View source for Non-linear dynamics

Many dynamical systems (cf. [[Dynamical system|Dynamical system]]) are described by difference equations (mappings) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n1100701.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n1100702.png" />, or by autonomous systems of differential equations (cf. [[Autonomous system|Autonomous system]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n1100703.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n1100704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n1100705.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n1100706.png" /> is a non-linear function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n1100707.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n1100708.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n1100709.png" /> is a non-linear function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007010.png" />, one calls the system of differential equations non-linear. A typical example is the pendulum described by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007013.png" /> is the displacement and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007014.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007015.png" />, whereas for the linear equation the amplitude is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007016.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007018.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007019.png" />-matrix with constant coefficients, the solution of the initial-value problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007020.png" /> exists for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007021.png" />. Among non-linear systems one finds the behaviour that the solution only exists for finite times. An example is the differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007022.png" />. The solution of the initial-value problem has a singularity at a finite time. Autonomous non-linear dynamical systems in the plane, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007024.png" /> can show limit cycle behaviour (cf. [[Limit cycle|Limit cycle]]). An example is the [[Van der Pol equation|van der Pol equation]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007025.png" /></td> </tr></table>

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007026.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007028.png" /> is an [[Analytic function|analytic function]], and non-linear discrete systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110070/n11007030.png" /> is an analytic function, can be linearized via Carleman linearization to infinite-dimensional linear dynamical systems.

====References====
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