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A short summary of this paper. Order and chaos in classical models of spin chains. Order and chaos in classical models of spin chains P. Belobrov, V. Beloshapkin, G. Zaslavskii, and A. Tret'yakov L. We construct a mapping which connects the spin states at neighboring chain sites and we consider its properties in various limiting cases.
We show that for a sufficiently high anisotropy field chaotic structures may appear. We give the spectral characteristics of these structures for the ferro- and antiferromagnetic cases. On the other hand, the formation of an incommen- Experimental and theoretical studies of the last decade surate structure always precedes the occurrence of amor- have shown that under well defined conditions there may phous structures. The appearance of incommensurate structures some general conclusions.
Firstly, it so far. For the equilibrium positions one can tures. In particular, we shall elucidate the conditions under write down equations connecting their values at neighboring which in the XYDA model an amorphous magnetic structure chain sites. These coupling equations are in the form of a appears. The mapping ob- 2. Its trajectories give the We consider a one-dimensional chain of classical spins coordinates of the atoms at the chain sites. Let the vectors oilie corresponds to a well defined value of the free energy.
The in the XYplane at an angle Siwith the X axis. The Hamilton- ground state of the chain corresponds to those trajectories ian of the system with interactions between nearest neigh- for which the free energy has a minimum. Such an approach bors can then be written in the form enabled one to incorporate into the analysis of possible struc- tures of the chain modern methods in the theory of dynamic systems.
In particular, it was shown in Refs. Of course, the occurrence of such sets is possible only We rewrite 2. One can minimize expression 2. The result of minimization selects that trajectory which 2. We for all j. This gives shall in what follows give a map of lines of constant 9 illus- trating what we have just said.
I5There are 2. Reversible systems are Using the definitions 2. The value of the two constants ,,al for the orientation angles of the initial spins determines all possible sequences 3.
Using Eq. We can then write Eqs. On the other branch we have from 2. The set of all trajectories 1. The Io,9-, and the set p connected with possible bifurcations of period of the chain in this case becomes equal to four.
The the trajectories. One seeseasily from Eqs. Each pair of initial We can associate with the mapping 3. The index "p" labels one of these trajectories. The family of trajectories described here generates a set of values arctg a tg r dz 6 z-n , 3. We analyzed the general situation for arbitrary values FIG. Examples of different configurations with the same energy for o f a numerically. We note first of all the physical the point 0,O. We note the following nontrivial fact. It is clear from 2.
The values of a in the vicinity of unity standing the antiferromagnetic nature of the exchange inter- correspond to : a strong ferromagnetic interaction with an action in that range.
A chastic structure is observed. This, in particular, means the 3. The parameter 9, is arbitrary. This means a in the simplest form: continuous degeneracy with respect to aO. The two provinces became known as Mandata, and Endo. Racism had emerged as a serious problem for generations. Usually, the strongest and most well-known man is crowned when a bloodline dies out, but no side wanted to give the other side power, there was no leader.
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