# Dynamic system

A ( deterministic ) dynamic system is a mathematical model of a time-dependent process that is homogeneous with respect to time, the further course of which therefore only depends on the initial state , but not on the choice of the starting point in time. The term dynamic system in its current form goes back to the mathematicians Henri Poincaré and George David Birkhoff .

Dynamic systems have a wide range of applications in everyday processes and allow insights into many areas not only of mathematics (e.g. number theory , stochastics ), but also physics (e.g. pendulum movement , climate models ) or theoretical biology (e.g. . Predator-prey models ).

A distinction is made between discrete and continuous time development. In a time-discrete dynamic system, the states change in equidistant time leaps, i.e. H. in successive, always equally large time intervals, while the state changes of a time-continuous dynamic system take place in infinitesimally small time steps. The most important means of describing continuous-time dynamic systems are autonomous ordinary differential equations . A mixed system of continuous and discrete sub-systems with continuously discrete dynamics is also referred to as hybrid . Examples of such hybrid dynamics can be found in process engineering (e.g. dosing master systems).

Important questions in connection with dynamic systems primarily concern their long-term behavior (e.g. stability , periodicity , chaos and ergodicity ), system identification and their regulation .

## Introductory examples

### Exponential growth

Two exponentially growing populations x t (red) and y t (blue) with y 0 = x 3

A simple example of a dynamic system is the evolution over time of a quantity that is subject to exponential growth , such as a population of an unhindered bacterial culture . The state at a fixed point in time is given here by a non-negative real number, namely the stock size of the population, that is, the state space of the system is the set of non-negative real numbers. If one first considers the states at the discrete points in time , i.e. over the period , then the following applies with a constant growth factor . This results in the state at a point in time . ${\ displaystyle X = [0, \ infty)}$${\ displaystyle x_ {0}, x_ {1}, x_ {2}, \ dotsc}$${\ displaystyle t = 0,1,2, \ dotsc}$ ${\ displaystyle T = \ mathbb {N} _ {0}}$${\ displaystyle x_ {t + 1} = ax_ {t}}$${\ displaystyle a}$${\ displaystyle t \ in T}$${\ displaystyle x_ {t} = a ^ {t} x_ {0}}$

The characteristic property of a dynamic system is that the state depends on the elapsed time and the initial value , but not on the choice of the initial point in time. Let another exponentially growing population with the same growth factor but with the initial value be given. At one point in time then applies ${\ displaystyle t \ in T}$${\ displaystyle x_ {0} \ in X}$${\ displaystyle y_ {0}, y_ {1}, y_ {2}, \ ldots}$${\ displaystyle a}$${\ displaystyle y_ {0} = x_ {t}}$${\ displaystyle s \ in T}$

${\ displaystyle y_ {s} = a ^ {s} y_ {0} = a ^ {s} a ^ {t} x_ {0} = a ^ {s + t} x_ {0} = x_ {s + t }}$.

The second population grows in the same period as the first in the period . This behavior can be or otherwise express: The so-called flow function , which any time and any initial state the state at the time associates, in this case , satisfied for all and all , the equation ${\ displaystyle [0, s]}$${\ displaystyle [t, s + t]}$ ${\ displaystyle \ Phi \ colon T \ times X \ to X}$${\ displaystyle t \ in T}$${\ displaystyle x \ in X}$${\ displaystyle \ Phi (t, x)}$${\ displaystyle t}$${\ displaystyle \ Phi (t, x) = a ^ {t} x}$${\ displaystyle s, t \ in T}$${\ displaystyle x \ in X}$

${\ displaystyle \ Phi {\ bigl (} s, \ Phi (t, x) {\ bigr)} = \ Phi (s + t, x)}$.

This is the so-called semigroup property of the flow of a dynamic system.

### Spring pendulum

Another source for dynamic systems is the mathematical modeling of mechanical systems, in the simplest case the movement of a point of mass under the influence of a force that depends on the location and speed, but not explicitly on time. The state of such a system at a point in time is given as the ordered pair , consisting of the location and the speed . In particular, the entire sequence of movements is then clearly determined by specifying an initial position together with an initial speed . In the case of a one-dimensional movement, this is the state space . ${\ displaystyle t \ in T = [0, \ infty)}$ ${\ displaystyle (x (t), v (t))}$${\ displaystyle x (t)}$${\ displaystyle v (t)}$${\ displaystyle x (0) = x_ {0}}$${\ displaystyle v (0) = v_ {0}}$${\ displaystyle X = \ mathbb {R} ^ {2}}$

 Damped oscillation and path in the state space

As a concrete example, a spring pendulum shall be considered, on whose mass the restoring force of the spring and possibly a speed-dependent frictional force acts with the mass . If one denotes the total force with , then the ordinary differential equation system results for the state${\ displaystyle m}$${\ displaystyle F (x (t), v (t))}$

{\ displaystyle {\ begin {aligned} {\ dot {x}} (t) & = v (t), \\ {\ dot {v}} (t) & = {\ frac {1} {m}} F (x (t), v (t)), \ end {aligned}}}

where the point above the variable denotes the derivative according to the - in this example continuous - time. The first equation says that the velocity is the derivative of the location with respect to time, and the second results directly from Newton's second axiom , according to which mass times acceleration is equal to the total force acting on the mass point.

It can be shown that in this system too the flow

${\ displaystyle \ Phi \ colon T \ times X \ to X, \ quad \ Phi (t, x_ {0}, v_ {0}) = {\ bigl (} x (t), v (t) {\ bigr )}}$

fulfills the semi-group property. If one looks at the course of the system state in the state space , i.e. the so-called path , a trajectory emerges in the case of a damped oscillation of the spring pendulum , which spirals towards the rest position . ${\ displaystyle X = \ mathbb {R} ^ {2}}$ ${\ displaystyle \ {(x (t), v (t)) \ in \ mathbb {R} ^ {2} \ mid t \ geq 0 \}}$${\ displaystyle (0,0)}$

## Definitions

A dynamical system is a triple consisting of a set or the period , a non-empty set , the state space (the phase space ), and an operation from on so that for all states and all times : ${\ displaystyle (T, X, \ Phi),}$ ${\ displaystyle T = \ mathbb {N} _ {0}, \ mathbb {Z}, \ mathbb {R} _ {0} ^ {+}}$${\ displaystyle \ mathbb {R},}$${\ displaystyle X}$ ${\ displaystyle \ Phi \ colon \, T \ times X \ to X}$${\ displaystyle T}$${\ displaystyle X,}$ ${\ displaystyle x \ in X}$ ${\ displaystyle t, s \ in T}$

1. ${\ displaystyle \ Phi (0, x) = x}$   ( Identity property ) and
2. ${\ displaystyle \ Phi (s, \ Phi (t, x)) = \ Phi (s + t, x)}$   ( Semigroup property ).

If or is, then it means time-discrete or discrete for short , and with or is called time-continuous or continuous . is also referred to as a discrete or continuous dynamic system for real time or as invertible if or applies. ${\ displaystyle T = \ mathbb {N} _ {0}}$${\ displaystyle T = \ mathbb {Z}}$${\ displaystyle (T, X, \ Phi)}$ ${\ displaystyle T = \ mathbb {R} _ {0} ^ {+}}$${\ displaystyle T = \ mathbb {R}}$${\ displaystyle (T, X, \ Phi)}$ ${\ displaystyle (T, X, \ Phi)}$${\ displaystyle T = \ mathbb {Z}}$${\ displaystyle T = \ mathbb {R}}$

For each the mapping is called the movement of , and the set is called the orbit (the (full) orbit , the trajectory , the phase curve , the trajectory curve , the solution curve ) of . The positive half-orbit or forward orbit of is and, if invertible, is the negative half-orbit or reverse orbit of . ${\ displaystyle x \ in X}$${\ displaystyle \ beta _ {x} \ colon \, T \ to X, \, t \ mapsto \ beta _ {x} (t): = \ Phi (t, x)}$${\ displaystyle x = \ beta _ {x} (0)}$${\ displaystyle O (x): = \ {\ beta _ {x} (t) \ mid t \ in T \}}$${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle O ^ {+} (x): = \ {\ beta _ {x} (t) \ mid t \ in T \ cap \ mathbb {R} _ {0} ^ {+} \}}$${\ displaystyle (T, X, \ Phi)}$${\ displaystyle O ^ {-} (x): = \ {\ beta _ {x} (t) \ mid -t \ in T \ cap \ mathbb {R} _ {0} ^ {+} \}}$${\ displaystyle x}$

A discrete dynamic system is continuous if its state space is a (non-empty) metric space and if every transformation belonging to a point in time is continuous . A continuous dynamic system is called continuous or a semi-flow if its state space is a metric space and if every transformation belonging to a point in time and every movement of a state is continuous. In addition, a continuous discrete dynamic system is also called a cascade and a half-flow is called a river . The state space of a continuous dynamic system is also called the phase space and of each of the orbit as the phase curve or trajectory of which is simply written with . ${\ displaystyle (T, X, \ Phi)}$${\ displaystyle X}$${\ displaystyle t \ in T}$ ${\ displaystyle \ varphi _ {t} \ colon \, X \ to X, \, x \ mapsto \ varphi _ {t} (x): = \ Phi (t, x),}$ ${\ displaystyle (T, X, \ Phi)}$ ${\ displaystyle X}$${\ displaystyle (\ mathbb {Z}, X, \ Phi)}$${\ displaystyle (\ mathbb {R}, X, \ Phi)}$${\ displaystyle x_ {0} \ in X}$${\ displaystyle x_ {0}}$${\ displaystyle x \ colon \, t \ mapsto x (t)}$${\ displaystyle x (0) = x_ {0}}$

Coupled to continuous and given case additional discrete dynamical systems into a single system, it is called this a continuous-discrete it or hybrid it dynamic system.

### Remarks

• In the literature, a distinction is often not made between dynamic systems and continuous dynamic systems or flows, and a flow is often understood as a differentiable flow (see below). There are also more general definitions of continuous dynamic systems in which z. B. a topological manifold , a (possibly compact ) Hausdorff space or even just a topological space is taken as phase space .
• Instead of the left operation as in the definition above, dynamic systems are often defined with a right operation on , the order of the arguments then reverses accordingly.${\ displaystyle \ Phi}$ ${\ displaystyle \ Phi ^ {*} \ colon \, X \ times T \ to X}$${\ displaystyle X}$
• In the definition, the identity property of the operation is required because every state should not change as long as no time passes (i.e. for ). This property means that the corresponding transformation is the identical mapping to :  ${\ displaystyle \ Phi}$${\ displaystyle x}$${\ displaystyle t = 0}$${\ displaystyle 0}$${\ displaystyle X}$${\ displaystyle \ varphi _ {0} = \ operatorname {id} _ {X}.}$
• The semigroup property makes the dynamic system homogeneous with respect to time: You first get from state to state in time units and then from there in time units to state , i.e. H. to the same state that one comes to directly from the state in units of time. The all time points belonging transformations form a commutative semigroup with the composition as a link and a neutral element , as well as the picture is a semi group homomorphism :  for all this transformation semigroup is in invertible dynamical systems even a group , because for all is the inverse element to${\ displaystyle t}$${\ displaystyle x}$${\ displaystyle \ Phi (t, x)}$${\ displaystyle s}$${\ displaystyle \ Phi (s + t, x)}$${\ displaystyle x}$${\ displaystyle s + t}$${\ displaystyle t}$${\ displaystyle \ varphi _ {t} \ colon \, X \ to X, \, x \ mapsto \ varphi _ {t} (x): = \ Phi (t, x),}$ ${\ displaystyle \ circ}$ ${\ displaystyle \ varphi _ {0}}$${\ displaystyle T \ to X ^ {X} \!, \, t \ mapsto \ varphi _ {t},}$${\ displaystyle \ varphi _ {s + t} = \ varphi _ {s} \ circ \ varphi _ {t}}$${\ displaystyle s, t \ in T.}$${\ displaystyle t \ in T}$${\ displaystyle \ varphi _ {- t}}$${\ displaystyle \ varphi _ {t}.}$
• A dynamic system with or can be exactly invertible then to a dynamic system to continue when to belong transform an inverse function has. There are then and recursive for all. If continuous, then are uniquely given by for all with and also all transformations belonging to negative times. An operation of on is explained with so precisely that the invertible continuation of is.${\ displaystyle (T, X, \ Phi)}$${\ displaystyle T = \ mathbb {N} _ {0}}$${\ displaystyle T = \ mathbb {R} _ {0} ^ {+}}$${\ displaystyle (T ', X, \ Phi')}$${\ displaystyle (T '\ cap \ mathbb {R} _ {0} ^ {+}, X, \ Phi' | _ {(T '\ cap \ mathbb {R} _ {0} ^ {+}) \ times X}) = (T, X, \ Phi)}$ ${\ displaystyle 1}$${\ displaystyle \ varphi _ {1}}$ ${\ displaystyle (\ varphi _ {1}) ^ {- 1}}$${\ displaystyle \ varphi _ {- 1}: = (\ varphi _ {1}) ^ {- 1}}$ ${\ displaystyle \ varphi _ {- (n + 1)}: = \ varphi _ {- 1} \ circ \ varphi _ {- n}}$${\ displaystyle n \ in \ mathbb {N}.}$${\ displaystyle (T, X, \ Phi)}$${\ displaystyle \ varphi _ {- t}: = \ varphi _ {1-s} \ circ \ varphi _ {- (n + 1)}}$${\ displaystyle t = n + s \ in \ mathbb {R} _ {0} ^ {+}}$${\ displaystyle n \ in \ mathbb {N} _ {0}}$${\ displaystyle s \ in [\, 0,1)}$${\ displaystyle T ': = T \ cup \ {- t \ mid t \ in T \}}$${\ displaystyle \ Phi '\ colon \, T' \ times X \ to X, \, (t, x) \ mapsto \ Phi '(t, x): = \ varphi _ {t} (x),}$${\ displaystyle T '}$${\ displaystyle X}$${\ displaystyle (T ', X, \ Phi')}$${\ displaystyle (T, X, \ Phi)}$
• Because of the semigroup property, every discrete dynamic system or as an iterative application of the corresponding transformation with the points in time as iteration indices can be understood:  for all and with is additionally for all therefore is already clearly defined by and is easier to write.${\ displaystyle (\ mathbb {N} _ {0}, X, \ Phi)}$${\ displaystyle (\ mathbb {Z}, X, \ Phi)}$${\ displaystyle 1}$${\ displaystyle \ varphi: = \ varphi _ {1}}$${\ displaystyle \ varphi _ {t + 1} = \ varphi \ circ \ varphi _ {t}}$${\ displaystyle t \ in \ mathbb {N} _ {0}}$${\ displaystyle (\ mathbb {Z}, X, \ Phi)}$${\ displaystyle \ varphi _ {t-1} = \ varphi ^ {- 1} \ circ \ varphi _ {t}}$${\ displaystyle -t \ in \ mathbb {N} _ {0}.}$${\ displaystyle (T, X, \ Phi)}$${\ displaystyle \ varphi}$${\ displaystyle (X, \ varphi)}$
• If you limit the time to a continuous dynamic system , then a discrete dynamic system always results . On the one hand, this discretization is widely used in numerics , e.g. B. in backward analysis. On the other hand, there are natural and technical systems that are characterized by non-continuous changes in state and that can be modeled in a direct way by discrete dynamic systems.${\ displaystyle (T, X, \ Phi),}$${\ displaystyle T \ cap \ mathbb {Z}}$${\ displaystyle (T \ cap \ mathbb {Z}, X, \ Phi | _ {(T \ cap \ mathbb {Z}) \ times X})}$
• Differentiable (half) flows are (half) flows in which every transformation belonging to a point in time is differentiable . In particular, each of these transformations of a differentiable flow is a diffeomorphism .${\ displaystyle (T, X, \ Phi)}$
• In the theory of dynamic systems one is particularly interested in the behavior of trajectories for . Here, limit sets and their stability is of great importance. The simplest limit sets are fixed points , these are those points with for all , i.e. those states whose orbit is the one-element set . One is also interested in points whose path converges towards a fixed point. In addition to fixed points, the most important limit quantities are the periodic orbits . However, especially in non-linear systems, one also encounters complex non-periodic limit sets. In the theory of non-linear systems, fixed points, periodic orbits and general non-periodic limit sets are subsumed under the umbrella term attractor (or repeller , if repulsive, see also strange attractor ). These are examined in detail in chaos theory .${\ displaystyle t \ to \ pm \ infty}$${\ displaystyle x \ in X}$${\ displaystyle \ Phi (t, x) = x}$${\ displaystyle t \ in T}$${\ displaystyle x}$${\ displaystyle \ {x \}}$${\ displaystyle t \ to + \ infty}$

## Important special cases

### Ordinary differential equations

Continuous dynamic systems occur mainly in connection with ordinary differential equations . The autonomous differential equation is given

${\ displaystyle {\ dot {x}} (t) = f (x (t))}$

with a vector field in a field . If the equation for all the initial values of one for all defined and unique solution with possesses, then with a continuous dynamic system. The trajectories of the system are therefore the solution curves of the differential equation. The fixed points here are those with ; they are also called stationary or critical points of the vector field. ${\ displaystyle f \ colon X \ to \ mathbb {R} ^ {n}}$ ${\ displaystyle X \ subseteq \ mathbb {R} ^ {n}}$${\ displaystyle x_ {0} \ in X}$${\ displaystyle t \ in \ mathbb {R}}$${\ displaystyle \ beta _ {x_ {0}} \ colon \ mathbb {R} \ to X}$${\ displaystyle \ beta _ {x_ {0}} (0) = x_ {0}}$${\ displaystyle (\ mathbb {R}, X, \ Phi)}$${\ displaystyle \ Phi (t, x): = \ beta _ {x} (t)}$${\ displaystyle x \ in X}$${\ displaystyle f (x) = 0}$

### iteration

Discrete dynamic systems are closely related to the iteration of functions. If there is a self-mapping of an arbitrary set , i.e. a function that assigns an element to each , then one can consider the recursively defined sequence for an initial value . With -fold consecutive execution ( times) then applies . The equation shows that order with a discrete dynamic system. Conversely, for a dynamic system by an image with defined. The fixed points of such a system are those with . ${\ displaystyle g \ colon X \ to X}$${\ displaystyle X}$${\ displaystyle x \ in X}$${\ displaystyle g (x) \ in X}$${\ displaystyle x_ {0} \ in X}$${\ displaystyle x_ {n + 1} = g (x_ {n})}$${\ displaystyle n \ in \ mathbb {N} _ {0}}$${\ displaystyle n}$ ${\ displaystyle g ^ {n} = g \ circ g \ circ \ ldots \ circ g}$${\ displaystyle n}$${\ displaystyle x_ {n} = g ^ {n} (x_ {0})}$${\ displaystyle g ^ {m + n} = g ^ {m} \ circ g ^ {n}}$${\ displaystyle (\ mathbb {N} _ {0}, X, \ Phi)}$${\ displaystyle \ Phi (n, x_ {0}) = g ^ {n} (x_ {0})}$${\ displaystyle (\ mathbb {N} _ {0}, X, \ Phi)}$${\ displaystyle g (x): = \ Phi (1, x)}$${\ displaystyle g \ colon X \ to X}$${\ displaystyle \ Phi (n, x_ {0}) = g ^ {n} (x_ {0})}$${\ displaystyle x \ in X}$${\ displaystyle g (x) = x}$

Examples of this are Markov chains in discrete time with finite state space . The state space for the purposes of a dynamic system are then all likelihood vectors to the time is , and the iteration is given by the left multiplication of the likelihood vector with the transition matrix . The fixed points are then the stationary distributions . ${\ displaystyle S}$${\ displaystyle S}$${\ displaystyle T = \ mathbb {N}}$${\ displaystyle x_ {n}}$ ${\ displaystyle M}$

## literature

• Herbert Amann: Ordinary differential equations . 2nd Edition. de Gruyter, Berlin 1995, ISBN 3-11-014582-0 .
• George David Birkhoff : Dynamical Systems. Rev. Ed. AMS, Providence, RI, 1966.
• Manfred Denker: Introduction to the Analysis of Dynamic Systems . Springer, Berlin a. a. 2005, ISBN 3-540-20713-9 .
• John Guckenheimer , Philip Holmes: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Corr. 3rd printing. Springer, New York 1990, ISBN 3-540-90819-6 .
• Diederich Hinrichsen, Anthony J. Pritchard: Mathematical Systems Theory I - Modeling, State Space Analysis, Stability and Robustness. Springer, 2005.
• Wolfgang Metzler: Nonlinear Dynamics and Chaos , BG Teubner, Stuttgart / Leipzig 1998, ISBN 3-519-02391-1 .
• Gerald Teschl : Ordinary Differential Equations and Dynamical Systems . American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( free online version ).
• J. de Vries: Elements of Topological Dynamics. Springer, 1993.