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Dynamical Systems Notes Dynamical Systems Notes These are notes takento prepare for an oral exam in Dynamical Systems. The content comes from various sources, including from notes from a class taught by Arnd Scheel. Overview - Definitions : Dynamical System : Mathematical formalization for any fixed "rule" which describes the time dependence of a point’s position in its ambient space. The concept unifies very different types of such "rules" in mathematics: the different choices made for how time is measured and the special properties of the ambient space may give an idea of the vastness of the class of objects described by this concept. 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 ambient space may be simply a set, without the need of a smooth space-time structure defined on it. 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. The measure theoretical definitions assume the existence of a measure-preserving transformation. This appears to exclude dissipative systems, as in a dissipative system a small region of phase space shrinks under time evolution. A simple construction (sometimes called the Krylov–Bogolyubov theorem) shows that 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. The difficulty in constructing the natural measure for a dynamical system 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. A dynamical system is a manifold M called the phase (or state ) space endowed with a family of smooth evolution functions Φt that for any element of t 5 T, the time, maps a point of the phase space back into the phase space. The notion of smoothness changes with applications and the type of manifold. There are several choices for the set T. When T is taken to be the reals, the dynamical system is called a flow ; and if T is restricted to the non-negative reals, then the dynamical system is a semi -flow . When T is taken to be the integers, it is a cascade or a map ; and the restriction to the non-negative integers is a semi -cascade . 6 The evolution function çt is often the solution of a differential equation of motion : x: vx. The equation gives the time derivative of a trajectory xt on the phase space starting at some point x0. The vector field vx is a smooth function that at every point of the phase space M provides the velocity vector of the dynamical system at that point. (These vectors t are not vectors in the phase space M, but in the tangent space TxM of the point x.) Given a smooth ç , an autonomous vector field can be derived from it. There’s no need for higher order derivatives in the equation, nor for time dependence in vx, these can be eliminated by 6 considering systems of higher dimension. Other types of differential equations than x : vx can be used to define the evolution: Gx,x% : 0 is an example arising from modeling of mechanical systems with complicated constraints. The differential equations determining the evolution çt are often ODEs; in this case the phase space M is a finite dimensional manifold. Geometrical Definition A dynamical system is the tuple MMM, f, TTT , with MMM a manifold (locally a Banach space or Euclidean space), TTT the domain for time (non-negative reals, the integers, ...) and f an evolution rule t f t (with t 5 TTT ) such that f t is a diffeomorphism of the manifold to itself. So, f is a mapping of the time-domain TTT into the space of diffeomorphisms of the manifold to itself. In other terms, ft is a diffeomorphism, for every time t in the domain TTT . Measure theoretical definition A dynamical system may be defined formally, as a measure -preserving transformation of a sigma -algebra , the quadruplet X,B,µ,^. Here, X is a set, and B is a sigma-algebra on X, so that the pair X,B is a measurable space . µ is a finite measure on the sigma-algebra, so that the triplet X,B,µ is a probability space . A map ^ : X X is said to be 4/22/2020 JodinMorey 1 B-measurable if and only if, for every ] 5 B, one has ^?1] 5 B. A map ^ is said to be a measure preserving map if and only if, for every ] 5 B, one has S^?1] : S]. Combining the above, a map ^ is said to be a measure -preserving transformation of X , if it is a map from X to itself, it is B-measurable, and is measure-preserving. The quadruple X,B,µ,^, for such a ^, is then defined to be a dynamical system . The map ^ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates ^n : ^ E ^ E E ^ for integer n are studied. For continuous dynamical systems, the map ^ is understood to be a finite time evolution map and the construction is more complicated. Examples : Logisticmap Complexquadraticpolynomial Dyadictransformation Tent map Double pendulum Arnold ’s cat map Horseshoe map Baker ’s map ; a chaotic piecewise linear map Billiards and outer billiards Hénonmap Lorenzsystem Circle map Rössler map Kaplan –Yorke map List of chaotic maps : Swinging Atwood’s machine, Quadratic map simulation system, Bouncing ball dynamics. Definitions r n A dynamical system often has an evolution rule of the form: x : fx,t, x 5 R , with an init. cond. xt0 : x0. 1 Lyapunov Stability : If the solutions that start out near an equilibrium point xe stay near xe forever, then xe is Lyapunov stable. Asymptotically Stable : if xe is Lyapunov stable and all solutions that start out near xe converge to xe, then xe is asymptotically stable. Exponential Stability : An equilibrium point xe : 0 is an exponentially stable equilibrium point of (1) if there exist ?αt?t0 constants m,α ; 0 and L ; 0 such that for all |xt0| L and t t0, we have: |xt| me |xt0|. The largest constant α which may be utilized is called the rate of convergence . Exponential Asymptotic Lyapunov Structural Stability : Fundamental property of some dynamical systems which implies that the qualitative behavior of the trajectories are unaffected by small perturbations (to be exact, C1-small perturbations). Examples of such qualitative properties are numbers of fixed points and periodic orbits (but not their periods). Unlike Lyapunov stability, which considers perturbations of initial conditions for a fixed system, structural stability deals with perturbations of the system itself. Phase -Space Distribution Function : A function of seven variables, f x,y,z,t; vx,vy,vz : [p,q;t, which gives the number of particles per unit volume in single-particle phase space. It is the number of particles per unit volume having approximately the velocity v : vx,vy,vz near the position r : x,y,z and time t. Liouville ’s Theorem : The phase-space distribution function is constant along the trajectories of the system—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability . The Liouville Equation : Describes the time evolution of the phase space distribution function [p,q;t: 6 6 d[ : /[ + n /[ q + /[ p . Time derivatives are evaluated according to Hamilton’s equations for the system. dt [t >i:1 /qi i /pi i This equation demonstrates the conservation of density in phase space. Liouville ’s Theorem states "The distribution function is constant along any trajectory in phase space." Poincaré Recurrence Theorem : Any dynamical system defined by an ordinary differential equation determines a flow map ft mapping phase space on itself. The system is said to be volume-preserving if the volume of a set in phase space is invariant under the flow. For instance, all Hamiltonian systems are volume-preserving because of Liouville’s theorem. The theorem is then: If a flow preserves volume and has only bounded orbits, then for each open set there exist orbits that 4/22/2020 JodinMorey 2 intersect the set infinitely often. Ergodic Theory : A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the phase space eventually revisit the set. More precise information is provided by various ergodic theorems which assert that, under certain conditions, the time average of a function along the trajectories exists almost everywhere and is related to the space average. For ergodic systems , this time average is the same for almost all initial points: statistically speaking, the system that evolves for a long time "forgets" its initial state. Bifurcation Theory : 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. 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.
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