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Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it.

At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manifold. The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval only one future state follows from the current state. However, some systems are stochastic, in that random events also affect the evolution of the state variables.

In physics, a dynamical system is described as a "particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives". In order to make a prediction about the system's future behavior, an analytical solution of such equations or their integration over time through computer simulation is realized.

The study of dynamical systems is the focus of dynamical systems theory, which has applications to a wide variety of fields such as mathematics, physics, biology, chemistry, engineering, economics, history, and medicine. Dynamical systems are a fundamental part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly and self-organization processes, and the edge of chaos concept.

Overview
The concept of a dynamical system has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is an implicit relation that gives the state of the system for only a short time into the future. (The relation is either a differential equation, difference equation or other time scale.) To determine the state for all future times requires iterating the relation many times—each advancing time a small step. The iteration procedure is referred to as solving the system or integrating the system. If the system can be solved, then, given an initial point, it is possible to determine all its future positions, a collection of points known as a trajectory or orbit.

Before the advent of computers, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system.

For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because:
 * The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions.  To address these questions several notions of stability have been introduced in the study of dynamical systems, such as Lyapunov stability or structural stability.  The stability of the dynamical system implies that there is a class of models or initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their equivalence changes with the different notions of stability.
 * The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic, whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class.  Classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes. Linear dynamical systems and systems that have two numbers describing a state are examples of dynamical systems where the possible classes of orbits are understood.
 * The behavior of trajectories as a function of a parameter may be what is needed for an application. As a parameter is varied, the dynamical systems may have bifurcation points where the qualitative behavior of the dynamical system changes. For example, it may go from having only periodic motions to apparently erratic behavior, as in the transition to turbulence of a fluid.
 * The trajectories of the system may appear erratic, as if random. In these cases it may be necessary to compute averages using one very long trajectory or many different trajectories.  The averages are well defined for ergodic systems and a more detailed understanding has been worked out for hyperbolic systems. Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of statistical mechanics and of chaos.

History
Many people regard French mathematician Henri Poincaré as the founder of dynamical systems. Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). These papers included the Poincaré recurrence theorem, which states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state.

Aleksandr Lyapunov developed many important approximation methods. His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamical system.

In 1913, George David Birkhoff proved Poincaré's "Last Geometric Theorem", a special case of the three-body problem, a result that made him world-famous. In 1927, he published his Dynamical Systems. Birkhoff's most durable result has been his 1931 discovery of what is now called the ergodic theorem. Combining insights from physics on the ergodic hypothesis with measure theory, this theorem solved, at least in principle, a fundamental problem of statistical mechanics. The ergodic theorem has also had repercussions for dynamics.

Stephen Smale made significant advances as well. His first contribution was the Smale horseshoe that jumpstarted significant research in dynamical systems. He also outlined a research program carried out by many others.

Oleksandr Mykolaiovych Sharkovsky developed Sharkovsky's theorem on the periods of discrete dynamical systems in 1964. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.

In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity. Palestinian mechanical engineer Ali H. Nayfeh applied nonlinear dynamics in mechanical and engineering systems. His pioneering work in applied nonlinear dynamics has been influential in the construction and maintenance of machines and structures that are common in daily life, such as ships, cranes, bridges, buildings, skyscrapers, jet engines, rocket engines, aircraft and spacecraft.

Formal definition
In the most general sense, a dynamical system is a tuple (T, X, Φ) where T is a monoid, written additively, X is a non-empty set and Φ is a function
 * $$\Phi: U \subseteq (T \times X) \to X$$

with
 * $$\mathrm{proj}_{2}(U) = X$$ (where $$\mathrm{proj}_{2}$$ is the 2nd projection map)

and for any x in X:
 * $$\Phi(0,x) = x$$
 * $$\Phi(t_2,\Phi(t_1,x)) = \Phi(t_2 + t_1, x),$$

for $$\, t_1,\, t_2 + t_1 \in I(x)$$ and $$\ t_2 \in I(\Phi(t_1, x)) $$, where we have defined the set $$ I(x) := \{ t \in T : (t,x) \in U \}$$ for any x in X.

In particular, in the case that $$ U = T \times X $$ we have for every x in X that $$ I(x) = T $$ and thus that Φ defines a monoid action of T on X.

The function Φ(t,x) is called the evolution function of the dynamical system: it associates to every point x in the set X a unique image, depending on the variable t, called the evolution parameter. X is called phase space or state space, while the variable x represents an initial state of the system.

We often write
 * $$\Phi_x(t) \equiv \Phi(t,x)$$
 * $$\Phi^t(x) \equiv \Phi(t,x)$$

if we take one of the variables as constant. The function
 * $$\Phi_x:I(x) \to X$$

is called the flow through x and its graph is called the trajectory through x. The set
 * $$\gamma_x \equiv\{\Phi(t,x) : t \in I(x)\}$$

is called the orbit through x. The orbit through x is the image of the flow through x. A subset S of the state space X is called Φ-invariant if for all x in S and all t in T
 * $$\Phi(t,x) \in S.$$

Thus, in particular, if S is Φ-invariant, $$I(x) = T$$ for all x in S. That is, the flow through x must be defined for all time for every element of S.

More commonly there are two classes of definitions for a dynamical system: one is motivated by ordinary differential equations and is geometrical in flavor; and the other is motivated by ergodic theory and is measure theoretical in flavor.

Geometrical definition
In the geometrical definition, a dynamical system is the tuple $$ \langle \mathcal{T}, \mathcal{M}, f\rangle $$. $$\mathcal{T}$$ is the domain for time – there are many choices, usually the reals or the integers, possibly restricted to be non-negative. $$\mathcal{M}$$ is a manifold, i.e. locally a Banach space or Euclidean space, or in the discrete case a graph. f is an evolution rule t → ft (with $$t\in\mathcal{T}$$) such that ft is a diffeomorphism of the manifold to itself. So, f is a "smooth" mapping of the time-domain $$ \mathcal{T}$$ into the space of diffeomorphisms of the manifold to itself. In other terms, f(t) is a diffeomorphism, for every time t in the domain $$ \mathcal{T}$$.

Real dynamical system
A real dynamical system, real-time dynamical system, continuous time dynamical system, or flow is a tuple (T, M, Φ) with T an open interval in the real numbers R, M a manifold locally diffeomorphic to a Banach space, and Φ a continuous function. If Φ is continuously differentiable we say the system is a differentiable dynamical system. If the manifold M is locally diffeomorphic to Rn, the dynamical system is finite-dimensional; if not, the dynamical system is infinite-dimensional. This does not assume a symplectic structure. When T is taken to be the reals, the dynamical system is called global or a flow; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow.

Discrete dynamical system
A discrete dynamical system, discrete-time dynamical system is a tuple (T, M, Φ), where M is a manifold locally diffeomorphic to a Banach space, and Φ is a function. When T is taken to be the integers, it is a cascade or a map. If T is restricted to the non-negative integers we call the system a semi-cascade.

Cellular automaton
A cellular automaton is a tuple (T, M, Φ), with T a lattice such as the integers or a higher-dimensional integer grid, M is a set of functions from an integer lattice (again, with one or more dimensions) to a finite set, and Φ a (locally defined) evolution function. As such cellular automata are dynamical systems. The lattice in M represents the "space" lattice, while the one in T represents the "time" lattice.

Multidimensional generalization
Dynamical systems are usually defined over a single independent variable, thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called multidimensional systems. Such systems are useful for modeling, for example, image processing.

Compactification of a dynamical system
Given a global dynamical system (R, X, Φ) on a locally compact and Hausdorff topological space X, it is often useful to study the continuous extension Φ* of Φ to the one-point compactification X* of X. Although we lose the differential structure of the original system we can now use compactness arguments to analyze the new system (R, X*, Φ*).

In compact dynamical systems the limit set of any orbit is non-empty, compact and simply connected.

Measure theoretical definition
A dynamical system may be defined formally as a measure-preserving transformation of a measure space, the triplet (T, (X, Σ, μ), Φ). Here, T is a monoid (usually the non-negative integers), X is a set, and (X, Σ, μ) is a probability space, meaning that Σ is a sigma-algebra on X and μ is a finite measure on (X, Σ). A map Φ: X → X is said to be Σ-measurable if and only if, for every σ in Σ, one has $$\Phi^{-1}\sigma \in \Sigma$$. A map Φ is said to preserve the measure if and only if, for every σ in Σ, one has $$\mu(\Phi^{-1}\sigma ) = \mu(\sigma)$$. Combining the above, a map Φ is said to be a measure-preserving transformation of X , if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The triplet (T, (X, Σ, μ), Φ), for such a Φ, is then defined to be a dynamical system.

The map Φ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates $$\Phi^n = \Phi \circ \Phi \circ \dots \circ \Phi$$ for every integer n are studied. For continuous dynamical systems, the map Φ is understood to be a finite time evolution map and the construction is more complicated.

Relation to geometric definition
The measure theoretical definition assumes the existence of a measure-preserving transformation. Many different invariant measures can be associated to any one evolution rule. If the dynamical system is given by a system of differential equations the appropriate measure must be determined. This makes it difficult to develop ergodic theory starting from differential equations, so it becomes convenient to have a dynamical systems-motivated definition within ergodic theory that side-steps the choice of measure and assumes the choice has been made. A simple construction (sometimes called the Krylov–Bogolyubov theorem) shows that for a large class of systems it is always possible to construct a measure so as to make the evolution rule of the dynamical system a measure-preserving transformation. In the construction a given measure of the state space is summed for all future points of a trajectory, assuring the invariance.

Some systems have a natural measure, such as the Liouville measure in Hamiltonian systems, chosen over other invariant measures, such as the measures supported on periodic orbits of the Hamiltonian system. For chaotic dissipative systems the choice of invariant measure is technically more challenging. The measure needs to be supported on the attractor, but attractors have zero Lebesgue measure and the invariant measures must be singular with respect to the Lebesgue measure. A small region of phase space shrinks under time evolution.

For hyperbolic dynamical systems, the Sinai–Ruelle–Bowen measures appear to be the natural choice. They are constructed on the geometrical structure of stable and unstable manifolds of the dynamical system; they behave physically under small perturbations; and they explain many of the observed statistics of hyperbolic systems.

Construction of dynamical systems
The concept of evolution in time is central to the theory of dynamical systems as seen in the previous sections: the basic reason for this fact is that the starting motivation of the theory was the study of time behavior of classical mechanical systems. But a system of ordinary differential equations must be solved before it becomes a dynamic system. For example consider an initial value problem such as the following:

 * $$\dot{\boldsymbol{x}}=\boldsymbol{v}(t,\boldsymbol{x})$$
 * $$\boldsymbol{x}|_=\boldsymbol{x}_0$$

where
 * $$\dot{\boldsymbol{x}}$$ represents the velocity of the material point x
 * M is a finite dimensional manifold
 * v: T × M → TM is a vector field in Rn or Cn and represents the change of velocity induced by the known forces acting on the given material point in the phase space M. The change is not a vector in the phase space M, but is instead in the tangent space TM.

There is no need for higher order derivatives in the equation, nor for the parameter t in v(t,x), because these can be eliminated by considering systems of higher dimensions.

Depending on the properties of this vector field, the mechanical system is called
 * autonomous, when v(t, x) = v(x)
 * homogeneous when v(t, 0) = 0 for all t

The solution can be found using standard ODE techniques and is denoted as the evolution function already introduced above

 * $$\boldsymbol(t)=\Phi(t,\boldsymbol_0)$$

The dynamical system is then (T, M, Φ).

Some formal manipulation of the system of differential equations shown above gives a more general form of equations a dynamical system must satisfy

 * $$\dot{\boldsymbol{x}}-\boldsymbol{v}(t,\boldsymbol{x})=0 \qquad\Leftrightarrow\qquad \mathfrak\left(t,\Phi(t,\boldsymbol_0)\right)=0$$

where $$\mathfrak{G}:{{(T\times M)}^M}\to\mathbf{C}$$ is a functional from the set of evolution functions to the field of the complex numbers.

This equation is useful when modeling mechanical systems with complicated constraints.

Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally Banach spaces—in which case the differential equations are partial differential equations.

Examples

 * Arnold's cat map
 * Baker's map is an example of a chaotic piecewise linear map
 * Billiards and outer billiards
 * Bouncing ball dynamics
 * Circle map
 * Complex quadratic polynomial
 * Double pendulum
 * Dyadic transformation
 * Hénon map
 * Irrational rotation
 * Kaplan–Yorke map
 * List of chaotic maps
 * Lorenz system
 * Quadratic map simulation system
 * Rössler map
 * Swinging Atwood's machine
 * Tent map

Linear dynamical systems
Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the N-dimensional Euclidean space, so any point in phase space can be represented by a vector with N numbers. The analysis of linear systems is possible because they satisfy a superposition principle: if u(t) and w(t) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will u(t) + w(t).

Flows
For a flow, the vector field v(x) is an affine function of the position in the phase space, that is,
 * $$ \dot{x} = v(x) = A x + b,$$

with A a matrix, b a vector of numbers and x the position vector. The solution to this system can be found by using the superposition principle (linearity). The case b ≠ 0 with A = 0 is just a straight line in the direction of b:

 * $$\Phi^t(x_1) = x_1 + b t. $$

When b is zero and A ≠ 0 the origin is an equilibrium (or singular) point of the flow, that is, if x0 = 0, then the orbit remains there. For other initial conditions, the equation of motion is given by the exponential of a matrix: for an initial point x0,

 * $$\Phi^t(x_0) = e^{t A} x_0. $$

When b = 0, the eigenvalues of A determine the structure of the phase space. From the eigenvalues and the eigenvectors of A it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin.

The distance between two different initial conditions in the case A ≠ 0 will change exponentially in most cases, either converging exponentially fast towards a point, or diverging exponentially fast. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for chaotic behavior.

Maps
A discrete-time, affine dynamical system has the form of a matrix difference equation:
 * $$ x_{n+1} = A x_n + b, $$

with A a matrix and b a vector. As in the continuous case, the change of coordinates x → x + (1 − A)–1b removes the term b from the equation. In the new coordinate system, the origin is a fixed point of the map and the solutions are of the linear system Anx0. The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map.

As in the continuous case, the eigenvalues and eigenvectors of A determine the structure of phase space. For example, if u1 is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along α u1, with α ∈ R, is an invariant curve of the map. Points in this straight line run into the fixed point.

There are also many other discrete dynamical systems.

Local dynamics
The qualitative properties of dynamical systems do not change under a smooth change of coordinates (this is sometimes taken as a definition of qualitative):  a singular point of the vector field (a point where v(x) = 0) will remain a singular point under smooth transformations; a periodic orbit is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates (usually unspecified, but computable) that makes the dynamical system as simple as possible.

Rectification
A flow in most small patches of the phase space can be made very simple. If y is a point where the vector field v(y) ≠ 0, then there is a change of coordinates for a region around y where the vector field becomes a series of parallel vectors of the same magnitude. This is known as the rectification theorem.

The rectification theorem says that away from singular points the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space M the dynamical system is integrable. In most cases the patch cannot be extended to the entire phase space. There may be singular points in the vector field (where v(x) = 0); or the patches may become smaller and smaller as some point is approached. The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one. If the next time the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches.

Near periodic orbits
In general, in the neighborhood of a periodic orbit the rectification theorem cannot be used. Poincaré developed an approach that transforms the analysis near a periodic orbit to the analysis of a map. Pick a point x0 in the orbit γ and consider the points in phase space in that neighborhood that are perpendicular to v(x0). These points are a Poincaré section S(γ, x0), of the orbit. The flow now defines a map, the Poincaré map F : S → S, for points starting in S and returning to S. Not all these points will take the same amount of time to come back, but the times will be close to the time it takes x0.

The intersection of the periodic orbit with the Poincaré section is a fixed point of the Poincaré map F. By a translation, the point can be assumed to be at x = 0. The Taylor series of the map is F(x) = J · x + O(x2), so a change of coordinates h can only be expected to simplify F to its linear part

 * $$ h^{-1} \circ F \circ h(x) = J \cdot x.$$

This is known as the conjugation equation. Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. Poincaré first approached it assuming all functions to be analytic and in the process discovered the non-resonant condition. If λ1, ..., λν are the eigenvalues of J they will be resonant if one eigenvalue is an integer linear combination of two or more of the others. As terms of the form λi – Σ (multiples of other eigenvalues) occurs in the denominator of the terms for the function h, the non-resonant condition is also known as the small divisor problem.

Conjugation results
The results on the existence of a solution to the conjugation equation depend on the eigenvalues of J and the degree of smoothness required from h. As J does not need to have any special symmetries, its eigenvalues will typically be complex numbers. When the eigenvalues of J are not in the unit circle, the dynamics near the fixed point x0 of F is called hyperbolic and when the eigenvalues are on the unit circle and complex, the dynamics is called elliptic.

In the hyperbolic case, the Hartman–Grobman theorem gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map J · x. The hyperbolic case is also structurally stable. Small changes in the vector field will only produce small changes in the Poincaré map and these small changes will reflect in small changes in the position of the eigenvalues of J in the complex plane, implying that the map is still hyperbolic.

The Kolmogorov–Arnold–Moser (KAM) theorem gives the behavior near an elliptic point.

Bifurcation theory
When the evolution map Φt (or the vector field it is derived from) depends on a parameter μ, the structure of the phase space will also depend on this parameter. Small changes may produce no qualitative changes in the phase space until a special value μ0 is reached. At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation.

Bifurcation theory considers a structure in phase space (typically a fixed point, a periodic orbit, or an invariant torus) and studies its behavior as a function of the parameter μ. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems.

The bifurcations of a hyperbolic fixed point x0 of a system family Fμ can be characterized by the eigenvalues of the first derivative of the system DFμ(x0) computed at the bifurcation point. For a map, the bifurcation will occur when there are eigenvalues of DFμ on the unit circle. For a flow, it will occur when there are eigenvalues on the imaginary axis. For more information, see the main article on Bifurcation theory.

Some bifurcations can lead to very complicated structures in phase space. For example, the Ruelle–Takens scenario describes how a periodic orbit bifurcates into a torus and the torus into a strange attractor. In another example, Feigenbaum period-doubling describes how a stable periodic orbit goes through a series of period-doubling bifurcations.

Ergodic systems
In many dynamical systems, it is possible to choose the coordinates of the system so that the volume (really a ν-dimensional volume) in phase space is invariant. This happens for mechanical systems derived from Newton's laws as long as the coordinates are the position and the momentum and the volume is measured in units of (position) × (momentum). The flow takes points of a subset A into the points Φt(A) and invariance of the phase space means that
 * $$ \mathrm{vol} (A) = \mathrm{vol} ( \Phi^t(A) ). $$

In the Hamiltonian formalism, given a coordinate it is possible to derive the appropriate (generalized) momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the Liouville measure.

In a Hamiltonian system, not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible. The states with the same energy form an energy shell Ω, a sub-manifold of the phase space. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space. Then almost every point of A returns to A infinitely often. The Poincaré recurrence theorem was used by Zermelo to object to Boltzmann's derivation of the increase in entropy in a dynamical system of colliding atoms.

One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the ergodic hypothesis. The hypothesis states that the length of time a typical trajectory spends in a region A is vol(A)/vol(Ω).

The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. Koopman approached the study of ergodic systems by the use of functional analysis. An observable a is a function that to each point of the phase space associates a number (say instantaneous pressure, or average height). The value of an observable can be computed at another time by using the evolution function φt. This introduces an operator Ut, the transfer operator,

 * $$ (U^t a)(x) = a(\Phi^{-t}(x)). $$

By studying the spectral properties of the linear operator U it becomes possible to classify the ergodic properties of Φt. In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Φt gets mapped into an infinite-dimensional linear problem involving U.

The Liouville measure restricted to the energy surface Ω is the basis for the averages computed in equilibrium statistical mechanics. An average in time along a trajectory is equivalent to an average in space computed with the Boltzmann factor exp(−βH). This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) to a larger class of dynamical systems that includes dissipative systems. SRB measures replace the Boltzmann factor and they are defined on attractors of chaotic systems.

Nonlinear dynamical systems and chaos
Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This seemingly unpredictable behavior has been called chaos. Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit (the stable manifold) and another of the points that diverge from the orbit (the unstable manifold).

This branch of mathematics deals with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible attractors?" or "Does the long-term behavior of the system depend on its initial condition?"

The chaotic behavior of complex systems is not the issue. Meteorology has been known for years to involve complex—even chaotic—behavior. Chaos theory has been so surprising because chaos can be found within almost trivial systems. The logistic map is only a second-degree polynomial; the horseshoe map is piecewise linear.

Solutions of Finite Duration
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, meaning here that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after. These finite-duration solutions cannot be analytical functions on the whole real line, and because they are non-Lipschitz functions at their ending time, they are not unique solutions of Lipschitz differential equations.

As example, the equation:
 * $$y'= -\text{sgn}(y)\sqrt{|y|},\,\,y(0)=1$$

Admits the finite duration solution:
 * $$y(x)=\frac{1}{4}\left(1-\frac{x}{2}+\left|1-\frac{x}{2}\right|\right)^2$$

title: Dynamical system


text:
Talk:Dynamical system

Rewrite of 11 June 05
Dynamical systems is a vast subject and I could not finish this article. I ran out of steam. I have tried to write the article using a minimal amount of formalism. My hope is that the more technical stuff will be in the linked articles. Minimizing formalism turned out to be harder than I expected. For example, it took me close to an hour to write the definition in the first sentence:

 * A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space.

There are many missing parts to the article, most notably a discussion of hyperbolic systems, and of stable and unstable manifolds and their intersections. There is no history of the subject. The article also needs some illustrations. XaosBits 14:05, 11 June 2005 (UTC)

 * I am very impressed with your rewrite; well done! However, I don't understand the second sentence of the following fragment:
 * "Linear systems are simpler than more general dynamical systems. But it is not the linearity that makes them simple, it is the structure of the orbits in phase space.  There are coordinate transformations that would not change the structure of phase space, but could make it into a nonlinear system."
 * I thought that linear systems are so simple because one can solve them explicitly. They are not very simple; for instance, you can have a centre which is a rather non-generic structure. I am certain you have a good reason for what you wrote down, but I am just curious what it is, so could you please enlighten me (in technical language, if necessary)?
 * A small remark: my browser doesn't show ₀ (&#8320;). What is this supposed to be, and is it necessary to use it, as other people's browsers may have the same problem. I also notice that you did not include any examples. Was this a conscious decision or are they to be included in the illustrations you mention above?
 * Thanks again, and I look forward to reading more of your work. -- Jitse Niesen 08:20, 12 June 2005 (UTC)

 * Now that I re-read that sentence, I agree, its wrong (I'm redefining linear). I deleted it.  What I had in mind when I was writing that was to point out that it is the structure of the vector field that makes analysis difficult.  So if I pick a coordinate change that is nonlinear and apply it to a x-dot = v(x) system, I will end up with a non-linear system that is no more interesting than the linear one.
 * The lack of examples is lack of energy on my part. There should be examples and illustrations.
 * The &#8320; is a zero subscript. I've replaced them in the main article.
 * XaosBits 16:10, 12 June 2005 (UTC)

discrete dynamical systems?
The term discrete dynamical system redirects here, but there is (almost) no discussion of the topic here. Would be nice for somebody to write it. Dmharvey Talk 19:43, 5 July 2005 (UTC)

Definition, again
I changed the definition back. A dynamical system is a smooth function from a manifold to itself dependent on one parameter, time. While the manifold could be replaced by a more general set, loosing smoothness takes one to ergodic theory or symbolic dynamics. The parameter needs to be a monoid traditionally written in additive form. One could just formalize that definition, but the challenge is to re-write it so that someone with high school mathematics can follow the definition.

From what I replaced, I liked the use of relationship instead of function and responding to their own values, which emphasizes the mapping a point of the manifold back onto itself. But I could not get it to work. Maybe someone else could try. XaosBits (talk) 05:53, 29 December 2015 (UTC)

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Proposed merge of Dynamical system (definition) into Dynamical system
"definition" is the main aspect of any topic fgnievinski (talk) 21:03, 15 July 2021 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Dynamical system


text:
Dhimmi

Dhimmī (ذمي ḏimmī,, collectively أهل الذمة ʾahl aḏ-ḏimmah/dhimmah "the people of the covenant") or muʿāhid (معاهد) is a historical term for non-Muslims living in an Islamic state with legal protection. The word literally means "protected person", referring to the state's obligation under sharia to protect the individual's life, property, as well as freedom of religion, in exchange for loyalty to the state and payment of the jizya tax, in contrast to the zakat, or obligatory alms, paid by the Muslim subjects. Dhimmi were exempt from certain duties assigned specifically to Muslims if they paid the poll tax (jizya) but were otherwise equal under the laws of property, contract, and obligation.

Historically, dhimmi status was originally applied to Jews, Christians, and Sabians, who are considered "People of the Book" in Islamic theology. Later, this status was also applied to Zoroastrians, Sikhs, Hindus, Jains, and Buddhists. Jews and Christians were required to pay the jizyah while others, depending on the different rulings of the four Madhhabs, might be required to accept Islam, pay the jizya, be exiled, or be killed.

During the rule of al-Mutawakkil, the tenth Abbasid Caliph, numerous restrictions reinforced the second-class citizen status of dhimmīs and forced their communities into ghettos. For instance, they were required to distinguish themselves from their Muslim neighbors by their dress. They were not permitted to build new churches or synagogues or repair old churches according to the Pact of Umar.

Under Sharia, the dhimmi communities were usually governed by their own laws in place of some of the laws applicable to the Muslim community. For example, the Jewish community of Medina was allowed to have its own Halakhic courts, and the Ottoman millet system allowed its various dhimmi communities to rule themselves under separate legal courts. These courts did not cover cases that involved religious groups outside of their own communities, or capital offences. Dhimmi communities were also allowed to engage in certain practices that were usually forbidden for the Muslim community, such as the consumption of alcohol and pork.

Some Muslims reject the dhimma system by arguing that it is a system which is inappropriate in the age of nation-states and democracies. There is a range of opinions among 20th-century and contemporary Islamic theologians about whether the notion of dhimma is appropriate for modern times, and, if so, what form it should take in an Islamic state.

There are differences among the Islamic Madhhabs regarding which non-Muslims can pay jizya and have dhimmi status. The Hanafi and Maliki Madhabs generally allow non-Muslims to have dhimmi status. In contrast, the Shafi'i and Hanbali Madhabs only allow Christians, Jews and Zoroastrians to have dhimmi status, and they maintain that all other non-Muslims must either convert to Islam or be fought.

The "Dhimma contract"
Based on Quranic verses and Islamic traditions, sharia law distinguishes between Muslims, followers of other Abrahamic religions, and Pagans or people belonging to other polytheistic religions. As monotheists, Jews and Christians have traditionally been considered "People of the Book", and afforded a special legal status known as dhimmi derived from a theoretical contract—"dhimma" or "residence in return for taxes". Islamic legal systems based on sharia law incorporated the religious laws and courts of Christians, Jews, and Hindus, as seen in the early caliphate, al-Andalus, Indian subcontinent, and the Ottoman Millet system.

In Yemenite Jewish sources, a treaty was drafted between Muhammad and his Jewish subjects, known as kitāb ḏimmat al-nabi, written in the 17th year of the Hijra (638 CE), which gave express liberty to the Jews living in Arabia to observe the Sabbath and to grow-out their side-locks, but required them to pay the jizya (poll-tax) annually for their protection. Muslim governments in the Indus basin readily extended the dhimmi status to the Hindus and Buddhists of India. Eventually, the largest school of Islamic jurisprudence applied this term to all Non-Muslims living in Muslim lands outside the sacred area surrounding Mecca, Arabia.

In medieval Islamic societies, the qadi (Islamic judge) usually could not interfere in the matters of non-Muslims unless the parties voluntarily chose to be judged according to Islamic law, thus the dhimmi communities living in Islamic states usually had their own laws independent from the sharia law, as with the Jews who would have their own rabbinical courts. These courts did not cover cases that involved other religious groups, or capital offences or threats to public order. By the 18th century, however, dhimmi frequently attended the Ottoman Muslim courts, where cases were taken against them by Muslims, or they took cases against Muslims or other dhimmi. Oaths sworn by dhimmi in these courts were tailored to their beliefs. Non-Muslims were allowed to engage in certain practices (such as the consumption of alcohol and pork) that were usually forbidden by Islamic law, in point of fact, any Muslim who pours away their wine or forcibly appropriates it is liable to pay compensation. Some Islamic theologians held that Zoroastrian "self-marriages", considered incestuous under sharia, should also be tolerated. Ibn Qayyim Al-Jawziyya (1292–1350) opined that most scholars of the Hanbali school held that non-Muslims were entitled to such practices, as long as they were not presented to sharia courts and the religious minorities in question held them to be permissible. This ruling was based on the precedent that there were no records of the Islamic prophet Muhammad forbidding such self-marriages among Zoroastrians, despite coming into contact with Zoroastrians and knowing about this practice. Religious minorities were also free to do as they wished in their own homes, provided they did not publicly engage in illicit sexual activity in ways that could threaten public morals.

There are parallels for this in Roman and Jewish law. According to law professor H. Patrick Glenn of McGill University, "[t]oday it is said that the dhimmi are 'excluded from the specifically Muslim privileges, but on the other hand they are excluded from the specifically Muslim duties' while (and here there are clear parallels with western public and private law treatment of aliens—Fremdenrecht, la condition de estrangers), '[f]or the rest, the Muslim and the dhimmi are equal in practically the whole of the law of property and of contracts and obligations'." Quoting the Qur'anic statement, "Let Christians judge according to what We have revealed in the Gospel", Muhammad Hamidullah writes that Islam decentralized and "communalized" law and justice. However, the classical dhimma contract is no longer enforced. Western influence over the Muslim world has been instrumental in eliminating the restrictions and protections of the dhimma contract.

The Dhimma contract and Sharia law
The dhimma contract is an integral part of traditional Islamic law. From the 9th century AD, the power to interpret and refine law in traditional Islamic societies was in the hands of the scholars (ulama). This separation of powers served to limit the range of actions available to the ruler, who could not easily decree or reinterpret law independently and expect the continued support of the community. Through succeeding centuries and empires, the balance between the ulema and the rulers shifted and reformed, but the balance of power was never decisively changed. At the beginning of the 19th century, the Industrial Revolution and the French Revolution introduced an era of European world hegemony that included the domination of most of the Muslim lands. At the end of the Second World War, the European powers found themselves too weakened to maintain their empires. The wide variety in forms of government, systems of law, attitudes toward modernity and interpretations of sharia are a result of the ensuing drives for independence and modernity in the Muslim world.

Muslim states, sects, schools of thought and individuals differ as to exactly what sharia law entails. In addition, Muslim states today utilize a spectrum of legal systems. Most states have a mixed system that implements certain aspects of sharia while acknowledging the supremacy of a constitution. A few, such as Turkey, have declared themselves secular. Local and customary laws may take precedence in certain matters, as well. Islamic law is therefore polynormative, and despite several cases of regression in recent years, the trend is towards liberalization. Questions of human rights and the status of minorities cannot be generalized with regards to the Muslim world. They must instead be examined on a case-by-case basis, within specific political and cultural contexts, using perspectives drawn from the historical framework.

The end of the Dhimma contract
The status of the dhimmi "was for long accepted with resignation by the Christians and with gratitude by the Jews" but the rising power of Christendom and the radical ideas of the French Revolution caused a wave of discontent among Christian dhimmis. The continuing and growing pressure from the European powers combined with pressure from Muslim reformers gradually relaxed the inequalities between Muslims and non-Muslims.

On 18 February 1856, the Ottoman Reform Edict of 1856 (خط همايونى) was issued, building upon the 1839 edict. It came about partly as a result of pressure from and the efforts of the ambassadors of France, Austria and the United Kingdom, whose respective countries were needed as allies in the Crimean War. It again proclaimed the principle of equality between Muslims and non-Muslims, and produced many specific reforms to this end. For example, the jizya tax was abolished and non-Muslims were allowed to join the army.

According to some scholars, discrimination against dhimmis did not end with the Edict of 1856, and they remained second-class citizens at least until the end of World War I. H.E.W. Young, the British Council in Mosul, wrote in 1909, "The attitude of the Muslims toward the Christians and the Jews is that of a master towards slaves, whom he treats with a certain lordly tolerance so long as they keep their place. Any sign of pretension to equality is promptly repressed."

Views of modern Islamic scholars on the status of non-Muslims in an Islamic society

 * The Iranian Shi'a Muslim Ayatollah Ruhollah Khomeini indicates in his book Islamic Government: Governance of the Jurist that non-Muslims should be required to pay the poll tax, in return for which they would profit from the protection and services of the state; they would, however, be excluded from all participation in the political process. Bernard Lewis remarks about Khomeini that one of his main grievances against the Shah, Mohammad Reza Pahlavi, was that his legislation allowed the theoretical possibility of non-Muslims exercising political or judicial authority over Muslims.
 * The Egyptian theologian Yusuf al-Qaradawi, chairman of the International Union of Muslim Scholars, has stated in his Al Jazeera program Sharia and Life, which has an estimated audience of 35 to 60 million viewers: "When we say dhimmis (ahl al-dhimma) it means that [...] they are under the covenant of God and His Messenger and the Muslim community and their responsibility (ḍamān), and it is everyone's duty to protect them, and this is what is intended by the word. At present many of our brethren are offended by the word dhimmis, and I have stated in what I wrote in my books that I don't see anything to prevent contemporary Islamic ijtihad from discarding this word dhimmis and calling them non-Muslim citizens."
 * Another Egyptian Islamist, Mohammad Salim al-Awa argued the concept of dhimmi must be re-interpreted in the context of Egyptian nationalism. Al-Awa and other Muslim scholars based this on the idea that while the previous dhimma condition result from the Islamic conquest, the modern Egyptian state results from a joint Muslim-Christian campaign to end the British occupation of Egypt. In modern-day Egypt, he argues, the constitution replaces the dhimma contract.
 * Muhammad Husayn Tabataba'i, a 20th-century Shia scholar writes that dhimmis should be treated "in a good and decent manner". He addresses the argument that good treatment of dhimmis was abrogated by Quranic verse 9:29 by stating that, in the literal sense, this verse is not in conflict with good treatment of dhimmis.
 * Javed Ahmad Ghamidi, a Pakistani theologian, writes in Mizan that certain directives of the Quran were specific only to Muhammad against peoples of his times, besides other directives, the campaign involved asking the polytheists of Arabia for submission to Islam as a condition for exoneration and the others for jizya and submission to the political authority of the Muslims for exemption from capital punishment and for military protection as the dhimmis of the Muslims. Therefore, after Muhammad and his companions, there is no concept in Islam obliging Muslims to wage war for propagation or implementation of Islam.
 * The Iranian Shia jurist Grand Ayatollah Naser Makarem Shirazi states in Selection of the Tafsir Nemooneh that the main philosophy of jizya is that it is only a financial aid to those Muslims who are in the charge of safeguarding the security of the state and dhimmis' lives and properties on their behalf.
 * Prominent Islamic thinkers like Fahmi Huwaidi and Tarek El-Bishry have based their justification for full citizenship of non-Muslims in an Islamic states on the precedent set by Muhammad in the Constitution of Medina. They argue that in this charter the People of Book, have the status of citizens (muwatinun) rather than dhimmis, sharing equal rights and duties with Muslims.
 * Legal scholar L. Ali Khan also points to the Constitution of Medina as a way forward for Islamic states in his 2006 paper titled The Medina Constitution. He suggests this ancient document, which governed the status of religions and races in the first Islamic state, in which Jewish tribes are "placed on an equal footing with [...] Muslims" and granted "the freedom of religion," can serve as a basis for the protection of minority rights, equality, and religious freedom in the modern Islamic state.
 * Tariq Ramadan, Professor of Islamic Studies at the University of Oxford, advocates the inclusion of academic disciplines and Islamic society, along with traditional Islamic scholars, in an effort to reform Islamic law and address modern conditions. He speaks of remaining faithful to the higher objectives of sharia law. He posits universal rights of dignity, welfare, freedom, equality and justice in a religiously and culturally pluralistic Islamic (or other) society, and proposes a dialogue regarding the modern term "citizenship," although it has no clear precedent in classical fiqh. He further includes the terms "non-citizen", "foreigner", "resident" and "immigrant" in this dialogue, and challenges not only Islam, but modern civilization as a whole, to come to terms with these concepts in a meaningful way with regards to problems of racism, discrimination and oppression.

Dhimmi communities
Jews and Christians living under early Muslim rule were considered dhimmis, a status that was later also extended to other non-Muslims like Hindus and Buddhists. They were allowed to "freely practice their religion, and to enjoy a large measure of communal autonomy" and guaranteed their personal safety and security of property, in return for paying tribute and acknowledging Muslim rule. Islamic law and custom prohibited the enslavement of free dhimmis within lands under Islamic rule. Taxation from the perspective of dhimmis who came under the Muslim rule, was "a concrete continuation of the taxes paid to earlier regimes" (but much lower under the Muslim rule ). They were also exempted from the zakat tax paid by Muslims. The dhimmi communities living in Islamic states had their own laws independent from the Sharia law, such as the Jews who had their own Halakhic courts. The dhimmi communities had their own leaders, courts, personal and religious laws, and "generally speaking, Muslim tolerance of unbelievers was far better than anything available in Christendom, until the rise of secularism in the 17th century". "Muslims guaranteed freedom of worship and livelihood, provided that they remained loyal to the Muslim state and paid a poll tax". "Muslim governments appointed Christian and Jewish professionals to their bureaucracies", and thus, Christians and Jews "contributed to the making of the Islamic civilization".

However, dhimmis faced social and symbolic restrictions, and a pattern of stricter, then more lax, enforcement developed over time. Marshall Hodgson, a historian of Islam, writes that during the era of the High Caliphate (7th–13th Centuries), zealous Shariah-minded Muslims gladly elaborated their code of symbolic restrictions on the dhimmis.

From an Islamic legal perspective, the pledge of protection granted dhimmis the freedom to practice their religion and spared them forced conversions. The dhimmis also served a variety of useful purposes, mostly economic, which was another point of concern to jurists. Religious minorities were free to do whatever they wished in their own homes, but could not "publicly engage in illicit sex in ways that threaten public morals". In some cases, religious practices that Muslims found repugnant were allowed. One example was the Zoroastrian practice of incestuous "self-marriage" where a man could marry his mother, sister or daughter. According to the famous Islamic legal scholar Ibn Qayyim Al-Jawziyya (1292–1350), non-Muslims had the right to engage in such religious practices even if it offended Muslims, under the conditions that such cases not be presented to Islamic Sharia courts and that these religious minorities believed that the practice in question is permissible according to their religion. This ruling was based on the precedent that Muhammad did not forbid such self-marriages among Zoroastrians despite coming in contact with them and having knowledge of their practices.

The Arabs generally established garrisons outside towns in the conquered territories, and had little interaction with the local dhimmi populations for purposes other than the collection of taxes. The conquered Christian, Jewish, Mazdean and Buddhist communities were otherwise left to lead their lives as before.

Christians
According to historians Lewis and Stillman, local Christians in Syria, Iraq, and Egypt were non-Chalcedonians and many may have felt better off under early Muslim rule than under that of the Byzantine Orthodox of Constantinople. In 1095, Pope Urban II urged western European Christians to come to the aid of the Christians of Palestine. The subsequent Crusades brought Roman Catholic Christians into contact with Orthodox Christians whose beliefs they discovered to differ from their own perhaps more than they had realized, and whose position under the rule of the Muslim Fatimid Caliphate was less uncomfortable than had been supposed. Consequently, the Eastern Christians provided perhaps less support to the Crusaders than had been expected. When the Arab East came under Ottoman rule in the 16th century, Christian populations and fortunes rebounded significantly. The Ottomans had long experience dealing with Christian and Jewish minorities, and were more tolerant towards religious minorities than the former Muslim rulers, the Mamluks of Egypt.

However, Christians living under Islamic rule have suffered certain legal disadvantages and at times persecution. In the Ottoman Empire, in accordance with the dhimmi system implemented in Muslim countries, they, like all other Christians and also Jews, were accorded certain freedoms. The dhimmi system in the Ottoman Empire was largely based upon the Pact of Umar. The client status established the rights of the non-Muslims to property, livelihood and freedom of worship but they were in essence treated as second-class citizens in the empire and referred to in Turkish as gavours, a pejorative word meaning "infidel" or "unbeliever". The clause of the Pact of Umar which prohibited non-Muslims from building new places of worship was historically imposed on some communities of the Ottoman Empire and ignored in other cases, at discretion of the local authorities. Although there were no laws mandating religious ghettos, this led to non-Muslim communities being clustered around existing houses of worship.

In addition to other legal limitations, dhimmis, including the Christians among them, were not considered equals to Muslims and several prohibitions were placed on them. Their testimony against Muslims was inadmissible in courts of law wherein a Muslim could be punished; this meant that their testimony could only be considered in commercial cases. They were forbidden to carry weapons or ride atop horses and camels. Their houses could not overlook those of Muslims; and their religious practices were severely circumscribed (e.g., the ringing of church bells was strictly forbidden).

Jews
Because the early Islamic conquests initially preserved much of the existing administrative machinery and culture, in many territories they amounted to little more than a change of rulers for the subject populations, which "brought peace to peoples demoralized and disaffected by the casualties and heavy taxation that resulted from the years of Byzantine-Persian warfare".

María Rosa Menocal, argues that the Jewish dhimmis living under the caliphate, while allowed fewer rights than Muslims, were still better off than in the Christian parts of Europe. Jews from other parts of Europe made their way to al-Andalus, where in parallel to Christian sects regarded as heretical by Catholic Europe, they were not just tolerated, but where opportunities to practice faith and trade were open without restriction save for the prohibitions on proselytization.

Bernard Lewis states:

"Generally, the Jewish people were allowed to practice their religion and live according to the laws and scriptures of their community. Furthermore, the restrictions to which they were subject were social and symbolic rather than tangible and practical in character. That is to say, these regulations served to define the relationship between the two communities, and not to oppress the Jewish population."

Professor of Jewish medieval history at Hebrew University of Jerusalem, Hayim Hillel Ben-Sasson, notes:

"The legal and security situation of the Jews in the Muslim world was generally better than in Christendom, because in the former, Jews were not the sole 'infidels', because in comparison to the Christians, Jews were less dangerous and more loyal to the Muslim regime, and because the rapidity and the territorial scope of the Muslim conquests imposed upon them a reduction in persecution and a granting of better possibility for the survival of members of other faiths in their lands."

According to the French historian Claude Cahen, Islam has "shown more toleration than Europe towards the Jews who remained in Muslim lands."

Comparing the treatment of Jews in the medieval Islamic world and medieval Christian Europe, Mark R. Cohen notes that, in contrast to Jews in Christian Europe, the "Jews in Islam were well integrated into the economic life of the larger society", and that they were allowed to practice their religion more freely than they could do in Christian Europe.

According to the scholar Mordechai Zaken, tribal chieftains (also known as aghas) in tribal Muslim societies such as the Kurdish society in Kurdistan would tax their Jewish subjects. The Jews were in fact civilians protected by their chieftains in and around their communities; in return they paid part of their harvest as dues, and contributed their skills and services to their patron chieftain.

Hindus and Buddhists
By the 10th century, the Turks of Central Asia had invaded the Indic plains, and spread Islam in Northwestern parts of India. At the end of the 12th century, the Muslims advanced quickly into the Ganges Plain. In one decade, a Muslim army led by Turkic slaves consolidated resistance around Lahore and brought northern India, as far as Bengal, under Muslim rule. From these Turkic slaves would come sultans, including the founder of the sultanate of Delhi. By the 15th century, major parts of Northern India was ruled by Muslim rulers, mostly descended from invaders. In the 16th century, India came under the influence of the Mughals. Babur, the first ruler of the Mughal empire, established a foothold in the north which paved the way for further expansion by his successors. Although the Mughal emperor Akbar has been described as a universalist, most Mughal emperors were oppressive of native Hindu, Buddhist and later Sikh populations. Aurangzeb specifically was inclined towards a highly fundamentalist approach.

Restrictions
There were a number of restrictions on dhimmis. In a modern sense the dhimmis would be described as second-class citizens. According to historian Marshall Hodgson, from very early times Muslim rulers would very often humiliate and punish dhimmis (usually Christians or Jews that refused to convert to Islam). It was official policy that dhimmis should “feel inferior and to know ‘their place".

Although dhimmis were allowed to perform their religious rituals, they were obliged to do so in a manner not conspicuous to Muslims. Loud prayers were forbidden, as were the ringing of church bells and the blowing of the shofar. They were also not allowed to build or repair churches and synagogues without Muslim consent. Moreover, dhimmis were not allowed to seek converts among Muslims. In the Mamluk Egypt, where non-Mamluk Muslims were not allowed to ride horses and camels, dhimmis were prohibited even from riding donkeys inside cities. Sometimes, Muslim rulers issued regulations requiring dhimmis to attach distinctive signs to their houses.

Most of the restrictions were social and symbolic in nature, and a pattern of stricter, then more lax, enforcement developed over time. The major financial disabilities of the dhimmi were the jizya poll tax and the fact dhimmis and Muslims could not inherit from each other. That would create an incentive to convert if someone from the family had already converted. Ira M. Lapidus states that the "payment of the poll tax seems to have been regular, but other obligations were inconsistently enforced and did not prevent many non-Muslims from being important political, business, and scholarly figures. In the late ninth and early tenth centuries, Jewish bankers and financiers were important at the 'Abbasid court." The jurists and scholars of Islamic sharia law called for humane treatment of the dhimmis.

A Muslim man may marry a Jewish or Christian dhimmī woman, who may keep her own religion (though her children were automatically considered Muslims and had to be raised as such), but a Muslim woman cannot marry a dhimmī man unless he converts to Islam. Dhimmīs are prohibited from converting Muslims under severe penalties, while Muslims are encouraged to convert dhimmīs.

Jizya tax
Payment of the jizya obligated Muslim authorities to protect dhimmis in civil and military matters. Sura 9 (At-Tawba), verse 29 stipulates that jizya be exacted from non-Muslims as a condition required for jihad to cease. Islamic jurists required adult, free, healthy males among the dhimma community to pay the jizya, while exempting women, children, the elderly, slaves, those affected by mental or physical handicaps, and travelers who did not settle in Muslim lands. According to Abu Yusuf dhimmi should be imprisoned until they pay the jizya in full. Other jurists specified that dhimmis who don't pay jizya should have their heads shaved and made to wear a dress distinctive from those dhimmis who paid the jizya and Muslims.

Lewis states there are varying opinions among scholars as to how much of a burden jizya was. According to Norman Stillman: "jizya and kharaj were a "crushing burden for the non-Muslim peasantry who eked out a bare living in a subsistence economy." Both agree that ultimately, the additional taxation on non-Muslims was a critical factor that drove many dhimmis to leave their religion and accept Islam. However, in some regions the jizya on populations was significantly lower than the zakat, meaning dhimmi populations maintained an economic advantage. According to Cohen, taxation, from the perspective of dhimmis who came under Muslim rule, was "a concrete continuation of the taxes paid to earlier regimes". Lewis observes that the change from Byzantine to Arab rule was welcomed by many among the dhimmis who found the new yoke far lighter than the old, both in taxation and in other matters, and that some, even among the Christians of Syria and Egypt, preferred the rule of Islam to that of Byzantines. Montgomery Watt states, "the Christians were probably better off as dhimmis under Muslim-Arab rulers than they had been under the Byzantine Greeks." In some places, for example Egypt, the jizya was a tax incentive for Christians to convert to Islam.

Some scholars have tried compute the relative taxation on Muslims vs non-Muslims in the early Abbasid period. According to one estimate, Muslims had an average tax rate of 17–20 dirhams per person, which rose to 30 dirhams per person when in kind levies are included. Non-Muslims paid either 12, 24 or 48 dirhams per person, depending on their taxation category, though most probably paid 12.

The importance of dhimmis as a source of revenue for the Rashidun Caliphate is illustrated in a letter ascribed to Umar I and cited by Abu Yusuf: "if we take dhimmis and share them out, what will be left for the Muslims who come after us? By God, Muslims would not find a man to talk to and profit from his labors."

The early Islamic scholars took a relatively humane and practical attitude towards the collection of jizya, compared to the 11th century commentators writing when Islam was under threat both at home and abroad.

The jurist Abu Yusuf, the chief judge of the caliph Harun al-Rashid, rules as follows regarding the manner of collecting the jizya

"No one of the people of the dhimma should be beaten in order to exact payment of the jizya, nor made to stand in the hot sun, nor should hateful things be inflicted upon their bodies, or anything of that sort. Rather they should be treated with leniency."

In the border provinces, dhimmis were sometimes recruited for military operations. In such cases, they were exempted from jizya for the year of service.

Administration of law
Religious pluralism existed in medieval Islamic law and ethics. The religious laws and courts of other religions, including Christianity, Judaism and Hinduism, were usually accommodated within the Islamic legal framework, as exemplified in the Caliphate, Al-Andalus, Ottoman Empire and Indian subcontinent. In medieval Islamic societies, the qadi (Islamic judge) usually could not interfere in the matters of non-Muslims unless the parties voluntarily chose to be judged according to Islamic law. The dhimmi communities living in Islamic states usually had their own laws independent from the Sharia law, such as the Jews who had their own Halakha courts.

Dhimmis were allowed to operate their own courts following their own legal systems. However, dhimmis frequently attended the Muslim courts in order to record property and business transactions within their own communities. Cases were taken out against Muslims, against other dhimmis and even against members of the dhimmi's own family. Dhimmis often took cases relating to marriage, divorce or inheritance to the Muslim courts so these cases would be decided under sharia law. Oaths sworn by dhimmis in the Muslim courts were sometimes the same as the oaths taken by Muslims, sometimes tailored to the dhimmis' beliefs.

Muslim men could generally marry dhimmi women who are considered People of the Book, however Islamic jurists rejected the possibility any non-Muslim man might marry a Muslim woman. Bernard Lewis notes that "similar position existed under the laws of Byzantine Empire, according to which a Christian could marry a Jewish woman, but a Jew could not marry a Christian woman under pain of death".

Quranic verses as a basis for Islamic policies toward dhimmis
Lewis states
 * The phrase "Let there be no compulsion in religion: ...", from has sometimes been interpreted in the Islamic legal and theological traditions to mean followers of other religions should not be forced to adopt Islam
 * The phrase "Unto you your religion, and unto me my religion.", from has been used as a "proof-text for pluralism and coexistence".
 * has served to justify the tolerated position accorded to the followers of Christianity, Judaism, and Sabianism under Muslim rule.

Hadith
A hadith by Muhammad, "Whoever killed a muʿāhid (a person who is granted the pledge of protection by the Muslims) shall not smell the fragrance of Paradise though its fragrance can be smelt at a distance of forty years (of traveling).",  is cited as a foundation for the right of non-Muslim citizens to live peacefully and undisturbed in an Islamic state. Anwar Shah Kashmiri writes in his commentary on Sahih al-Bukhari Fayd al-Bari on this hadith: "You know the gravity of sin for killing a Muslim, for its odiousness has reached the point of disbelief, and it necessitates that [the killer abides in Hell] forever. As for killing a non-Muslim citizen [muʿāhid], it is similarly no small matter, for the one who does it will not smell the fragrance of Paradise."

A similar hadith in regard to the status of the dhimmis: "Whoever wrongs one with whom a compact (treaty) has been made [i.e., a dhimmi] and lays on him a burden beyond his strength, I will be his accuser."

Constitution of Medina
The Constitution of Medina, a formal agreement between Muhammad and all the significant tribes and families of Medina (including Muslims, Jews and pagans), declared that non-Muslims in the Ummah had the following rights:
 * 1) The security (dhimma) of God is equal for all groups,
 * 2) Non-Muslim members have equal political and cultural rights as Muslims. They will have autonomy and freedom of religion.
 * 3) Non-Muslims will take up arms against the enemy of the Ummah and share the cost of war. There is to be no treachery between the two.
 * 4) Non-Muslims will not be obliged to take part in religious wars of the Muslims.

Khaybar agreement
A precedent for the dhimma contract was established with the agreement between Muhammad and the Jews after the Battle of Khaybar, an oasis near Medina. Khaybar was the first territory attacked and conquered by Muslims. When the Jews of Khaybar surrendered to Muhammad after a siege, Muhammad allowed them to remain in Khaybar in return for handing over to the Muslims one half their annual produce.

Pact of Umar
The Pact of Umar, traditionally believed to be between caliph Umar and the conquered Jerusalem Christians in the seventh century, was another source of regulations pertaining to dhimmis. However, Western orientalists doubt the authenticity of the pact, arguing it is usually the victors and not the vanquished who impose rather than propose, the terms of peace, and that it is highly unlikely that the people who spoke no Arabic and knew nothing of Islam could draft such a document. Academic historians believe the Pact of Umar in the form it is known today was a product of later jurists who attributed it to Umar in order to lend greater authority to their own opinions. The similarities between the Pact of Umar and the Theodosian and Justinian Codes of the Eastern Roman Empire suggest that perhaps much of the Pact of Umar was borrowed from these earlier codes by later Islamic jurists. At least some of the clauses of the pact mirror the measures first introduced by the Umayyad caliph Umar II or by the early Abbasid caliphs.

Cultural interactions and cultural differences
During the Middle Ages, local associations known as futuwwa clubs developed across the Islamic lands. There were usually several futuwwah in each town. These clubs catered to varying interests, primarily sports, and might involve distinctive manners of dress and custom. They were known for their hospitality, idealism and loyalty to the group. They often had a militaristic aspect, purportedly for the mutual protection of the membership. These clubs commonly crossed social strata, including among their membership local notables, dhimmi and slaves – to the exclusion of those associated with the local ruler, or amir.

Muslims and Jews were sometimes partners in trade, with the Muslim taking days off on Fridays and Jews taking off on Saturdays.

Andrew Wheatcroft describes how some social customs such as different conceptions of dirt and cleanliness made it difficult for the religious communities to live close to each other, either under Muslim or under Christian rule.

In modern times
The dhimma and the jizya poll tax are no longer imposed in Muslim majority countries. In the 21st century, jizya is widely regarded as being at odds with contemporary secular conceptions of citizens' civil rights and equality before the law, although there have been occasional reports of religious minorities in conflict zones and areas subject to political instability being forced to pay jizya.

In 2009 it was claimed that a group of militants that referred to themselves as the Taliban imposed the jizya on Pakistan's minority Sikh community after occupying some of their homes and kidnapping a Sikh leader.

In 2013, the Muslim Brotherhood in Egypt occupied the town of Dalga immediately following the overthrow of Mohammed Morsi on 3 July, and reportedly imposed jizya on the 15,000 Christian Copts living there. However, in autumn of that same year Egyptian authorities were able to retake control of the town following two prior failed attempts.

In February 2014, the Islamic State of Iraq and the Levant (ISIL) announced that it intended to extract jizya from Christians in the city of Raqqa, Syria, which it controlled at the time. Christians who refused to accept the dhimma contract and pay the tax were to have to either convert to Islam, leave or be executed. Wealthy Christians would have to pay half an ounce of gold, the equivalent of $664 twice a year; middle-class Christians were to have to pay half that amount and poorer ones were to be charged one-fourth that amount. In June 2014 the Institute for the Study of War reported that ISIL claims to have collected jizya and fay. On 18 July 2014 ISIL ordered the Christians in Mosul to accept the dhimma contract and pay the jizya or convert to Islam. If they refused to accept either of the options they would be killed.

title: Dhimmi


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Doctor V64

The Doctor V64 (also referred to simply as the V64) is a development and backup device made by Bung Enterprises Ltd that is used in conjunction with the Nintendo 64. The Doctor V64 also had the ability to play video CDs and audio CDs. Additionally, it could apply stereo 3D effects to the audio.

History
The V64 was released in 1996 and was priced around US$450. It was the first commercially-available backup device for the Nintendo 64 unit. The Partner N64 development kit, which was manufactured by Silicon Graphics and sold officially by Nintendo, was a comparatively expensive development machine. The V64 served as a lower-cost development machine, though its unofficial status would later lead to conflict with Nintendo. Some third-party developers used a number of V64s in their development process.

Specifications
The CPU of the V64 is a 6502, and the operating system is contained in a BIOS.

The V64 unit contains a CD-ROM drive which sits underneath the Nintendo 64 and plugs into the expansion slot on the underside of the Nintendo 64. The expansion slot is essentially a mirror image of the cartridge slot on the top of the unit, with the same electrical connections; thus, the Nintendo 64 reads data from the Doctor V64 in the same manner as it would from a cartridge plugged into the normal slot.

Game booting
In order to get around Nintendo's lockout chip, when using the V64, a game cartridge is plugged into the Nintendo 64 through an adaptor which connects only the lockout chip. The game cart used for the operation had to contain the same lockout chip used by the game back up.

Saving game progress
The second problem concerned saving progress. Most N64 games are saved to the cart itself instead of external memory cards. If the player wanted to keep their progress, then the cartridge used had to have the same type of non-volatile memory hardware. Alternatively, Bung produced the "DX256" and "DS1" add-ons to allow (EEPROM and SRAM respectively) saves to be made without using the inserted cartridge. These devices were inserted into the top-slot of the N64 with the game cartridge being then inserted into the top of them to just provide the security bypass. Save slots on the DX256 were selected using an alpha and numeric encoder knobs on the front of the device.

Uploading game images
The Doctor V64 could be used to read the data from a game cartridge and transfer the data to a PC via the parallel port. This allowed developers and homebrew programmers to upload their game images to the Doctor V64 without having to create a CD backup each time. It also allowed users to upload game images taken from the Internet.

Doctor V64 Jr.
Following the Doctor V64's success, Bung released the Doctor V64 Jr. in December 1998. This was a condensed, cost-efficient version of the original V64. The Doctor V64 Jr. has no CD drive and plugs into the normal cartridge slot on the top of the Nintendo 64. Data is loaded into the Doctor V64 Jr.'s battery-backed RAM from a PC via a parallel port connection. The Doctor V64 Jr. has up to 512 megabits (64 MB) of memory storage. This was done to provide for future Nintendo 64 carts that employed larger memory storage, but the high costs associated with ordering large storage carts kept this occurrence at a minimum. Only a handful of 512-megabit games were released for the Nintendo 64 system.

Promotions
In 1998 and 1999, there was a homebrew competition known as "Presence of Mind" (POM), an N64 demo competition led by dextrose.com. The contest consisted of submitting a user-developed N64 program, game, or utility. Bung Enterprises promoted the event and supplied prizes (usually Doctor V64 related accessories). Though a contest was planned for 2000, the interest in the N64 was already fading, and so did the event. POM contest demo entries can still be found on the Internet.

Role in piracy
The Doctor V64 unit was the first commercially available backup device for the Nintendo 64 unit. Though the unit was sold as a development machine, it could be modified to enable the creation and use of commercial game backups. Unlike official development units, the purchase of V64s was not restricted to software companies only. For this reason, the unit became a popular choice among those looking to proliferate unlicensed copies of games.

Original Doctor V64 units sold by Bung did not allow the playing of backups. A person would have to modify the unit by themselves in order to make it backup friendly. This usually required a user to download and install a modified Doctor V64 BIOS. Additionally, the cartridge adapter had to be opened and soldered in order to allow for the operational procedure. Though Bung never sold backup enabled V64s, many re-sellers would modify the units themselves.

Conflicts with Nintendo
During the N64's lifetime, Nintendo revised the N64's model, making the serial port area smaller. This slight change in the N64's plastic casing made the connection to the Doctor V64 difficult to achieve without user modification. This revision may have been a direct reaction from Nintendo to discourage the use of V64 devices, and may also explain why Bung decided to discontinue the use of this port in the later Doctor V64 Jr. models.

Nintendo made many legal efforts worldwide in order to stop the sale of Doctor V64 units. They sued Bung directly as well as specific store retailers in Europe and North America for copyright infringement. Eventually, Nintendo managed to have the courts prohibit the sale of Doctor V64 units in the United States.

Main menu

The Doctor V64 implemented text-based menu-driven screens. The menus consisted of white text superimposed over a black background. Utilizing the buttons on the V64 unit, a user would navigate the menus and issue commands. Though the menu was mainly designed for game developers, it is possible to back up cartridges with it (through the use of an unofficial V64 BIOS). Some of the menu items related to game backups were removed from the V64's BIOS near the end of its life due to pressure from Nintendo. These items are only available by obtaining a patched V64 BIOS.

CD-ROM access speed
Most early V64 models shipped with a standard IDE 8X CD-ROM. During the manufacturing lifetime of the device, latter V64 models shipped with 16X and eventually 20X drives. V64 units could be purchased without a CD-ROM drive. It is possible to replace the unit with a faster IDE CD-ROM unit (such as the 52X model in the image on this page).

Many Doctor V64s shipped internationally were ordered without an installed CD-ROM drive, to save on shipping costs associated with weight, to avoid import duty on the drive, and to allow users to customize the units in response to the ever-increasing speeds of drives available. The variance in the power draw of different manufacturers drives at different speeds caused issues with disc spin-ups exceeding the wattage rating of the included Bung PSU. This led to users swapping out the Bung PSU for a more powerful model, or selecting low draw drives (mainly Panasonic drives sometimes badged as Creative).

CD-Media
V64s can read CD-Rs and CD-RWs (provided the installed CD-ROM unit supports rewritable media). Supported media has to be recorded in Mode 1, ISO 9660 format. Doctor V64s only support the 8.3 DOS naming convention. As such, Joliet file system is not supported.

RAM
Depending on the model, V64s came with either 128 megabits (16 MB) or 256 megabits (32 MB) of RAM. Original V64 units shipped with 128 megabits of RAM. V64 units started shipping with 256 megabits when developers started using bigger sized memory carts for their games. Users had the option of buying a memory upgrade from Bung and other re-sellers.

Power supply
The Doctor V64 uses a 4 Pin MiniDIN jack (as used for S-Video) for connecting the power supply cord. Power supplies included with Doctor V64s were very unreliable. Bung replaced the power supply with a sturdier version in later V64 units. Replacing broken power supplies became one of the most common maintenance problems with the V64. It is possible to modify an AT PC power supply for V64 use. Only 4 cables have to be connected to the V64 for it to function.

Additional information

 * The ROM extensions ".v64" and ".z64" started out as the preferred naming conventions by Doctor V64 and Z64 users, respectively. It would also imply the file's "endianness" as those units employed little endian (V64) and big endian (Z64) byte alignment. ".n64" was used as well but not as much (it became more popular as N64 emulators began to appear). The terms ".v64" and ".z64" are still widely used today by the emulation community.
 * Acclaim Entertainment subsidiary Iguana Entertainment used Doctor V64 units as their development hardware of choice during the N64 era. They were best known for developing the Turok, NBA Jam, NFL Quarterback Club, and South Park video games.

title: Doctor V64