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.
In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensional subspace, called the Poincaré section, transversal to the flow of the system. More precisely, one considers a periodic orbit with initial conditions within a section of the space, which leaves that section afterwards, and observes the point at which this orbit first returns to the section. One then creates a map to send the first point to the second, hence the name first recurrence map. The transversality of the Poincaré section means that periodic orbits starting on the subspace flow through it and not parallel to it.
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.
In mathematics, the Selberg trace formula, introduced by Selberg (1956), is an expression for the character of the unitary representation of a Lie group G on the space L2(Γ\G) of square-integrable functions, where Γ is a cofinite discrete group. The character is given by the trace of certain functions on G.
In differential geometry, pushforward is a linear approximation of smooth maps on tangent spaces. Suppose that is a smooth map between smooth manifolds; then the differential of at a point , denoted , is, in some sense, the best linear approximation of near . It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, the differential is a linear map from the tangent space of at to the tangent space of at , . Hence it can be used to push tangent vectors on forward to tangent vectors on . The differential of a map is also called, by various authors, the derivative or total derivative of .
In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow is a group action of the real numbers on a set.
In dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. This is the opposite of a conservative system, to which the Poincaré recurrence theorem applies. Intuitively, the connection between wandering sets and dissipation is easily understood: if a portion of the phase space "wanders away" during normal time-evolution of the system, and is never visited again, then the system is dissipative. The language of wandering sets can be used to give a precise, mathematical definition to the concept of a dissipative system. The notion of wandering sets in phase space was introduced by Birkhoff in 1927.
In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms.
The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz. Given certain conditions on a function, the theorem demonstrates the existence of a saddle point. The theorem is unusual in that there are many other theorems regarding the existence of extrema, but few regarding saddle points.
In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.
In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can be used to understand the long term behavior of a dynamical system. A system that has reached its limiting set is said to be at equilibrium.
In mathematics, the Melnikov method is a tool to identify the existence of chaos in a class of dynamical systems under periodic perturbation.
In physics, the Toda oscillator is a special kind of nonlinear oscillator. It represents a chain of particles with exponential potential interaction between neighbors. These concepts are named after Morikazu Toda. The Toda oscillator is used as a simple model to understand the phenomenon of self-pulsation, which is a quasi-periodic pulsation of the output intensity of a solid-state laser in the transient regime.
In mathematics, Ratner's theorems are a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner around 1990. The theorems grew out of Ratner's earlier work on horocycle flows. The study of the dynamics of unipotent flows played a decisive role in the proof of the Oppenheim conjecture by Grigory Margulis. Ratner's theorems have guided key advances in the understanding of the dynamics of unipotent flows. Their later generalizations provide ways to both sharpen the results and extend the theory to the setting of arbitrary semisimple algebraic groups over a local field.
The uncertainty theory invented by Baoding Liu is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms.
The Zwanzig projection operator is a mathematical device used in statistical mechanics. This projection operator acts in the linear space of phase space functions and projects onto the linear subspace of "slow" phase space functions. It was introduced by Robert Zwanzig to derive a generic master equation. It is mostly used in this or similar context in a formal way to derive equations of motion for some "slow" collective variables.
In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system.
In mathematics, a harmonic morphism is a (smooth) map :(M^{m},g)\to (N^{n},h)} between Riemannian manifolds that pulls back real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphisms form a special class of harmonic maps i.e. those that are horizontally (weakly) conformal.
Phase reduction is a method used to reduce a multi-dimensional dynamical equation describing a nonlinear limit cycle oscillator into a one-dimensional phase equation. Many phenomena in our world such as chemical reactions, electric circuits, mechanical vibrations, cardiac cells, and spiking neurons are examples of rhythmic phenomena, and can be considered as nonlinear limit cycle oscillators.
