Lyapunov exponent. In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation diverge (provided that the divergence can be treated within the linearized approximation) at a rate given by where is the Lyapunov exponent. The rate of separation can be different for different orientations of initial separation vector. The exponent is named after Aleksandr Lyapunov. Definition of the maximal Lyapunov exponent[edit] The maximal Lyapunov exponent can be defined as follows: The limit ensures the validity of the linear approximation at any time.[1] For discrete time system (maps or fixed point iterations) , for an orbit starting with this translates into: Definition of the Lyapunov spectrum[edit] For a dynamical system with evolution equation in an n–dimensional phase space, the spectrum of Lyapunov exponents .
The . . A.M. Center for Nonlinear Dynamics. Lotka–Volterra equation. The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations: where, x is the number of prey (for example, rabbits);y is the number of some predator (for example, foxes); and represent the growth rates of the two populations over time;t represents time; and, , and are parameters describing the interaction of the two species. The Lotka–Volterra system of equations is an example of a Kolmogorov model,[1][2][3] which is a more general framework that can model the dynamics of ecological systems with predator-prey interactions, competition, disease, and mutualism.
History[edit] The Lotka–Volterra predator–prey model was initially proposed by Alfred J. C.S. In economics[edit] Physical meanings of the equations[edit] and. Complex dynamics. Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions.
Techniques[1][edit] Parts[edit] Holomorphic dynamics ( dynamics of holomorphic functions )[3]in one complex variablein several complex variablesConformal dynamics unites holomorphic dynamics in one complex variable with differentiable dynamics in one real variable. See also[edit] References[edit] Complex dynamics, Lennart Carleson, Theodore W. Logistic map. Where: is a number between zero and one, and represents the ratio of existing population to the maximum possible population at year n, and hence x0 represents the initial ratio of population to max. population (at year 0) r is a positive number, and represents a combined rate for reproduction and starvation. This nonlinear difference equation is intended to capture two effects. However, as a demographic model the logistic map has the pathological problem that some initial conditions and parameter values lead to negative population sizes. The r=4 case of the logistic map is a nonlinear transformation of both the bit shift map and the case of the tent map.
Behavior dependent on r[edit] By varying the parameter r, the following behavior is observed: For any value of r there is at most one stable cycle. A bifurcation diagram summarizes this. Chaos and the logistic map[edit] Two- and three-dimensional phase diagrams show the stretching-and-folding structure of the logistic map . Is given by . For . . Cobweb plot. Construction of a cobweb plot of the logistic map, showing an attracting fixed point. An animated cobweb diagram of the logistic map, showing chaotic behaviour for most values of r > 3.57.
A cobweb plot, or Verhulst diagram is a visual tool used in the dynamical systems field of mathematics to investigate the qualitative behaviour of one-dimensional iterated functions, such as the logistic map. Using a cobweb plot, it is possible to infer the long term status of an initial condition under repeated application of a map. Method[edit] For a given iterated function f: R → R, the plot consists of a diagonal (x = y) line and a curve representing y = f(x). , apply the following steps. Find the point on the function curve with an x-coordinate of . Interpretation[edit] See also[edit] Jones diagram – similar plotting technique. Hyperbolic equilibrium point. A stable manifold and an unstable manifold exist,Shadowing occurs,The dynamics on the invariant set can be represented via symbolic dynamics,A natural measure can be defined,The system is structurally stable. Orbits near a two-dimensional saddle point, an example of a hyperbolic equilibrium.
Maps[edit] Since the eigenvalues are given by Flows[edit] Let F : Rn → Rn be a C1 (that is, continuously differentiable) vector field with a critical point p and let J denote the Jacobian matrix of F at p. If the matrix J has no eigenvalues with zero real parts then p is called hyperbolic. Example[edit] Consider the nonlinear system (0, 0) is the only equilibrium point. The eigenvalues of this matrix are . [edit] In the case of an infinite dimensional system - for example systems involving a time delay - the notion of the "hyperbolic part of the spectrum" refers to the above property. See also[edit] Notes[edit] Jump up ^ Strogatz, Steven (2001). References[edit] Pitchfork bifurcation. In bifurcation theory, a field within mathematics, a pitchfork bifurcation is a particular type of local bifurcation.
Pitchfork bifurcations, like Hopf bifurcations have two types - supercritical or subcritical. In continuous dynamical systems described by ODEs—i.e. flows—pitchfork bifurcations occur generically in systems with symmetry. Supercritical case[edit] Supercritical case: solid lines represent stable points, while dotted line represents unstable one. For negative values of , there is one stable equilibrium at . There is an unstable equilibrium at , and two stable equilibria at Subcritical case[edit] Subcritical case: solid line represents stable point, while dotted lines represent unstable ones. In this case, for the equilibrium at is stable, and there are two unstable equilbria at is unstable. Formal definition[edit] An ODE described by a one parameter function with satisfying: (f is an odd function), has a pitchfork bifurcation at . References[edit] See also[edit] Hopf bifurcation. For a more general survey on Hopf bifurcation and dynamical systems in general, see.[1][2][3][4][5] Overview[edit] Supercritical / subcritical Hopf bifurcations[edit] The limit cycle is orbitally stable if a specific quantity called the first Lyapunov coefficient is negative, and the bifurcation is supercritical.
Otherwise it is unstable and the bifurcation is subcritical. where z, b are both complex and λ is a parameter. The number α is called the first Lyapunov coefficient. If α is negative then there is a stable limit cycle for λ > 0: where The bifurcation is then called supercritical. If α is positive then there is an unstable limit cycle for λ < 0. Remarks[edit] The "smallest chemical reaction with Hopf bifurcation" was found in 1995 in Berlin, Germany.[6] The same biochemical system has been used in order to investigate how the existence of a Hopf bifurcation influences our ability to reverse-engineer dynamical systems.[7] Example[edit] The Selkov model is Theorem (see section 11.2 of [3]).
Saddle-node bifurcation. If the phase space is one-dimensional, one of the equilibrium points is unstable (the saddle), while the other is stable (the node). Here is the state variable and is the bifurcation parameter. If there are two equilibrium points, a stable equilibrium point at and an unstable one at .At (the bifurcation point) there is exactly one equilibrium point. At this point the fixed point is no longer hyperbolic. Saddle node bifurcation A saddle-node bifurcation occurs in the consumer equation (see transcritical bifurcation) if the consumption term is changed from to , that is the consumption rate is constant and not in proportion to resource Saddle-node bifurcations may be associated with hysteresis loops and catastrophes.
Example[edit] Phase portrait showing Saddle-node bifurcation. An example of a saddle-node bifurcation in two-dimensions occurs in the two-dimensional dynamical system: As can be seen by the animation obtained by plotting phase portraits by varying the parameter See also[edit] Bifurcation theory. Phase portrait showing saddle-node bifurcation Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, 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 behaviour.[1] Bifurcations occur in both continuous systems (described by ODEs, DDEs or PDEs), 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.[2] Henri Poincaré also later named various types of stationary points and classified them.
Bifurcation types[edit] It is useful to divide bifurcations into two principal classes: Local bifurcations[edit] Poincaré–Bendixson theorem. Theorem[edit] Moreover, there is at most one orbit connecting different fixed points in the same direction. However, there could be countably many homoclinic orbits connecting one fixed point.
A weaker version of the theorem was originally conceived by Henri Poincaré, although he lacked a complete proof which was later given by Ivar Bendixson (1901). Discussion[edit] The condition that the dynamical system be on the plane is necessary to the theorem. On a torus, for example, it is possible to have a recurrent non-periodic orbit. In particular, chaotic behaviour can only arise in continuous dynamical systems whose phase space has three or more dimensions. Applications[edit] One important implication is that a two-dimensional continuous dynamical system cannot give rise to a strange attractor. References[edit] Jump up ^ Teschl, Gerald (2012).
Limit set. 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. Types[edit] In general limits sets can be very complicated as in the case of strange attractors, but for 2-dimensional dynamical systems the Poincaré–Bendixson theorem provides a simple characterization of all possible limit sets as a union of fixed points and periodic orbits.
Definition for iterated functions[edit] Let be a metric space, and let be a continuous function. -limit set of , denoted by , is the set of cluster points of the forward orbit of the iterated function . If and only if there is a strictly increasing sequence of natural numbers such that as . Where denotes the closure of set . If is a homeomorphism (that is, a bicontinuous bijection), then the Both sets are in R so that and. List of chaotic maps. List of chaotic maps[edit] List of fractals[edit]
Orbit (dynamics) For discrete-time dynamical systems the orbits are sequences, for real dynamical systems the orbits are curves and for holomorphic dynamical systems the orbits are Riemann surfaces. Diagram showing the periodic orbit of a mass-spring system in simple harmonic motion. (Here the velocity and position axes have been reversed from the standard convention in order to align the two diagrams) Given a dynamical system (T, M, Φ) with T a group, M a set and Φ the evolution function where we define then the set is called orbit through x. For every point x on the orbit.
Given a real dynamical system (R, M, Φ), I(x)) is an open interval in the real numbers, that is . Is called positive semi-orbit through x and is called negative semi-orbit through x. For discrete time dynamical system : forward orbit of x is a set : backward orbit of x is a set : and orbit of x is a set : where : Usually different notation is used : is written as where is in the above notation. acting on a probability space is a lattice inside. Limit-cycle. Behavior in a nonlinear system In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity.
Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912). We consider a two-dimensional dynamical system of the form where is a smooth function. With values in which satisfies this differential equation. Such that for all . By the Jordan curve theorem, every closed trajectory divides the plane into two regions, the interior and the exterior of the curve.
Given a limit cycle and a trajectory in its interior that approaches the limit cycle for time approaching . , and also for trajectories in the exterior approaching the limit cycle. Computational Methods Dynamical Systems. Lagrangian. Linear dynamical system. Dynamical system. List of dynamical systems and differential equations topics. Dynamical systems theory.