Bialternate matrix products and its application in bifurcation theoryVeronika Hajnov\303\241, Masaryk University, Brno, Czech RepublicThe central theorems in bifurcation theory are normal form theorems. The structure of all the theorems is the same. It claims, under certain assumptions, an arbitrary system of differential, resp, difference, equations is locally topologically equivalent to the normal form. One type of assumption can be formulated as equalities. For generic one-parameter bifurcations, there is always only one equality assumption. It stands as a condition for eigenvalues of the Jacobi matrix of the system. Those assumptions, so-called test functions, are formulated in section Bifurcation of this sheet. Bialternate product is a matrix product, which allows expressing test functions for Hopf and Neimark-Sacker bifurcations detection and continuation.References:
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<Text-field style="Heading 1" layout="Heading 1">Bialternate product <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkNGLw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkNGLw==</Equation><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkNGLw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkNGLw==</Equation></Text-field>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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkNGLw==
<Text-field style="Heading 1" layout="Heading 1">Bialternate product - examples</Text-field>QyQ+SSJBRzYiLUknTWF0cml4R0YlNiUiIiNGKTcmJkkiYUdGJTYjIiM2JkYsNiMiIzcmRiw2IyIjQCZGLDYjIiNBIiIiLUkzYmlhbHRlcm5hdGVwcm9kdWN0RzYiNiQsJEkiQUdGJCIiIy0mSS5MaW5lYXJBbGdlYnJhR0YkNiNJL0lkZW50aXR5TWF0cml4R0YkNiMtJkYrNiNJLVJvd0RpbWVuc2lvbkdGJDYjRic=QyQtSTNiaWFsdGVybmF0ZXByb2R1Y3RHNiI2JEkiQUdGJUYnIiIiQyQ+SSJCRzYiLUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiUiIiRGLDcrJkkiYUdGJTYjIiM2JkYvNiMiIzcmRi82IyIjOCZGLzYjIiNAJkYvNiMiI0EmRi82IyIjQiZGLzYjIiNKJkYvNiMiI0smRi82IyIjTCIiIg==LUkzYmlhbHRlcm5hdGVwcm9kdWN0RzYiNiQsJEkiQkdGJCIiIy0mSS5MaW5lYXJBbGdlYnJhRzYkSShfc3lzbGliR0YkJSpwcm90ZWN0ZWRHNiNJL0lkZW50aXR5TWF0cml4R0YkNiMtJkYrNiNJLVJvd0RpbWVuc2lvbkdGJDYjRic=QyQtSTNiaWFsdGVybmF0ZXByb2R1Y3RHNiI2JEkiQkdGJUYnIiIiQyQ+SSJDRzYiLUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiUiIiVGLDcyJkkiYUdGJTYjIiM2JkYvNiMiIzcmRi82IyIjOCZGLzYjIiM5JkYvNiMiI0AmRi82IyIjQSZGLzYjIiNCJkYvNiMiI0MmRi82IyIjSiZGLzYjIiNLJkYvNiMiI0wmRi82IyIjTSZGLzYjIiNUJkYvNiMiI1UmRi82IyIjViZGLzYjIiNXIiIiQyQtSTNiaWFsdGVybmF0ZXByb2R1Y3RHNiI2JCwkSSJDR0YlIiIjLSZJLkxpbmVhckFsZ2VicmFHNiRJKF9zeXNsaWJHRiUlKnByb3RlY3RlZEc2I0kvSWRlbnRpdHlNYXRyaXhHRiU2Iy0mRiw2I0ktUm93RGltZW5zaW9uR0YlNiNGKCIiIg==QyQtSTNiaWFsdGVybmF0ZXByb2R1Y3RHNiI2JEkiQ0dGJUYnIiIiQyg+SSJBRzYiLkYkISIiPkkiQkdGJS5GKUYnPkkiQ0dGJS5GLEYn
<Text-field style="Heading 1" layout="Heading 1">Bifurcations</Text-field>QyQ+SStkZXRlY3RIb3BmRzYiZio2JEkqZnVuY3Rpb25zR0YlSSp2YXJpYWJsZXNHRiU2I0kiQUdGJUYlRiVDJD44JC0mSS9WZWN0b3JDYWxjdWx1c0dGJTYjSSlKYWNvYmlhbkdGJTYkOSQ5JS0mSS5MaW5lYXJBbGdlYnJhRzYkSShfc3lzbGliR0YlJSpwcm90ZWN0ZWRHNiNJLERldGVybWluYW50R0YlNiMtSTNiaWFsdGVybmF0ZXByb2R1Y3RHRiU2JCwkRi4iIiMtJkY5NiNJL0lkZW50aXR5TWF0cml4R0YlNiMtJkY5NiNJLVJvd0RpbWVuc2lvbkdGJTYjRi5GJUYlRiUhIiI=QyQ+SStkZXRlY3Rmb2xkRzYiZio2JEkqZnVuY3Rpb25zR0YlSSp2YXJpYWJsZXNHRiU2I0kiQUdGJUYlRiVDJD44JC0mSS9WZWN0b3JDYWxjdWx1c0dGJTYjSSlKYWNvYmlhbkdGJTYkOSQ5JS0mSS5MaW5lYXJBbGdlYnJhRzYkSShfc3lzbGliR0YlJSpwcm90ZWN0ZWRHNiNJLERldGVybWluYW50R0YlNiNGLkYlRiVGJSEiIg==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
<Text-field style="Heading 1" layout="Heading 1">Bifurcations - examples</Text-field>
<Text-field style="Heading 2" layout="Heading 2">Selkov model</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkNGLw==QyY+SSNmMUc2ImYqNiRJInhHRiVJInlHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwoOSQhIiIqJkkiYUdGJSIiIjklRjJGMiomRi4iIiNGM0YyRjJGJUYlRiVGMj5JI2YyR0YlZipGJ0YlRipGJSwoSSJiR0YlRjJGMEYvRjRGL0YlRiVGJUYyQyQ+SSNmcEc2IiYtSSZzb2x2ZUdGJTYkNyQtSSNmMUdGJTYkSSJ4R0YlSSJ5R0YlLUkjZjJHRiVGLTckRi5GLzYjIiIiRjQ=QyQ+SSpwb2x5X2hvcGZHNiItSStkZXRlY3RIb3BmR0YlNiQ3JC1JI2YxR0YlNiRJInhHRiVJInlHRiUtSSNmMkdGJUYsNyRGLUYuIiIiQyQ+SSVob3BmRzYiLUkpc2ltcGxpZnlHRiU2Iy1JJXN1YnNHJSpwcm90ZWN0ZWRHNiRJI2ZwR0YlSSpwb2x5X2hvcGZHRiUiIiI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkNGLw==
<Text-field style="Heading 2" layout="Heading 2">Lorenz system</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkNGLw==Qyg+SSNmMUc2ImYqNiVJInhHRiVJInlHRiVJInpHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSomSSZzaWdtYUdGJSIiIiwmOSVGMDkkISIiRjBGJUYlRiVGMD5JI2YyR0YlZipGJ0YlRitGJSwmKiZGM0YwLCZJInJHRiVGMDkmRjRGMEYwRjJGNEYlRiVGJUYwPkkjZjNHRiVmKkYnRiVGK0YlLCYqJkYzRjBGMkYwRjAqJkkiYkdGJUYwRjxGMEY0RiVGJUYlRjA=QyY+SS1maXhlZF9wb2ludHNHNiItSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkNyUtSSNmMUdGJTYlSSJ4R0YlSSJ5R0YlSSJ6R0YlLUkjZjJHRiVGLy1JI2YzR0YlRi83JUYwRjFGMiEiIj5JJGZwMUdGJSZGJDYjIiIiRj0=QyQ+SSpwb2x5X2hvcGZHNiItSStkZXRlY3RIb3BmR0YlNiQ3JS1JI2YxR0YlNiVJInhHRiVJInlHRiVJInpHRiUtSSNmMkdGJUYsLUkjZjNHRiVGLDclRi1GLkYvIiIiQyQ+SSVob3BmRzYiLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Iy1JJXN1YnNHRik2JEkkZnAxR0YlSSpwb2x5X2hvcGZHRiUiIiI=
<Text-field style="Heading 2" layout="Heading 2">LPA model (cEL=0)</Text-field>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QyQ+SSpwb2x5X2ZvbGRHNiItSS1kZXRlY3Rmb2xkX2RHRiU2JDclLUkjZjFHRiU2JUkiTEdGJUkiUEdGJUkiQUdGJS1JI2YyR0YlRiwtSSNmM0dGJUYsNyVGLUYuRi8iIiI=QyQ+SSJmRzYiLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Iy1JJXN1YnNHRik2JEkkZnAxR0YlSSpwb2x5X2ZvbGRHRiUiIiI=PkkqcG9seV9mbGlwRzYiLUkrZGV0ZWN0ZmxpcEdGJDYkNyUtSSNmMUdGJDYlSSJMR0YkSSJQR0YkSSJBR0YkLUkjZjJHRiRGKy1JI2YzR0YkRis3JUYsRi1GLg==QyQ+SSNwZEc2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMtSSVzdWJzR0YpNiRJJGZwMkdGJUkqcG9seV9mbGlwR0YlIiIiPkkocG9seV9OU0c2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMtSTRkZXRlY3ROZWltYXJrU2Fja2VyR0YkNiQ3JS1JI2YxR0YkNiVJIkxHRiRJIlBHRiRJIkFHRiQtSSNmMkdGJEYxLUkjZjNHRiRGMTclRjJGM0Y0QyQ+SSNuc0c2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMtSSVzdWJzR0YpNiRJJGZwMkdGJUkocG9seV9OU0dGJSIiIg==Qyg+SSh0d29fcGFyRzYiLUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JDcnLCYtSSNmMUdGJTYlSSJMR0YlSSJQR0YlSSJBR0YlIiIiRjEhIiIsJi1JI2YyR0YlRjBGNEYyRjUsJi1JI2YzR0YlRjBGNEYzRjVJKnBvbHlfZmxpcEdGJUkocG9seV9OU0dGJTcnRjFGMkYzSSJiR0YlSSRjRUFHRiVGNT5JJXBkbnNHRiU3JC1JKXNpbXBsaWZ5R0YoNiMmJkYkNiNGNDYjIiIlLUZFNiMmRkg2IyIiJkY0PkkjcjJHRiU3JC1GRTYjJiZGJDYjIiIjRkotRkU2IyZGV0ZPRjQ=QyQ+SSRKYWNHNiItJkkvVmVjdG9yQ2FsY3VsdXNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kpSmFjb2JpYW5HRiU2JDclLUkjZjFHRiU2JUkiTEdGJUkiUEdGJUkiQUdGJS1JI2YyR0YlRjItSSNmM0dGJUYyNyVGM0Y0RjUiIiI=QyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLSZJLkxpbmVhckFsZ2VicmFHNiRGJ0YmNiNJLEVpZ2VudmFsdWVzR0YoNiMtRiQ2JC1JJXN1YnNHRiY2JEklcGRuc0dGKC1GNDYkJkktZml4ZWRfcG9pbnRzR0YoNiMiIiNJJEphY0dGKC9JJ2Fzc3VtZUdGKEkpcG9zaXRpdmVHRiZGPiIiIg==QyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLSZJLkxpbmVhckFsZ2VicmFHNiRGJ0YmNiNJLEVpZ2VudmFsdWVzR0YoNiMtRiQ2JC1JJXN1YnNHRiY2JEkjcjJHRigtRjQ2JCZJLWZpeGVkX3BvaW50c0dGKDYjIiIjSSRKYWNHRigvSSdhc3N1bWVHRiVJKXBvc2l0aXZlR0YmRj4iIiI=Qyo+SSN1TEc2IiQiJTg7ISIlISIiPkkjdUFHRiUkIiN2ISIjRik+SSRjRUxHRiUiIiFGKT5JJGNQQUdGJSQiJVtWISInRik=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QyQ+SSNwM0c2Ii0mSSZwbG90c0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSS1pbXBsaWNpdHBsb3RHRiU2K0kjbnNHRiUvSSRjRUFHRiU7IiIhJCIiJiEiIy9JImJHRiU7IiIiIiM6L0kqbnVtcG9pbnRzR0YlIigrKysiL0kmY29sb3JHRiVRJXBpbmtGJS9JJXZpZXdHRiU3JEYyO0YzRjsvSSVheGVzR0YlSSRib3hHRiUvSSdsYWJlbHNHRiU3JFEhRiVGTC9JJ2xlZ2VuZEdGJVEvTmVpbWFyay1TYWNrZXJGJSEiIg==LSZJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJKGRpc3BsYXlHRig2IzwlSSNwMUdGKEkjcDJHRihJI3AzR0YoQyRJJXBkbnNHNiIiIiI=QyRJI3IyRzYiIiIiQzQ+SSJiRzYiLkYkISIiPkkkY0VBR0YlLkYpRic+SSRjUEFHRiUuRixGJz5JJGNFTEdGJS5GL0YnPkkjdUFHRiUuRjJGJz5JI3VMR0YlLkY1Ric+SSNwMUdGJS5GOEYnPkkjcDJHRiUuRjtGJz5JI3AzR0YlLkY+Ric=
<Text-field style="Heading 2" layout="Heading 2">LPA model 2 (cPA=0)</Text-field>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QyQ+SSpwb2x5X2ZvbGRHNiItSS1kZXRlY3Rmb2xkX2RHRiU2JDclLUkjZjFHRiU2JUkiTEdGJUkiUEdGJUkiQUdGJS1JI2YyR0YlRiwtSSNmM0dGJUYsNyVGLUYuRi8iIiI=QyQ+SSJmRzYiLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Iy1JJXN1YnNHRik2JEkkZnAxR0YlSSpwb2x5X2ZvbGRHRiUiIiI=PkkqcG9seV9mbGlwRzYiLUkrZGV0ZWN0ZmxpcEdGJDYkNyUtSSNmMUdGJDYlSSJMR0YkSSJQR0YkSSJBR0YkLUkjZjJHRiRGKy1JI2YzR0YkRis3JUYsRi1GLg==QyQ+SSNwZEc2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMtSSVzdWJzR0YpNiRJJGZwMkdGJUkqcG9seV9mbGlwR0YlIiIiPkkocG9seV9OU0c2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMtSTRkZXRlY3ROZWltYXJrU2Fja2VyR0YkNiQ3JS1JI2YxR0YkNiVJIkxHRiRJIlBHRiRJIkFHRiQtSSNmMkdGJEYxLUkjZjNHRiRGMTclRjJGM0Y0QyQ+SSNuc0c2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMtSSVzdWJzR0YpNiRJJGZwMkdGJUkocG9seV9OU0dGJSIiIg==Qyg+SSh0d29fcGFyRzYiLUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JDcnLCYtSSNmMUdGJTYlSSJMR0YlSSJQR0YlSSJBR0YlIiIiRjEhIiIsJi1JI2YyR0YlRjBGNEYyRjUsJi1JI2YzR0YlRjBGNEYzRjVJKnBvbHlfZmxpcEdGJUkocG9seV9OU0dGJTcnRjFGMkYzSSJiR0YlSSRjRUFHRiVGNT5JJXBkbnNHRiU3JC1JKXNpbXBsaWZ5R0YoNiMmJkYkNiNGNDYjIiIlLUZFNiMmRkg2IyIiJkY0PkkjcjJHRiU3JC1GRTYjJiZGJDYjIiIjRkotRkU2IyZGV0ZPRjQ=QyQ+SSRKYWNHNiItJkkvVmVjdG9yQ2FsY3VsdXNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kpSmFjb2JpYW5HRiU2JDclLUkjZjFHRiU2JUkiTEdGJUkiUEdGJUkiQUdGJS1JI2YyR0YlRjItSSNmM0dGJUYyNyVGM0Y0RjUiIiI=QyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLSZJLkxpbmVhckFsZ2VicmFHNiRGJ0YmNiNJLEVpZ2VudmFsdWVzR0YoNiMtRiQ2JC1JJXN1YnNHRiY2JEklcGRuc0dGKC1GNDYkJkktZml4ZWRfcG9pbnRzR0YoNiMiIiNJJEphY0dGKC9JJ2Fzc3VtZUdGJUkpcG9zaXRpdmVHRiZGPiIiIg==QyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLSZJLkxpbmVhckFsZ2VicmFHNiRGJ0YmNiNJLEVpZ2VudmFsdWVzR0YoNiMtRiQ2JC1JJXN1YnNHRiY2JEkjcjJHRigtRjQ2JCZJLWZpeGVkX3BvaW50c0dGKDYjIiIjSSRKYWNHRigvSSdhc3N1bWVHRiVJKXBvc2l0aXZlR0YmRj4iIiI=Qyo+SSN1TEc2IiQiJTg7ISIlISIiPkkjdUFHRiUkIiMnKiEiI0YpPkkkY0VMR0YlJCIlJlEiISImRik+SSRjUEFHRiUiIiFGKQ==QyQ+SSNwMUc2Ii0mSSZwbG90c0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSS1pbXBsaWNpdHBsb3RHRiU2K0kiZkdGJS9JJGNFQUdGJTsiIiEkIiIjISIiL0kiYkdGJTtGMyIjOi9JKm51bXBvaW50c0dGJSImKysiL0kmY29sb3JHRiVRJmdyZWVuRiUvSSV2aWV3R0YlNyRGMkY5L0klYXhlc0dGJUkkYm94R0YlL0knbGFiZWxzR0YlNyRRIUYlRkovSSdsZWdlbmRHRiVRL1RyYW5za3JpdGlja3xed3xcdUYlRjY=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QyRJI3IyRzYiIiIiQzQ+SSJiRzYiLkYkISIiPkkkY0VBR0YlLkYpRic+SSRjUEFHRiUuRixGJz5JJGNFTEdGJS5GL0YnPkkjdUFHRiUuRjJGJz5JI3VMR0YlLkY1Ric+SSNwMUdGJS5GOEYnPkkjcDJHRiUuRjtGJz5JI3AzR0YlLkY+Ric=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkNGLw==
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<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Bifurcations and Gr\303\266bner basis - example</Font></Text-field>
<Text-field style="Heading 2" size="12" bold="false" layout="Heading 2"><Font bold="true" size="16">Spruce budworm model</Font></Text-field>Qyo+SSJLRzYiLkYkISIiPkknTGFtYmRhR0YlLkYpRic+SSJyR0YlLkYsRic+SSJmR0YlZio2I0kiTkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqKEYsIiIiOSRGOCwmRjhGOComRjlGOEYkRidGJ0Y4RjgqKEYpRjhGOSIiIywmRjhGOCokRjlGPUY4RidGJ0YlRiVGJUY4QyQ+SSNwMUc2Ii1JJm51bWVyRyUqcHJvdGVjdGVkRzYjLUkrZGV0ZWN0Zm9sZEdGJTYkNyMtSSJmR0YlNiNJIk5HRiU3I0YxIiIiQyQ+SSNwMkc2Ii1JI29wRyUqcHJvdGVjdGVkRzYkIiIkLUkmbnVtZXJHRig2Iy1JImZHRiU2I0kiTkdGJSIiIg==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Qyo+SSJLRzYiLkYkISIiPkknTGFtYmRhR0YlLkYpRic+SSJyR0YlLkYsRictJkkmcGxvdHNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kvaW1wbGljaXRwbG90M2RHRiU2KkklZm9sZEdGJS9GLDsiIiEkIiIoRicvRiQ7RjoiJCsiL0YpO0Y6IiIiL0kqbnVtcG9pbnRzR0YlIicrKzUvSSZjb2xvckdGJS5JJWdyYXlHRiUvSSZzdHlsZUdGJUkoc3VyZmFjZUdGJS9JLXRyYW5zcGFyZW5jeUdGJSQiIiZGJ0ZC
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8" size="16">Rosenzweig\305\257v\342\200\223MacArthur\305\257v model (Holli</Font><Font size="16">ng 3)</Font></Text-field>QzQ+SSJLRzYiLkYkISIiPkknbGFtYmRhR0YlLkYpRic+SSJyR0YlLkYsRic+SSJoR0YlLkYvRic+SSJlR0YlLkYyRic+SSJtR0YlLkY1Ric+SSJnR0YlZio2JEkiTkdGJUkiUEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKihGKSIiIjkkIiIjLCZGQUZBKihGKUZBRi9GQUZCRkNGQUYnRiVGJUYlRkE+SSNmMUdGJWYqRjpGJUY9RiUsJiooRixGQUZCRkEsJkZBRkEqJkZCRkFGJEYnRidGQUZBKiYtRjg2JEZCOSVGQUZQRkFGJ0YlRiVGJUZBPkkjZjJHRiVmKkY6RiVGPUYlLCYqKEYyRkFGTkZBRlBGQUZBKiZGNUZBRlBGQUYnRiVGJUYlRkE=QyQ+SSNwMUc2Ii1JJm51bWVyRyUqcHJvdGVjdGVkRzYjLUkrZGV0ZWN0SG9wZkdGJTYkNyQtSSNmMUdGJTYkSSJOR0YlSSJQR0YlLUkjZjJHRiVGMDckRjFGMiIiIg==QyQ+SSNwMkc2Ii1JJm51bWVyRyUqcHJvdGVjdGVkRzYjLUkrZGV0ZWN0Zm9sZEdGJTYkNyQtSSNmMUdGJTYkSSJOR0YlSSJQR0YlLUkjZjJHRiVGMDckRjFGMiIiIg==QyQ+SSNwM0c2IiomLUkmbnVtZXJHJSpwcm90ZWN0ZWRHNiMtSSdmYWN0b3JHNiRGKUkoX3N5c2xpYkdGJTYjLUkjZjFHRiU2JEkiTkdGJUkiUEdGJSIiIkYzISIiRjU=QyQ+SSNwNEc2IiomLUkmbnVtZXJHJSpwcm90ZWN0ZWRHNiMtSSdmYWN0b3JHNiRGKUkoX3N5c2xpYkdGJTYjLUkjZjJHRiU2JEkiTkdGJUkiUEdGJSIiIkY0ISIiRjU=
<Text-field style="Heading 2" layout="Heading 2">Classical approach</Text-field>QyQ+SSNlcUc2Ii1JKmFsbHZhbHVlc0dGJTYjJi1JJnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQ3JC1JI2YxR0YlNiRJIk5HRiVJIlBHRiUtSSNmMkdGJUYzNyRGNEY1NiMiIiQiIiI=QyQ+SSdob3BmXzFHNiItSSZudW1lckclKnByb3RlY3RlZEc2Iy1JKXNpbXBsaWZ5RzYkRihJKF9zeXNsaWJHRiU2Iy1JJXN1YnNHRig2JCZJI2VxR0YlNiMiIiJJI3AxR0YlRjU=QyQ+SSdob3BmXzJHNiItSSZudW1lckclKnByb3RlY3RlZEc2Iy1JKXNpbXBsaWZ5RzYkRihJKF9zeXNsaWJHRiU2Iy1JJXN1YnNHRig2JCZJI2VxR0YlNiMiIiNJI3AxR0YlIiIiQyQtSSZudW1lckclKnByb3RlY3RlZEc2Iy1JJ2ZhY3Rvckc2JEYlSShfc3lzbGliRzYiNiMtSSdleHBhbmRHRiU2IyomSSdob3BmXzFHRisiIiJJJ2hvcGZfMkdGK0YyRjI=QyQ+SSdmb2xkXzFHNiItSSZudW1lckclKnByb3RlY3RlZEc2Iy1JKXNpbXBsaWZ5RzYkRihJKF9zeXNsaWJHRiU2Iy1JJXN1YnNHRig2JCZJI2VxR0YlNiMiIiJJI3AyR0YlRjU=QyQ+SSdmb2xkXzJHNiItSSZudW1lckclKnByb3RlY3RlZEc2Iy1JKXNpbXBsaWZ5RzYkRihJKF9zeXNsaWJHRiU2Iy1JJXN1YnNHRig2JCZJI2VxR0YlNiMiIiNJI3AyR0YlIiIiQyQtSSZudW1lckclKnByb3RlY3RlZEc2Iy1JJ2ZhY3Rvckc2JEYlSShfc3lzbGliRzYiNiMtSSdleHBhbmRHRiU2IyomSSdmb2xkXzFHRisiIiJJJ2ZvbGRfMkdGK0YyRjI=Qy4+SSJLRzYiIiIpISIiPkknbGFtYmRhR0YlIiIiRic+SSJyR0YlIiIjRic+SSJoR0YlLkYvRic+SSJlR0YlRipGJz5JIm1HRiUuRjRGJw==QyQtJkkmcGxvdHNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0ktaW1wbGljaXRwbG90R0YpNik3JUknaG9wZl8xR0YpSSdob3BmXzJHRilJJ2ZvbGRfMUdGKS9JIm1HRik7IiIhIiIkL0kiaEdGKUYzL0kqbnVtcG9pbnRzR0YpIicrKzUvSSV2aWV3R0YpNyRGM0YzL0kmY29sb3JHRik3JS5JJmJsYWNrR0YpLkkkcmVkR0YpLkklYmx1ZUdGKS9JJWF4ZXNHRilJJmJveGVkR0YpIiIiQy4+SSJLRzYiLkYkISIiPkknbGFtYmRhR0YlLkYpRic+SSJyR0YlLkYsRic+SSJoR0YlLkYvRic+SSJlR0YlLkYyRic+SSJtR0YlLkY1Ric=
<Text-field style="Heading 2" layout="Heading 2">Groebner basis 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TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTc3MDM4Njg2WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQsJCosJSJiRyIiIiUiQUdGKS1JJGV4cEc2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IywkKiYlJGNFQUdGKUYqRikhIiJGKSUkY0VMR0YpLUYsNiMsJComRjVGKSUiTEdGKUY0RilGNCwmRilGKSUjdUxHRjQiIiFGPUY9RiksJiooRihGKUYrRilGNkYpRikqLEYoRilGKkYpRjNGKUYrRilGNkYpRjRGPSwmRilGKSUjdUFHRjRGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTc3MDMwODU0WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiJCIkISIiLCQqJiwmJSN1QUciIiIhIiNGK0YmLCoqJEYqIiIjRiZGKiIiJCEiJEYrKiR6LTYjRiolRiskISInIyIjNiJGLCEhIigjRitGL0YrRitGOCwkKiYsKkYuRitGMkYrRipGMUYwRitGK0YpRiYjRiZGL0YlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTc3MDIyNjYyWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiJCIkLCQqJnopNiMlI3VBRyMiIiIiISIlISIiJUYrLCZGKkYrISIkRishIiJGMEYwRjBGJQ==