.

The United Kingdom

The British courts were largely deferential towards their attitudes against the government before the 1960s. Since then, judicial activism has been well established throughout the UK. One of the first cases for this activism to be present was the Conway v Rimmer (1968). Previously, a claim like this would be defined as definitive, but the judges had slowly begun to adopt more of an activist line approach. This had become more prominent in which government actions were overturned by the courts. This can inevitably lead to clashes between the courts against the government as shown in the Miller case consisting of the 2016 Conservative government. The perceptions of judicial activism derived from the number of applications for judicial review made to the courts. This can be seen throughout the 1980s, where there about 500 applications within a year. This number dramatically increased as by 2013, there were 15,594 applications. This trend has become more frequent as time passes along, possibly pointing to a greater influence in the UK courts against the government. Along side with the amount of applications submitted to the courts, in some instances it has attracted media attention. For instance, in 1993, Jacob Rees-Mogg had challenged the Conservative government to ratify the Maastricht Treaty, which eventually had formed into the European Union. This was rejected by the Divisional Court and attracted large amounts of media attention to this case. Through these components it is largely evident that judicial activism should not be exaggerated. Ultimately, Judicial activism is greatly established throughout the UK as the courts are becoming more frequent to scrutinise at their own will, and at times, reject government legislation that the deem to be not within balance to the UK constitution and, becoming more visible.

Attractor

From Wikipedia, the free encyclopedia

Visual representation of a strange attractor.

In the mathematical field of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed.

In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space. In physical systems, the n dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in economic systems, they may be separate variables such as the inflation rate and the unemployment rate.

If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented geometrically in two or three dimensions, (as for example in the three-dimensional case depicted to the right). An attractor can be a point, a finite set of points, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor (see strange attractor below). If the variable is a scalar, the attractor is a subset of the real number line. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory.

A trajectory of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor, forward in time. The trajectory may be periodic or chaotic. If a set of points is periodic or chaotic, but the flow in the neighborhood is away from the set, the set is not an attractor, but instead is called a repeller (or repellor).

Motivation of attractors

A dynamical system is generally described by one or more differential or difference equations. The equations of a given dynamical system specify its behavior over any given short period of time. To determine the system's behavior for a longer period, it is often necessary to integrate the equations, either through analytical means or through iteration, often with the aid of computers.

Dynamical systems in the physical world tend to arise from dissipative systems: if it were not for some driving force, the motion would cease. (Dissipation may come from internal friction, thermodynamic losses, or loss of material, among many causes.) The dissipation and the driving force tend to balance, killing off initial transients and settle the system into its typical behavior. The subset of the phase space of the dynamical system corresponding to the typical behavior is the attractor, also known as the attracting section or attractee.

Invariant sets and limit sets are similar to the attractor concept. An invariant set is a set that evolves to itself under the dynamics. Attractors may contain invariant sets. A limit set is a set of points such that there exists some initial state that ends up arbitrarily close to the limit set (i.e. to each point of the set) as time goes to infinity. Attractors are limit sets, but not all limit sets are attractors: It is possible to have some points of a system converge to a limit set, but different points when perturbed slightly off the limit set may get knocked off and never return to the vicinity of the limit set.

For example, the damped pendulum has two invariant points: the point x₀ of minimum height and the point x₁ of maximum height. The point x₀ is also a limit set, as trajectories converge to it; the point x₁ is not a limit set. Because of the dissipation due to air resistance, the point x₀ is also an attractor. If there was no dissipation, x₀ would not be an attractor. Aristotle believed that objects moved only as long as they were pushed, which is an early formulation of a dissipative attractor.

Some attractors are known to be chaotic (see strange attractor), in which case the evolution of any two distinct points of the attractor result in exponentially diverging trajectories, which complicates prediction when even the smallest noise is present in the system.

Mathematical definition

Let t represent time and let f(t, •) be a function which specifies the dynamics of the system. That is, if a is a point in an n-dimensional phase space, representing the initial state of the system, then f(0, a) = a and, for a positive value of t, f(t, a) is the result of the evolution of this state after t units of time. For example, if the system describes the evolution of a free particle in one dimension then the phase space is the plane R² with coordinates (x,v), where x is the position of the particle, v is its velocity, a = (x,v), and the evolution is given by

Attracting period-3 cycle and its immediate basin of attraction for a certain parametrization of f(z) = z² + c. The three darkest points are the points of the 3-cycle, which lead to each other in sequence, and iteration from any point in the basin of attraction leads to (usually asymptotic) convergence to this sequence of three points.

${\displaystyle f(t,(x,v))=(x+tv,v).\ }$

An attractor is a subset A of the phase space characterized by the following three conditions:

• A is forward invariant under f: if a is an element of A then so is f(t,a), for all t > 0.
• There exists a neighborhood of A, called the basin of attraction for A and denoted B(A), which consists of all points b that "enter A in the limit t → ∞". More formally, B(A) is the set of all points b in the phase space with the following property:

For any open neighborhood N of A, there is a positive constant T such that f(t,b) ∈ N for all real t > T.

• There is no proper (non-empty) subset of A having the first two properties.

Since the basin of attraction contains an open set containing A, every point that is sufficiently close to A is attracted to A. The definition of an attractor uses a metric on the phase space, but the resulting notion usually depends only on the topology of the phase space. In the case of Rⁿ, the Euclidean norm is typically used.

Many other definitions of attractor occur in the literature. For example, some authors require that an attractor have positive measure (preventing a point from being an attractor), others relax the requirement that B(A) be a neighborhood. 

Types of attractors

Attractors are portions or subsets of the phase space of a dynamical system. Until the 1960s, attractors were thought of as being simple geometric subsets of the phase space, like points, lines, surfaces, and simple regions of three-dimensional space. More complex attractors that cannot be categorized as simple geometric subsets, such as topologically wild sets, were known of at the time but were thought to be fragile anomalies. Stephen Smale was able to show that his horseshoe map was robust and that its attractor had the structure of a Cantor set.

Two simple attractors are a fixed point and the limit cycle. Attractors can take on many other geometric shapes (phase space subsets). But when these sets (or the motions within them) cannot be easily described as simple combinations (e.g. intersection and union) of fundamental geometric objects (e.g. lines, surfaces, spheres, toroids, manifolds), then the attractor is called a strange attractor.

Fixed point

Weakly attracting fixed point for a complex number evolving according to a complex quadratic polynomial. The phase space is the horizontal complex plane; the vertical axis measures the frequency with which points in the complex plane are visited. The point in the complex plane directly below the peak frequency is the fixed point attractor.

A fixed point of a function or transformation is a point that is mapped to itself by the function or transformation. If we regard the evolution of a dynamical system as a series of transformations, then there may or may not be a point which remains fixed under each transformation. The final state that a dynamical system evolves towards corresponds to an attracting fixed point of the evolution function for that system, such as the center bottom position of a damped pendulum, the level and flat water line of sloshing water in a glass, or the bottom center of a bowl contain a rolling marble. But the fixed point(s) of a dynamic system is not necessarily an attractor of the system. For example, if the bowl containing a rolling marble was inverted and the marble was balanced on top of the bowl, the center bottom (now top) of the bowl is a fixed state, but not an attractor. This is equivalent to the difference between stable and unstable equilibria. In the case of a marble on top of an inverted bowl (a hill), that point at the top of the bowl (hill) is a fixed point (equilibrium), but not an attractor (stable equilibrium).

In addition, physical dynamic systems with at least one fixed point invariably have multiple fixed points and attractors due to the reality of dynamics in the physical world, including the nonlinear dynamics of stiction, friction, surface roughness, deformation (both elastic and plasticity), and even quantum mechanics. In the case of a marble on top of an inverted bowl, even if the bowl seems perfectly hemispherical, and the marble's spherical shape, are both much more complex surfaces when examined under a microscope, and their shapes change or deform during contact. Any physical surface can be seen to have a rough terrain of multiple peaks, valleys, saddle points, ridges, ravines, and plains. There are many points in this surface terrain (and the dynamic system of a similarly rough marble rolling around on this microscopic terrain) that are considered stationary or fixed points, some of which are categorized as attractors.

Finite number of points

In a discrete-time system, an attractor can take the form of a finite number of points that are visited in sequence. Each of these points is called a periodic point. This is illustrated by the logistic map, which depending on its specific parameter value can have an attractor consisting of 1 point, 2 points, 2ⁿ points, 3 points, 3×2ⁿ points, 4 points, 5 points, or any given positive integer number of points.

Limit cycle

A limit cycle is a periodic orbit of a continuous dynamical system that is isolated. Examples include the swings of a pendulum clock, and the heartbeat while resting. (The limit cycle of an ideal pendulum is not an example of a limit cycle attractor because its orbits are not isolated: in the phase space of the ideal pendulum, near any point of a periodic orbit there is another point that belongs to a different periodic orbit, so the former orbit is not attracting).

Van der Pol phase portrait: an attracting limit cycle

Limit torus

There may be more than one frequency in the periodic trajectory of the system through the state of a limit cycle. For example, in physics, one frequency may dictate the rate at which a planet orbits a star while a second frequency describes the oscillations in the distance between the two bodies. If two of these frequencies form an irrational fraction (i.e. they are incommensurate), the trajectory is no longer closed, and the limit cycle becomes a limit torus. This kind of attractor is called an N_t -torus if there are N_t incommensurate frequencies. For example, here is a 2-torus:

A time series corresponding to this attractor is a quasiperiodic series: A discretely sampled sum of N_t periodic functions (not necessarily sine waves) with incommensurate frequencies. Such a time series does not have a strict periodicity, but its power spectrum still consists only of sharp lines.

Strange attractor

A plot of Lorenz's strange attractor for values ρ = 28, σ = 10, β = 8/3

An attractor is called strange if it has a fractal structure. This is often the case when the dynamics on it are chaotic, but strange nonchaotic attractors also exist. If a strange attractor is chaotic, exhibiting sensitive dependence on initial conditions, then any two arbitrarily close alternative initial points on the attractor, after any of various numbers of iterations, will lead to points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.

The term strange attractor was coined by David Ruelle and Floris Takens to describe the attractor resulting from a series of bifurcations of a system describing fluid flow. Strange attractors are often differentiable in a few directions, but some are like a Cantor dust, and therefore not differentiable. Strange attractors may also be found in the presence of noise, where they may be shown to support invariant random probability measures of Sinai–Ruelle–Bowen type.

Examples of strange attractors include the double-scroll attractor, Hénon attractor, Rössler attractor, and Lorenz attractor.

Attractors characterize the evolution of a system

Bifurcation diagram of the logistic map. The attractor(s) for any value of the parameter r are shown on the ordinate in the domain ${\displaystyle 0<x<1}$. The colour of a point indicates how often the point ${\displaystyle (r,x)}$ is visited over the course of 10⁶ iterations: frequently encountered values are coloured in blue, less frequently encountered values are yellow. A bifurcation appears around ${\displaystyle r\approx 3.0}$, a second bifurcation (leading to four attractor values) around ${\displaystyle r\approx 3.5}$. The behaviour is increasingly complicated for ${\displaystyle r>3.6}$, interspersed with regions of simpler behaviour (white stripes).

The parameters of a dynamic equation evolve as the equation is iterated, and the specific values may depend on the starting parameters. An example is the well-studied logistic map, ${\displaystyle x_{n+1}=rx_{n}(1-x_{n})}$, whose basins of attraction for various values of the parameter r are shown in the figure. If ${\displaystyle r=2.6}$, all starting x values of ${\displaystyle x<0}$ will rapidly lead to function values that go to negative infinity; starting x values of ${\displaystyle x>1}$ will go to infinity. But for ${\displaystyle 0<x<1}$ the x values rapidly converge to ${\displaystyle x\approx 0.615}$, i.e. at this value of r, a single value of x is an attractor for the function's behaviour. For other values of r, more than one value of x may be visited: if r is 3.2, starting values of ${\displaystyle 0<x<1}$ will lead to function values that alternate between ${\displaystyle x\approx 0.513}$ and ${\displaystyle x\approx 0.799}$. At some values of r, the attractor is a single point (a "fixed point"), at other values of r two values of x are visited in turn (a period-doubling bifurcation), or, as a result of further doubling, any number k × 2ⁿ values of x; at yet other values of r, any given number of values of x are visited in turn; finally, for some values of r, an infinitude of points are visited. Thus one and the same dynamic equation can have various types of attractors, depending on its starting parameters.

Basins of attraction

An attractor's basin of attraction is the region of the phase space, over which iterations are defined, such that any point (any initial condition) in that region will asymptotically be iterated into the attractor. For a stable linear system, every point in the phase space is in the basin of attraction. However, in nonlinear systems, some points may map directly or asymptotically to infinity, while other points may lie in a different basin of attraction and map asymptotically into a different attractor; other initial conditions may be in or map directly into a non-attracting point or cycle.

Linear equation or system

A single-variable (univariate) linear difference equation of the homogeneous form ${\displaystyle x_{t}=ax_{t-1}}$ diverges to infinity if |a| > 1 from all initial points except 0; there is no attractor and therefore no basin of attraction. But if |a| < 1 all points on the number line map asymptotically (or directly in the case of 0) to 0; 0 is the attractor, and the entire number line is the basin of attraction.

Likewise, a linear matrix difference equation in a dynamic vector X, of the homogeneous form ${\displaystyle X_{t}=AX_{t-1}}$ in terms of square matrix A will have all elements of the dynamic vector diverge to infinity if the largest eigenvalue of A is greater than 1 in absolute value; there is no attractor and no basin of attraction. But if the largest eigenvalue is less than 1 in magnitude, all initial vectors will asymptotically converge to the zero vector, which is the attractor; the entire n-dimensional space of potential initial vectors is the basin of attraction.

Similar features apply to linear differential equations. The scalar equation ${\displaystyle dx/dt=ax}$ causes all initial values of x except zero to diverge to infinity if a > 0 but to converge to an attractor at the value 0 if a < 0, making the entire number line the basin of attraction for 0. And the matrix system ${\displaystyle dX/dt=AX}$ gives divergence from all initial points except the vector of zeroes if any eigenvalue of the matrix A is positive; but if all the eigenvalues are negative the vector of zeroes is an attractor whose basin of attraction is the entire phase space.

Nonlinear equation or system

Equations or systems that are nonlinear can give rise to a richer variety of behavior than can linear systems. One example is Newton's method of iterating to a root of a nonlinear expression. If the expression has more than one real root, some starting points for the iterative algorithm will lead to one of the roots asymptotically, and other starting points will lead to another. The basins of attraction for the expression's roots are generally not simple—it is not simply that the points nearest one root all map there, giving a basin of attraction consisting of nearby points. The basins of attraction can be infinite in number and arbitrarily small. For example, for the function ${\displaystyle f(x)=x^{3}-2x^{2}-11x+12}$, the following initial conditions are in successive basins of attraction:

Basins of attraction in the complex plane for using Newton's method to solve x⁵ − 1 = 0. Points in like-colored regions map to the same root; darker means more iterations are needed to converge.

2.35287527 converges to 4;

2.35284172 converges to −3;

2.35283735 converges to 4;

2.352836327 converges to −3;

2.352836323 converges to 1.

Newton's method can also be applied to complex functions to find their roots. Each root has a basin of attraction in the complex plane; these basins can be mapped as in the image shown. As can be seen, the combined basin of attraction for a particular root can have many disconnected regions. For many complex functions, the boundaries of the basins of attraction are fractals.

Partial differential equations

Parabolic partial differential equations may have finite-dimensional attractors. The diffusive part of the equation damps higher frequencies and in some cases leads to a global attractor. The Ginzburg–Landau, the Kuramoto–Sivashinsky, and the two-dimensional, forced Navier–Stokes equations are all known to have global attractors of finite dimension.

For the three-dimensional, incompressible Navier–Stokes equation with periodic boundary conditions, if it has a global attractor, then this attractor will be of finite dimensions.