View source for Non-linear dynamics

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Many dynamical systems (cf. [[Dynamical system|Dynamical system]]) are described by difference equations (mappings)  $  \mathbf x _ {t + 1 }  = \mathbf f ( \mathbf x _ {t} ) $, 
where  $  t = 0, 1, \dots $, 
or by autonomous systems of differential equations (cf. [[Autonomous system|Autonomous system]])  $  d \mathbf u/dt = \mathbf F ( \mathbf u ) $, 
where  $  \mathbf x = ( x _ {1} \dots x _ {n} ) $
and  $  \mathbf u = ( u _ {1} \dots u _ {n} ) $. 
If  $  \mathbf f $
is a non-linear function of  $  \mathbf x $, 
the discrete dynamical system is called non-linear. Piecewise-linear systems are also regarded as being non-linear. A simple example is the logistic mapping  $  x _ {t + 1 }  = 4 x _ {t} ( 1 - x _ {t} ) $. 
If  $  \mathbf F $
is a non-linear function of  $  \mathbf u $, 
one calls the system of differential equations non-linear. A typical example is the pendulum described by  $  d \theta/dt = v $, 
$  dv/dt = - ( g/L )  \sin  \theta $, 
where  $  \theta $
is the displacement and  $  \omega  ^ {2} = g/L $. 
For small displacements the differential equation reduces to a linear differential equation (cf. also [[Linear differential operator|Linear differential operator]]). For the non-linear equation the amplitude depends on  $  \omega $, 
whereas for the linear equation the amplitude is independent of  $  \omega $.

Linear superposition no longer holds for non-linear dynamical systems. The behaviour of solutions of finite-dimensional non-linear difference and differential equations is much more complicated than that of finite-dimensional linear systems. For a linear system  $  d \mathbf u/dt = A \mathbf u $, 
where  $  A $
is an  $  ( n \times n ) $-
matrix with constant coefficients, the solution of the initial-value problem  $  \mathbf u ( t = 0 ) = \mathbf u _ {0} $
exists for  $  t \in [ 0, \infty ) $. 
Among non-linear systems one finds the behaviour that the solution only exists for finite times. An example is the differential equation  $  du/dt = u  ^ {2} $. 
The solution of the initial-value problem has a singularity at a finite time. Autonomous non-linear dynamical systems in the plane,  $  du _ {1} /dt = F _ {1} ( u _ {1} ,u _ {2} ) $, 
$  du _ {2} /dt = F _ {2} ( u _ {1} ,u _ {2} ) $
can show limit cycle behaviour (cf. [[Limit cycle|Limit cycle]]). An example is the [[Van der Pol equation|van der Pol equation]]

$$ 
{
\frac{du _ {1} }{dt }
 } = u _ {2} , \quad
{
\frac{du _ {2} }{dt }
 } = \epsilon ( 1 - u _ {1}  ^ {2} ) u _ {2} .
$$

For  $  n \geq  3 $, 
autonomous systems of first-order ordinary differential equations can show so-called chaotic behaviour (cf. [[Chaos|Chaos]]; [[Strange attractor|Strange attractor]]). Moreover, the attractor can have a non-integer fractal dimension (capacity and [[Hausdorff dimension|Hausdorff dimension]]). One of the tasks in non-linear dynamics is to classify the attractors (fixed points, limit cycles, tori, strange attractors). In a non-linear dynamical system, the domain of attraction can have a fractal boundary. Furthermore, certain particular solutions need not be included in the general solution. Most non-linear dynamical systems cannot be solved explicitly. Thus, qualitative methods are applied to study them (cf. [[Qualitative theory of differential equations|Qualitative theory of differential equations]]). One such method is the Poincaré section. Here, the non-linear differential equation is mapped to a discrete non-linear system. The study of non-linear discrete systems in the complex domain leads to Julia and Mandelbrot sets (cf. [[Julia set|Julia set]]). Non-linear finite-dimensional autonomous systems  $  d \mathbf u/dt = \mathbf F ( \mathbf u ) $, 
where  $  \mathbf F $
is an [[Analytic function|analytic function]], and non-linear discrete systems  $  \mathbf x _ {t + 1 }  = \mathbf f ( \mathbf x _ {t} ) $, 
where  $  \mathbf f $
is an analytic function, can be linearized via Carleman linearization to infinite-dimensional linear dynamical systems.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.J. Lichtenberg, M.A. Lieberman, "Regular and stochastic motion" , Springer (1983) {{MR|0681862}} {{ZBL|0506.70016}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Medved', "Fundamentals of dynamical systems and bifurcation theory" , Hilger (1992) {{MR|1170124}} {{ZBL|0777.58027}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> Yu.I. Neimark, P.S. Landa, "Stochastic and chaotic oscillations" , Kluwer Acad. Publ. (1992) {{MR|1174970}} {{ZBL|0756.58001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H.G. Schuster, "Deterministic chaos" , VCH (1988) (Edition: Second) {{MR|0968443}} {{MR|0935128}} {{ZBL|0707.58002}} {{ZBL|0707.58003}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M.W. Hirsch, S. Smale, "Differential equations, dynamical systems, and linear algebra" , Acad. Press (1974) {{MR|0486784}} {{ZBL|0309.34001}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> W.-H. Steeb, "A handbook of terms used in chaos and quantum chaos" , BI Wissenschaftsverlag (1991) {{MR|1166990}} {{ZBL|0743.58001}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> K. Kowalski, W.-H. Steeb, "Nonlinear dynamical systems and Carleman linearization" , World Sci. (1991) {{MR|1178493}} {{ZBL|0753.34003}} </TD></TR></table>