Dynamical systems in the context of Limit cycle

Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems. Population dynamics is a branch of mathematical biology, and uses mathematical techniques such as differential equations to model behaviour. Population dynamics is also closely related to other mathematical biology fields such as epidemiology, and also uses techniques from evolutionary game theory in its modelling.

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations occur in both continuous systems (described by ordinary, delay or partial differential equations) and discrete systems (described by maps).

The name "bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior.

Critical transitions are abrupt shifts in the state of ecosystems, the climate, financial and economic systems or other complex dynamical systems that may occur when changing conditions pass a critical or bifurcation point. As such, they are a particular type of regime shift. Recovery from such shifts may require more than a simple return to the conditions at which a transition occurred, a phenomenon called hysteresis. In addition to natural systems, critical transitions are also studied in psychology, medicine, economics, sociology, military, and several other disciplines.

In mathematics and particularly in dynamical systems, an initial condition is the initial value (often at time ${\displaystyle t=0}$) of a differential equation, difference equation, or other "time"-dependent equation which evolves in time. The most fundamental case, an ordinary differential equation of order k (the number of derivatives in the equation), generally requires k initial conditions to trace the equation's evolution through time. In other contexts, the term may refer to an initial value of a recurrence relation, discrete dynamical system, hyperbolic partial differential equation, or even a seed value of a pseudorandom number generator, at "time zero", enough such that the overall system can be evolved in "time", which may be discrete or continuous. The problem of determining a system's evolution from initial conditions is referred to as an initial value problem.

In physics and mathematics, in the area of dynamical systems, a double pendulum, also known as a chaotic pendulum, is a pendulum with another pendulum attached to its end, forming a complex physical system that exhibits rich dynamic behavior with a strong sensitivity to initial conditions. The motion of a double pendulum is governed by a pair of coupled ordinary differential equations and is chaotic.

In mathematics and physics, the Poincaré recurrence theorem states that certain dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (for discrete state systems), their initial state.

The Poincaré recurrence time is the length of time elapsed until the recurrence. This time may vary greatly depending on the exact initial state and required degree of closeness. The result applies to isolated mechanical systems subject to some constraints, e.g., all particles must be bound to a finite volume. The theorem is commonly discussed in the context of ergodic theory, dynamical systems and statistical mechanics. Systems to which the Poincaré recurrence theorem applies are called conservative systems.

In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form ${\displaystyle dr/dt=Ar}$, where r is the perturbation to the steady state, A is a linear operator whose spectrum contains eigenvalues with positive real part. If all the eigenvalues have negative real part, then the solution is called linearly stable. Other names for linear stability include exponential stability or stability in terms of first approximation. If there exists an eigenvalue with zero real part then the question about stability cannot be solved on the basis of the first approximation and we approach the so-called "centre and focus problem".

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.

A Poincaré map can be interpreted as a discrete dynamical system with a state space that is one dimension smaller than the original continuous dynamical system. Because it preserves many properties of periodic and quasiperiodic orbits of the original system and has a lower-dimensional state space, it is often used for analyzing the original system in a simpler way. In practice this is not always possible as there is no general method to construct a Poincaré map.

In the theory of dynamical systems and control theory, a linear time-invariant system is marginally stable if it is neither asymptotically stable nor unstable. Roughly speaking, a system is stable if it always returns to and stays near a particular state (called the steady state), and is unstable if it goes further and further away from any state, without being bounded. A marginal system, sometimes referred to as having neutral stability, is between these two types: when displaced, it does not return to near a common steady state, nor does it go away from where it started without limit.

Marginal stability, like instability, is a feature that control theory seeks to avoid; we wish that, when perturbed by some external force, a system will return to a desired state. This necessitates the use of appropriately designed control algorithms.

George David Birkhoff (March 21, 1884 – November 12, 1944) was one of the top American mathematicians of his generation. He made valuable contributions to the theory of differential equations, dynamical systems, the four-color problem, the three-body problem, and general relativity. Today, Birkhoff is best remembered for the ergodic theorem. The George D. Birkhoff House, his residence in Cambridge, Massachusetts, has been designated a National Historic Landmark.