1. Nonlinear Systems and Pattern Formation

See also Research

Pattern formation in nature and manmade systems is of fundamental importance. Among the observed patterns, dissipative structures play a special role. The latter can only be maintained, when being fed continuously with energy that is dissipated in the system and finally released. In addition, these systems are highly nonlinear, which makes a theoretical description difficult. Well known examples for such structures in nature are water waves, lightnings or cloud formation. However, the most exiting dissipative structures can be found in the biosphere. Despite this, at the time being, it is advisable to investigate first relatively simple physical systems, in order to reach a better understanding of the fundamentals of pattern formation.

The starting for the work in the field of Nonlinear Systems and Pattern Formation is the nonlinear 3-component reaction-diffusion equation of the general type

and related equations. In equation (1) u is referred to as activator and v as well as w as inhibitors. The S-shaped function f(u) is identical or similar to ?u-u³. As has been inferred from the electrical equivalent circuit for the 2-component version of equ. (1), the latter can be interpreted as a model equation for a two layer electrical transport system in 1- or 2-dimensional space, being equipped with a high ohmic layer where the nonlinearity is due to an S-shaped current voltage characteristic of some active medium [60, 61, 186]. In fact, equation (1) exhibits many solutions that correspond to patterns in spatially extended electrical transport systems when interpreting u as the normalized current density, v as the normalized voltage drop and the term ?u-u³ as resulting from an S-shaped current-voltage characteristic of some active material. As proposed in ref. [96], for the theoretical investigations we use the term dissipative solitons (DS) for well localize solutions of equ. (1). The same term is used for experimental current filaments. Here the latter generate spots on a line in quasi 1-dimensional systems, or in a plane in the case of quasi 2-dimensional systems. In both cases the line and the plane are oriented vertically to the main current direction.

On the described background, the activities in the field of Nonlinear Systems and Pattern Formation consist of the following topic:

analytical and numerical investigations of equ. (1),
experimental and theoretical investigations of electrical networks,
experimental and theoretical investigations of planar dc gas-discharge systems,
experimental and theoretical investigations of planar ac gas-discharge systems,
experimental and theoretical investigations of planar semiconductor devices.

Several system specific additional models have also been applied to semiconductor devices [56, 60, 63, 70, 86,117,149] and ac gas-discharge systems [120, 121, 176, 182].

It turns out, that besides the homogenous state seven types of fundamental patterns dominate:

periodic patterns in space,
fronts,
localized solitary pattern (dissipative solitons),
targets,
spirals,
Voronoi diagrams.

Among the highlights of this branch of activity we mention the following topic:

Analytical and numerical investigation of equ. (1), including its bifurcation behaviour.
Considering equ. (1) as a guide to possible patterns of a certain class of spacially extended nonlinear systems.
Quantitative agtreement between solutions of equ. (1) and the experimentally observed patterns on electrical networks.
System specific theoretical description of experimentally observerd current filaments in ac gas-discharge
Interpretation of current filaments in terms of dissipative solitons and theoretical and experimental manifestation of oscillatory tails of these objects.
Experimental and theoretical discovery of 'molecules' made of dissipative solitons that interact via oscilatory tails
Derivation of a particle equation for particle conserving DSs.

In what follows we use the following abbreviations: ann. = annihilation, bif. = bifurcation, breath. = breathing, com. = complex, conc. = concentric, DS = dissipative soliton, dyn. = dynamic; gen. = generation, hex. = hexagonal, hom. = homogeneous, inhom. = inhomogegeneous, interact. = interaction, mol. = molecule, oscill. = oscillation in time, period. = periodic in space, prop. = propagation, rock. = rocking, rot. = rotating, stat. = stationary, trans. = transition, trav. = travelling, d. p. = discharge plane, str. = stripe; unp. = unpublished, z.-z. = zigzag and 𝓡^1,2,3= quasi 1,2 or 3-dimensional systems or calculation.

Solutions of the reaction diffusion equation (1)

**Fig. 1.1-1**
Two independent stationary dissipative solitons in 2-dimensional space of the type shown on the left hand side, may form a stationary stable molecule due to their oscillatory tails [153].

**Fig. 1.1-2**
Rotating and outrunning spirals in 2-dimensional space [189].

Table 1.1-1

Solutions of equation (1) corresponding to the investigated experimental electrical networks. In general, the discretization is that of the experimental system and quantitative agreement is obtained between theory and experiment.

space	pattern in space in time	remark	ref.
𝓡¹	- period. stat.	periodicity slightly disturbed	[55], [60], [61]
𝓡¹	- hom. oscill.		[60]
𝓡¹	- hom. stat. - period. stat.	Turing-bif.: hom. stat. → period. stat.	[62], [64], [66]
𝓡¹	- front trav.	front: (hom. stat. high/ hom. stat. low)	unp.
𝓡¹	- front stat. - front trav.	bif. analysis of front motion: (period. stat./hom. oscill.)	[80]
𝓡¹	- front trav.	front: (hom. stat. high/hom. stat. low) bif: unidirectional prop.→ bidirectional prop.	[85]
𝓡¹	- front stat. - front trav.	fronts: (hom. stat. high/hom. stat. low); pinning	[98]
𝓡¹	- DS stat.	bif.: cascade of increasing number of stat. DSs	[64]
𝓡¹	- front stat. - front trav.	analytical treatment: spacial dependence of front velocity on impurities	[98]
𝓡²	- target stat.	nearly concentric rings	[60], [61], [62]

Table 1.1-2

Additional analytic and numerical solutions of equ.(1).

space	pattern in space in time	remark	Ref.
𝓡¹	- hom. oscill.		unp.
𝓡¹	- DS stat.	bif.: cascade of increasing number of stat. DSs	[60]
𝓡¹	- period. stat. - DS stat.	bif.: cascade of increasing number of stat. DSs	[61]
𝓡¹	- DS stat.	bif.: cascade of increasing number of DSs	[64]
𝓡¹	- DS stat. - DS trav. - DS breath. - DS rock.	analytical treatment in relation to p⁺n⁺pn^-semiconductor devices	[73]
𝓡¹	- DS rock. - DS stat.	bif.: cascade of increasing number of stat. DSs	[75]
𝓡¹	- front stat. - DS stat.	analytical treatment of front-antifront (stat. hom. high/ stat. hom. low) int.; construction of DSs from fronts and antifronts	[77]
𝓡¹	- DS trav. - DS breath - DS rock.	bif.: (DS, breath.) → (DS, rock.)	[78]
𝓡¹	- DS trav.	reflection at each other and at the boundary	[79]
𝓡¹	- hom. oscill. - period. stat. - front trav.	analytical treatment of the co-dimension-2 bif. analysis of fronts (period. stat./hom. oscill.)	[80]
𝓡¹	- front trav.	analytical treatment; bif.: travelling front (stat. hom. high/ stat. hom. low); bif.: uni-directional propagation → bi-directional propagation	[85]
𝓡¹	- front trav. - front pinned	analytical bif. analysis; trav. fronts (hom. high/ hom. low) in the presence of impurities	[94]
𝓡¹	- front stat. - front trav. - DS stat. - DS trav. - DS breath. - DS rock.	analytical treatment; fronts (hom. high/hom. low) and anti-front (hom. low/ hom. high); DSs generated from fronts and anti-fronts; cascade of increasing number of stat.	[96]
𝓡¹	- front stat. - front breath. - front rocking	analytical treatment of fronts (stat. hom. high/ stat. hom. low) and DSs; Hopf-bif.: stat. front → oscill. front; Hopf-bif.: DS stat. → DS breath. or DS stat. → DS rocking	[101]
𝓡²	- DS trav. - DS interact. - DS gen. - DS ann.	stabilization of multi-DS-solution in 𝓡ⁿ, n > 1 due to the third variable in equ. (1); oscill. tails of DSs; generation of DSs in the tails of existing ones; hex. pattern consisting of DSs;	[106]
𝓡²	- DS stat. - DS oscill. - DS trav.	bif: DS, stat. ? DS; elliptically oscill. ? DS, trav.	[107]
𝓡¹, 𝓡², 𝓡³	- DS stat. - DS trav.	analytical treatment; bif: DS stat. → DS trav.	[113]
𝓡²	- DS stat. - mol. stat.	analytical treatment; DS interaction to form ‘molecules’; reduction of the r-d-equation to a particle equation	[115]
𝓡³	- DS stat. - DS trav.	bif.: DS stat. → DS trav.; DS scattering; mol. formation; DS generation, annihilation, splitting	[124]
𝓡², 𝓡³	- period. stat. - DS stat. - DS trav.	DS reflection; evolution at const. parameters: period. stat → successive generation of DSs → final arrangement of DSs in a stat. hexagonal pattern (self-completion); alternative: self- completion by stating from a single stat. DS	[132]
𝓡², 𝓡³	- DS stat. - DS trav. - mol. stat. - mol. trav. - mol. rot.	analytical treatment; various kind of interaction of DSs; foundation of the particle concept of DSs by reducing equ. (1) to a particle equation	[137]
𝓡², 𝓡³	- DS	Self-completion in 𝓡², 𝓡³	[145]
	- DS stat. - DS trav. - mol. stat. - mol. rot.	analytical treatment; bif.: mol. stat. → mol. rot.	[153], [158]
	- mol. stat. - mol. rot.	analytical treatment; bif.: mol. stat. → mol. rot.; using the particle approach	[162]
𝓡²	- DS stat. - mol. stat.	oscill. tails and interaction law	[163]
𝓡²	- Voronoi diagrams		[164]
𝓡²	- DS stat. - DS trav.	analytical treatment; bif.: DS. stat. → DS. trav.; due to a change of shape	[166]
𝓡^1,2,3	- DS	comprehensive review focused on DSs; mathematical foundation of a particle approach	[171]
𝓡²	DS breath.	analytical treatment; bif.: DS stat. → DS breath.	[179]
𝓡^1,2,3	DS	comprehensive review focused on DSs	[186]

1.2 Experimental investigations of electrical networks

The experiments have been performed on quasi 1- and 2-dimensional discrete electrical networks of which the 1-dimensinal version is sketched in Fig. 1.2-1.

**Fig. 1.2-1**
Equivalent circuit for the 1-dimensional version of the experimental electrical network. The resistor R(I) has an S-shaped current-voltage-characteristic [55].

**Fig. 1.2-2**
Self- organized spatio-temporal voltage patterns overserved on a quasi-1-dimensional electric networks. Crosses indicate experimental results, continuous lines correspond to solutions of equ. (1). [55].

Tab. 1.2

Experimentally overserved self- organized patterns on quasi 1- and 2-dimensional electric networks

space	pattern in space in time	remark	ref.
𝓡¹	- nearly stat. period.		[55]
𝓡¹	- nearly stat. period. - hom. oscill. - inhom. stat. - domains oscill.	various inhom. stat. patterns depending on the transient voltage; almost periodic domains oscillating with slightly different frequency	[60]
𝓡¹	- period. stat.	almost periodic domains oscillating with slightly different frequency	[61]
𝓡¹	- period. stat. - hom. oscill. - domains oscill. - DS stat. - DS trav.	solitary trav. DSs correspond to Fitz-Hugh-Nagumo nerve pulses; bif.: cascade of increasing number of stat. DSs; various other bif. scenarios; Turing bif.: hom. stat. → period. stat.	[62]
𝓡¹	- period. stat. - DS stat.	Turing bif.: hom. stat. → period. stat.; bif.: cascade of increasing number of stat. DSs; various other bif. scenarios	[64]
𝓡¹	- front trav. - perid. stat. - DS stat.	fronts: (period. stat./hom. stat.); Turing bif.: hom. stat. → period. stat.; bif.: cascade of increasing number of stat.	[66]
𝓡¹	- front stat. - front trav.	trav. front (hom. oscill./period. stat.); measurement of the bifurcation behaviour	[80]
𝓡¹	front trav.	front: (hom. high/ hom. low); bif.: unidirectional prop. → bidirectional prop.	[85]
𝓡¹	front stat. front trav.	front: (hom. high/ hom. low); dependence of speed on inhom.	[98]
𝓡²	- hom. oscill. - period. stat. - target stat.		[60]

1.3 Dc gas-discharge systems

The 2-dimensionnal version of the experimental set-up is sketched in Fig. 1.3-1. The device reduces to a quasi 1-demsional system provided the planar electrodes effectively degenerate to a line.

**Fig. 1.3-2**
Experimental set-up of the 2-dimensiona DC-gas-discharge system [188].

**Fig. 1.3-2**
Some patterns being observed in quasi 2-dimensional dc-gas-discharge systems. [187]

Table 1.3

Experimentally observed self-organized patterns in quasi 1- and 2-dimensioal dc gas-discharge systems.

space	pattern in space in time	remark	Ref.
𝓡¹	- DS stat.	bif.: cascade of increasing number of stat. DS	[58], [60], [61], [64], [65]
𝓡¹	- DS stat. - DS dyn. - DS chaos - DS compl.	bif.: cascade of increasing number of stat. DS; repeated splitting with subsequent deletion of one DS; spatiotemporal chaotic behaviour; com. dynamics of DSs;	[68]
𝓡¹	- period. stat. - period. trav. - period. rock./ trav.	Turing-bif. and subritical bif.: hom. stat. → period. stat.; coexistence of (period. stat.) and (period. trav.); period. state: simultaneous rock. and trav.	[69], [70]
	- hom. stat. - period. stat.	Turing- bif.: (hom. stat.) → (period. stat.)	[71]
𝓡¹	- DS stat	detection of oscillating self-exited moving striations	[72]
𝓡¹	- DS trav. - mol. trav. - DS compl.	DS trav.: reflection at the boundary and at each other, generation, annihilation, repeated splitting followed by extinction of 1 DS; further comlex dynamics in the course of DS propagation	[74]
𝓡¹		reviewing preceding result	[76]
𝓡¹	- DS trav. - Mol. trav.	DS trav.: reflection at the boundary and at each other, mol. formation and decay	[79]
𝓡¹	- DS stat. - DS trav. - mol. trav. - DS rock.	bif.: cascade of increasing number of stat. DSs; DS trav.: reflection at the boundary and at each other, generation, annihilation	[96]
𝓡¹	- DS stat. - DS trav.	bif.: cascade of increasing number of stat. DSs; reflection at the boundary	[104]
𝓡¹		summary of results	[131]
𝓡¹		summary of results	[136]
𝓡²	- period./ stat. hex. - period./ stat. stripes	Turing bif.: (hom. stat.) → (stripes, stat.); hexagonal arrangement of DSs covering the whole discharge plane	[102]
𝓡²	- perid./ stat. hex. - period./ stat. stripes		[104]
𝓡²	- period./ stat. stripes	z.z. destabilization	[109]
𝓡²	- perid./ stat. hex. - period./ stat. stripes	bif.:( stripe and hexagonal pattern) → ((non stationary patterns)	[110]
𝓡²	- period./ stat. hex. - stripes stat. - target stat. - spirals. rot.	z.z. destabilisation of targets and spirals; coexistence of (period/hex) and spirals; bif.: (hom./stat.) → (hex./stat.) → (stripe/stat)	[114]
𝓡²	- DS stat. - DS/ stat. hex. - period. stat. stripes	bif.: cascade of increasing number of stat. DSs to hexagonal arrangement; defects in hex. pattern	[116]
𝓡²	- stripe stat. - stripe period. stat. - stripe stat zigzag	bif.: single stat. stripe → period. stripe; bif.: period. stripe → zigzag stripe	[126]
𝓡²	- domains stat.	dynamic interaction	[128]
𝓡²		summary of results	[131]
𝓡²	- DS stat. - DS trav. - DS trav. breath. - DS gen. - DS ann.	DS generation and annihilation; various many DS patterns: e.g. DS arranged on lines, gas-like pattern	[133]
𝓡²	- DS trav. DSs	generation, annihilation, mol. formation	[134]
𝓡²		summary of results	[136]
𝓡²	- DS	gas-like many DS patterns; bif.: increasing number of DS; DS domains oscillating with different frequency	[142]
𝓡²	- rot. waves		[150]
𝓡²	- DS stat. - DS trav.	bif.: (DS stat.) → (DS trav.); use of stochastic data analysis	[151], [152]
𝓡²	- DS stat.	bif.: (DS stat.) → (DS trav.)	[154]
𝓡²	- mol. stat. - mol rot.	bif.: (mol, stat.) → (mol, rot.)	[162]
𝓡²	- DS trav.	detection of of oscillatory tails and measuring the interaction law of DSs by using stochastic data analysis	[163]
𝓡²	- DS stat. - DS trav.	bif.: (DS, stat.) → (DS, trav.)	[166]
𝓡²		summary of results	[167]
𝓡²	- DS stat. - DS trav. - perid. stat. and stripes trav. - target outrun. - spiral rot.	many stat. DSs at relatively large distance; gas-like motion of many DSs; stat. and trav. stripes; non-periodic stripes	[168]
𝓡²	- DS trav.	spontaineous division	[177]
𝓡²		Summary of results	[180], [184]
𝓡²	- DS	stochastic data analysis of the experimentally observed DSs	[185]
𝓡¹, 𝓡²	- DS	comprehensive review	[186]
𝓡²		series of pictures of patterns in planar dc gas-discharge systems	[188]
𝓡¹, 𝓡²		comprehensive review with respect to patter formation in planar dc gas-discharge systems	[190]

1.4 Experimental overservation of ac gas-discharge systems

The 2-dimensionnal version of the experimental set-up is sketched in Fig. 1.4-1. The device reduces to a quasi 1-demsional system provided the planar electrodes effectively degenerate to a line.

**Fig. 1.4-1**
Experimental set-up of the 2-dimensional ac gas-discharge system [187].

**Fig. 1.4-2**
Some patterns being observed in quasi 2-dimensional ac gas-discharge systems. [187]

Table 1.4

Experimentally observed self-organized patterns in quasi 1- and 2-dimensioal dc gas-discharge systems. The term ‘stat.’ refers to stationary in the sense of avera-ging with respect to a time that I much large then the period of the driving voltage.

space	pattern in space in time	remark	Ref.
𝓡²	- period./ stat. hex.	bif.: cascade of increasing number of stat. DS	[68], [76]
𝓡²	- DS stat. - mol. stat. - period. / stat. stripes	bif.: (hom./stat.) → (period. stripes/stat.); (DS/stat.) surrounded by closed loops generated by DSs on a short timescale	[81]
𝓡²	- DS stat. - Mol. stat.	(DS/stat.) arranged on curved lines; bif.: cascade of decreasing number of DSs	[99]
𝓡²	- DS breath.		[119]
𝓡²	- DS stat. - DS trav. - period./ stat. hex. - period. stat. stripes	(period./hex.) patterns seem to consist of DSs	[120]
𝓡²	- DS weak motion - perid./ stat. hex.	(period./hex.) pattern seem to consist of DSs; domains made of DSs	[122]
𝓡²	- DS trav. - DS ann. - mol. trav.		[127]
𝓡²		summary of results	[136]
𝓡²	- Voronoi diagrams		[143]
𝓡²	- Target stat.	bif.: destabilization into DSs	[157]
𝓡²	- Voronoi diagrams		[164]
𝓡²	- period./ rot. hex.	period. pattern consisting of DSs	[165]
𝓡²	- DS stat. - DS trav. - mol. - period./ hex.	many DS patterns: gas-like, liquid-like, arranged on a circle, molecules	[170]
𝓡²	- perod./ stat. hex.	bif.:( hom./stat.) → (period./hex. stat.); description in terms of plasma specific transport equations	[176]
𝓡²		series of pictures of patttern in planar dc gas-discharge systems	[188]
𝓡²	- DS	transition from ‘bright’ to ‘dark spots’ and vice versa	[181]
𝓡²	- DS stat. - DS trav. - period./ hex.		[182]
𝓡²		Summary of results	[184]
𝓡1, 𝓡²	DS	comprehensive review	[186]
𝓡²		series of pictures of patterns in planar ac gas-discharge systems	[187]

1.5 Experimental investigations of semiconductors

An example of the 2-dimensionnal version of the experimental set-up is sketched in Fig. 1.5-1.

**Fig. 1.5-1**
Experimental set-up for the investigation of p⁺n⁺pn^-devices [75].

**Fig. 1.5-2**
Travelling dissipative soliton being reflected at the boundary (a) and breathing DS. (b) [75]

Table 1.5

space	pattern in space in time	remark	Ref.
𝓡²	- DS stat.	pin diode	[59]
𝓡¹	- DS rocking	npnp^-- device	[71]
𝓡¹	- DS stat. - DS trav. - DS rocking	p⁺n⁺p^--semiconductor device; also: DS reflection at the boundary	[73]
𝓡¹	- DS stat. - DS trav. - DS rocking	n⁺pnp^-and p⁺n⁺pn devices; essentially 𝓡¹; also: DS reflection at the boundary; bif.: cascade of increasing number of stat. DS	[75]
𝓡¹	- DS stat. - DS rock. - DS dyn.	p⁺n⁺pn^-device; complex dynamics of a single DS: period doubling cascade to chaos	[83]
𝓡¹	- DS trav. - DS rock.	p⁺n⁺pn^-device	[86]
𝓡¹	- DS trav.	p⁺n⁺pn^-device; also: complex dynamical behaviour of an isolated DS	[88]
𝓡¹	- DS	Summary of result	[89]
𝓡²	- DS stat.	n⁺pnp^- and p⁺n⁺pn^--device	[95]
𝓡²	- DS stat. - DS trav. - front stat. - strings	ac driven ZnS:Mn electroluminescence device; bif.: cascade of increasing number of stat. DS; global oscillations; complex patterns: developing of rings and decay to strings (irregular formed sections of stripes); interaction of strings; domains	[97]
𝓡²	- DS stat. - DS trav. - front stat. - front trav.	ac driven ZnS:Mn electroluminescence device; bif.: cascade of increasing number of stat. DS	[99]
𝓡¹	- DS stat. - DS dyn.	p⁺n⁺pn^-device; complex dynamics of a single DS by superimposing an ac driving voltage: period doubling cascade to chaotic motion when superimposing an ac driving voltage; Arnold tongues	[103]
𝓡¹	- DS	p⁺n⁺pn^-device; influence of laser irradiation	[105]
𝓡²	- DS stat. - strings trav.	ac driven ZnS:Mn electroluminescence devices ; also: filament clusters and coexistence of DSs and strings	[112]
	-	p⁺n⁺pn^-andac driven ZnS:Mn electroluminescence devices:summary of results	[117]
𝓡²	- DS stat. - DS trav. - strings - spirals*⁾	ac driven ZnS:Mn electroluminescence devices; summary of results *⁾ see ref. in the cited ref.	[123], [129], [161]

Institute of Applied Physics

Nonlinear Systems and Patternformation - Magnetism - Material Science - Applied Physics

Nonlinear Systems and Pattern Formation - Magnetism - Materials Science - Applied Physics