. . . . "\u76F8\u5716\u662F\u5728\u7528\u7E6A\u5716\u7684\u65B9\u5F0F\u5728\u76F8\u5E73\u9762\u4E0A\u8868\u793A\u52D5\u614B\u7CFB\u7D71\u7684\u8ECC\u8DE1\u3002\u6BCF\u4E00\u500B\u4E0D\u540C\u7684\u521D\u59CB\u689D\u4EF6\u90FD\u7528\u4E00\u689D\u66F2\u7DDA\uFF08\u6216\u662F\u4E00\u500B\u9EDE\uFF09\u8868\u793A\u3002 \u5728\u7814\u7A76\u52D5\u614B\u7CFB\u7D71\u6642\uFF0C\u76F8\u5716\u662F\u5F88\u91CD\u8981\u7684\u5DE5\u5177\u3002\u76F8\u5716\u662F\u7531\u5728\u76F8\u7A7A\u9593\u4E2D\u5404\u9EDE\u8ECC\u8DE1\u7684\u7D44\u6210\u3002\u76F8\u5716\u53EF\u4EE5\u770B\u51FA\u52D5\u614B\u7CFB\u7D71\u5728\u7D66\u5B9A\u7684\u53C3\u6578\u4E0B\uFF0C\u662F\u5426\u6709\u5438\u5F15\u5B50\u3001\u6392\u65A5\u5B50\u6216\u662F\u6781\u9650\u73AF\u3002\u7684\u6982\u5FF5\u5728\u70BA\u7CFB\u7D71\u884C\u70BA\u5206\u985E\u6642\u975E\u5E38\u91CD\u8981\uFF0C\u4F8B\u5982\u4E8C\u500B\u4E0D\u540C\u7684\u76F8\u5716\u53EF\u80FD\u6703\u51FA\u73FE\u76F8\u540C\u7684\u672C\u8CEA\u6027\u52D5\u614B\u7279\u6027\u3002 \u5728\u76F8\u5716\u4E2D\u6703\u63CF\u7E6A\u7CFB\u7D71\u7684\u8ECC\u8DE1\uFF08\u4EE5\u7BAD\u982D\u8868\u793A\uFF09\u3001\u7A69\u5B9A\u7A69\u614B\uFF08\u4EE5\u9ED1\u9EDE\u8868\u793A\uFF09\u53CA\u4E0D\u7A69\u5B9A\u7A69\u614B\uFF08\u4EE5\u5713\u5708\u9EDE\u8868\u793A\uFF09\uFF0C\u76F8\u5716\u7684\u8EF8\u5C0D\u61C9\u72C0\u614B\u8B8A\u6578\u3002"@zh . . . . . "Ritratto di fase"@it . "\u0635\u0648\u0631\u0629 \u0627\u0644\u0637\u0648\u0631 \u0647\u064A \u062A\u0645\u062B\u064A\u0644 \u0647\u0646\u062F\u0633\u064A \u0644\u0645\u0633\u0627\u0631\u0627\u062A \u0627\u0644\u0646\u0638\u0627\u0645 \u0627\u0644\u062F\u064A\u0646\u0627\u0645\u064A\u0643\u064A \u0641\u064A \u0645\u0633\u062A\u0648\u0649 \u0627\u0644\u0637\u0648\u0631. \u064A\u062A\u0645 \u062A\u0645\u062B\u064A\u0644 \u0643\u0644 \u0645\u062C\u0645\u0648\u0639\u0629 \u0645\u0646 \u0627\u0644\u0634\u0631\u0648\u0637 \u0627\u0644\u0623\u0648\u0644\u064A\u0629 \u0628\u0645\u0646\u062D\u0646\u0649 \u0623\u0648 \u0646\u0642\u0637\u0629 \u0645\u062E\u062A\u0644\u0641\u0629. \u062A\u0639\u062F \u0635\u0648\u0631 \u0627\u0644\u0637\u0648\u0631 \u0623\u062F\u0627\u0629 \u0644\u0627 \u062A\u0642\u062F\u0631 \u0628\u062B\u0645\u0646 \u0641\u064A \u062F\u0631\u0627\u0633\u0629 \u0627\u0644\u0623\u0646\u0638\u0645\u0629 \u0627\u0644\u062F\u064A\u0646\u0627\u0645\u064A\u0643\u064A\u0629. \u0648\u0647\u064A \u062A\u062A\u0643\u0648\u0646 \u0645\u0646 \u0631\u0633\u0645 \u0627\u0644\u0645\u0633\u0627\u0631\u0627\u062A \u0627\u0644\u0645\u062D\u062A\u0645\u0644\u0629 \u0644\u0644\u0645\u062A\u062D\u0648\u0644\u0627\u062A \u0641\u064A \u0641\u0636\u0627\u0621 \u0627\u0644\u062D\u0627\u0644\u0629. \u062A\u0643\u0634\u0641 \u0635\u0648\u0631\u0629 \u0627\u0644\u0637\u0648\u0631 \u0639\u0646 \u0645\u0639\u0644\u0648\u0645\u0627\u062A \u0645\u062B\u0644 \u0645\u0627 \u0625\u0630\u0627 \u0643\u0627\u0646 \u0627\u0644\u062C\u0627\u0630\u0628 \u0623\u0648 \u0627\u0644\u0637\u0627\u0631\u062F \u0623\u0648 \u062F\u0648\u0631\u0629 \u0627\u0644\u062D\u062F \u0645\u0648\u062C\u0648\u062F\u064B\u0627 \u0644\u0642\u064A\u0645\u0629 \u0627\u0644\u0645\u0639\u0644\u0645\u0629 \u0627\u0644\u0645\u062E\u062A\u0627\u0631\u0629. \u064A\u0645\u062B\u0644 \u0627\u0644\u062C\u0627\u0630\u0628 \u0646\u0642\u0637\u0629 \u0645\u0633\u062A\u0642\u0631\u0629 \u062A\u0633\u0645\u0649 \u0623\u064A\u0636\u064B\u0627 \u00AB\u0628\u0627\u0644\u0648\u0639\u0629\u00BB. \u0648\u064A\u0639\u062A\u0628\u0631 \u0627\u0644\u0637\u0627\u0631\u062F \u0646\u0642\u0637\u0629 \u063A\u064A\u0631 \u0645\u0633\u062A\u0642\u0631\u0629\u060C \u0648\u0627\u0644\u062A\u064A \u062A\u064F\u0639\u0631\u0641 \u0623\u064A\u0636\u064B\u0627 \u0628\u0627\u0633\u0645 \u00AB\u0627\u0644\u0645\u0635\u062F\u0631\u00BB. \u0625\u0646 \u0645\u0641\u0647\u0648\u0645 \u0627\u0644\u062A\u0643\u0627\u0641\u0624 \u0627\u0644\u0637\u0648\u0628\u0648\u0644\u0648\u062C\u064A \u0645\u0647\u0645 \u0641\u064A \u062A\u0635\u0646\u064A\u0641 \u0633\u0644\u0648\u0643 \u0627\u0644\u0623\u0646\u0638\u0645\u0629 \u0645\u0646 \u062E\u0644\u0627\u0644 \u062A\u062D\u062F\u064A\u062F \u0645\u062A\u0649 \u062A\u0645\u062B\u0644 \u0635\u0648\u0631\u062A\u064A\u0646 \u0645\u062E\u062A\u0644\u0641\u062A\u064A\u0646 \u0644\u0644\u0637\u0648\u0631 \u0646\u0641\u0633 \u0627\u0644\u0633\u0644\u0648\u0643 \u0627\u0644\u062F\u064A\u0646\u0627\u0645\u064A\u0643\u064A \u0627\u0644\u0646\u0648\u0639\u064A."@ar . . "Diagram fazowy (fizyka)"@pl . . . "4690"^^ . "In der Mathematik dient das Phasenportr\u00E4t (auch Phasenportrait) der Veranschaulichung einer autonomen Differentialgleichung. Das Phasenraumportr\u00E4t gibt eine M\u00F6glichkeit, die zeitlichen Entwicklungen dynamischer Systeme graphisch zu analysieren. Dazu werden nur die dynamischen Gleichungen des Systems ben\u00F6tigt, eine explizite Darstellung der Zeitentwicklung, etwa durch analytisches L\u00F6sen einer Differentialgleichung, ist nicht n\u00F6tig. Man betrachtet also eine Differentialgleichung erster Ordnung:"@de . . . . . . . "O retrato de fase \u00E9 uma de todas as trajet\u00F3rias de um sistema din\u00E2mico no plano. Cada curva representa um diferente condi\u00E7\u00E3o inicial. O retrato de fase \u00E9 uma ferramenta valiosa no estudo dos sistemas din\u00E2micos aut\u00F4nomos de segunda ordem. A configura\u00E7\u00E3o das curvas no espa\u00E7o de fase revela informa\u00F5es sobre a exist\u00EAncias de atratores, e . O conceito de desempenha um papel importante na classifica\u00E7\u00E3o dos sistemas din\u00E2micos ao especificar quando dois sistemas diferentes apresentam o mesmo comportamento qualitativo. x(t)=A sen (\u03C9 t- \u03B4) (1-a) \u1E8B(t)=A\u03C9 co s(\u03C9 t- \u03B4) (1-b) (9)"@pt . . "Un retrat de fase \u00E9s una representaci\u00F3 geom\u00E8trica de les traject\u00F2ries d'un sistema din\u00E0mic en el pla de fase. Cada conjunt de condicions inicials \u00E9s representat per una corba diferent, o per un punt. Els retrats de fase s\u00F3n una eina molt \u00FAtil en l'estudi de sistemes din\u00E0mics. Consisteixen en una representaci\u00F3 gr\u00E0fica de les traject\u00F2ries t\u00EDpiques en un espai d'estats. Aix\u00F2 revela informaci\u00F3 com ara si un punt d'equilibri \u00E9s atractor, repel\u00B7lent o cicle l\u00EDmit segons el valor que s'escolleixi d'un par\u00E0metre. El concepte d'equival\u00E8ncia topol\u00F2gica \u00E9s important en la classificaci\u00F3 del comportament dels sistemes per especificar quan dos retrats de fase diferents representen el mateix comportament din\u00E0mic qualitativament. Un atractor \u00E9s un punt estable que \u00E9s tamb\u00E9 anomenat \"embornal\" (en angl\u00E8s, "@ca . . . "\u0635\u0648\u0631\u0629 \u0627\u0644\u0637\u0648\u0631 \u0647\u064A \u062A\u0645\u062B\u064A\u0644 \u0647\u0646\u062F\u0633\u064A \u0644\u0645\u0633\u0627\u0631\u0627\u062A \u0627\u0644\u0646\u0638\u0627\u0645 \u0627\u0644\u062F\u064A\u0646\u0627\u0645\u064A\u0643\u064A \u0641\u064A \u0645\u0633\u062A\u0648\u0649 \u0627\u0644\u0637\u0648\u0631. \u064A\u062A\u0645 \u062A\u0645\u062B\u064A\u0644 \u0643\u0644 \u0645\u062C\u0645\u0648\u0639\u0629 \u0645\u0646 \u0627\u0644\u0634\u0631\u0648\u0637 \u0627\u0644\u0623\u0648\u0644\u064A\u0629 \u0628\u0645\u0646\u062D\u0646\u0649 \u0623\u0648 \u0646\u0642\u0637\u0629 \u0645\u062E\u062A\u0644\u0641\u0629. \u062A\u0639\u062F \u0635\u0648\u0631 \u0627\u0644\u0637\u0648\u0631 \u0623\u062F\u0627\u0629 \u0644\u0627 \u062A\u0642\u062F\u0631 \u0628\u062B\u0645\u0646 \u0641\u064A \u062F\u0631\u0627\u0633\u0629 \u0627\u0644\u0623\u0646\u0638\u0645\u0629 \u0627\u0644\u062F\u064A\u0646\u0627\u0645\u064A\u0643\u064A\u0629. \u0648\u0647\u064A \u062A\u062A\u0643\u0648\u0646 \u0645\u0646 \u0631\u0633\u0645 \u0627\u0644\u0645\u0633\u0627\u0631\u0627\u062A \u0627\u0644\u0645\u062D\u062A\u0645\u0644\u0629 \u0644\u0644\u0645\u062A\u062D\u0648\u0644\u0627\u062A \u0641\u064A \u0641\u0636\u0627\u0621 \u0627\u0644\u062D\u0627\u0644\u0629. \u062A\u0643\u0634\u0641 \u0635\u0648\u0631\u0629 \u0627\u0644\u0637\u0648\u0631 \u0639\u0646 \u0645\u0639\u0644\u0648\u0645\u0627\u062A \u0645\u062B\u0644 \u0645\u0627 \u0625\u0630\u0627 \u0643\u0627\u0646 \u0627\u0644\u062C\u0627\u0630\u0628 \u0623\u0648 \u0627\u0644\u0637\u0627\u0631\u062F \u0623\u0648 \u062F\u0648\u0631\u0629 \u0627\u0644\u062D\u062F \u0645\u0648\u062C\u0648\u062F\u064B\u0627 \u0644\u0642\u064A\u0645\u0629 \u0627\u0644\u0645\u0639\u0644\u0645\u0629 \u0627\u0644\u0645\u062E\u062A\u0627\u0631\u0629. \u064A\u0645\u062B\u0644 \u0627\u0644\u062C\u0627\u0630\u0628 \u0646\u0642\u0637\u0629 \u0645\u0633\u062A\u0642\u0631\u0629 \u062A\u0633\u0645\u0649 \u0623\u064A\u0636\u064B\u0627 \u00AB\u0628\u0627\u0644\u0648\u0639\u0629\u00BB. \u0648\u064A\u0639\u062A\u0628\u0631 \u0627\u0644\u0637\u0627\u0631\u062F \u0646\u0642\u0637\u0629 \u063A\u064A\u0631 \u0645\u0633\u062A\u0642\u0631\u0629\u060C \u0648\u0627\u0644\u062A\u064A \u062A\u064F\u0639\u0631\u0641 \u0623\u064A\u0636\u064B\u0627 \u0628\u0627\u0633\u0645 \u00AB\u0627\u0644\u0645\u0635\u062F\u0631\u00BB. \u0625\u0646 \u0645\u0641\u0647\u0648\u0645 \u0627\u0644\u062A\u0643\u0627\u0641\u0624 \u0627\u0644\u0637\u0648\u0628\u0648\u0644\u0648\u062C\u064A \u0645\u0647\u0645 \u0641\u064A \u062A\u0635\u0646\u064A\u0641 \u0633\u0644\u0648\u0643 \u0627\u0644\u0623\u0646\u0638\u0645\u0629 \u0645\u0646 \u062E\u0644\u0627\u0644 \u062A\u062D\u062F\u064A\u062F \u0645\u062A\u0649 \u062A\u0645\u062B\u0644 \u0635\u0648\u0631\u062A\u064A\u0646 \u0645\u062E\u062A\u0644\u0641\u062A\u064A\u0646 \u0644\u0644\u0637\u0648\u0631 \u0646\u0641\u0633 \u0627\u0644\u0633\u0644\u0648\u0643 \u0627\u0644\u062F\u064A\u0646\u0627\u0645\u064A\u0643\u064A \u0627\u0644\u0646\u0648\u0639\u064A. \u064A\u0635\u0648\u0631 \u0631\u0633\u0645 \u0628\u064A\u0627\u0646\u064A \u0637\u0648\u0631\u064A \u0644\u0646\u0638\u0627\u0645 \u062F\u064A\u0646\u0627\u0645\u064A\u0643\u064A \u0645\u0633\u0627\u0631\u0627\u062A \u0627\u0644\u0646\u0638\u0627\u0645 (\u0627\u0644\u0623\u0633\u0647\u0645) \u0648\u0627\u0644\u062D\u0627\u0644\u0627\u062A \u0627\u0644\u062B\u0627\u0628\u062A\u0629 \u0627\u0644\u0645\u0633\u062A\u0642\u0631\u0629 (\u0627\u0644\u0646\u0642\u0627\u0637) \u0648\u0627\u0644\u062D\u0627\u0644\u0627\u062A \u0627\u0644\u062B\u0627\u0628\u062A\u0629 \u063A\u064A\u0631 \u0627\u0644\u0645\u0633\u062A\u0642\u0631\u0629 (\u0627\u0644\u062F\u0648\u0627\u0626\u0631) \u0641\u064A \u0641\u0636\u0627\u0621 \u0627\u0644\u062D\u0627\u0644\u0629. \u0625\u0646 \u0627\u0644\u0645\u062D\u0627\u0648\u0631 \u0641\u064A \u0635\u0648\u0631\u0629 \u0627\u0644\u0637\u0648\u0631 \u0647\u064A \u0645\u062A\u063A\u064A\u0631\u0627\u062A \u0627\u0644\u062D\u0627\u0644\u0629."@ar . . . . . . . . "Phasenportr\u00E4t"@de . "O retrato de fase \u00E9 uma de todas as trajet\u00F3rias de um sistema din\u00E2mico no plano. Cada curva representa um diferente condi\u00E7\u00E3o inicial. O retrato de fase \u00E9 uma ferramenta valiosa no estudo dos sistemas din\u00E2micos aut\u00F4nomos de segunda ordem. A configura\u00E7\u00E3o das curvas no espa\u00E7o de fase revela informa\u00F5es sobre a exist\u00EAncias de atratores, e . O conceito de desempenha um papel importante na classifica\u00E7\u00E3o dos sistemas din\u00E2micos ao especificar quando dois sistemas diferentes apresentam o mesmo comportamento qualitativo. Sabendo as condi\u00E7\u00F5es iniciais deste sistema, podemos descreve-lo como uma fun\u00E7\u00E3o do tempo se as condi\u00E7\u00F5es de posi\u00E7\u00E3o inicial e velocidade inicial \u00E9 dada. Pode-se considerar das duas quantidades e como sendo coordenadas de um ponto em um espa\u00E7o bidimensional, que \u00E9 chamado de espa\u00E7o de fase. Um espa\u00E7o de fase pode ser constru\u00EDdo para sistemas com mais de uma dimens\u00E3o, entretanto se, por exemplo, se o sistema \u00E9 bidimensional o espa\u00E7o de fase ter\u00E1 quatro dimens\u00F5es. A rela\u00E7\u00E3o geral para um oscilador \u00E9 dada por: n graus de liberdade o espa\u00E7o de fase ter\u00E1 um espa\u00E7o 2n dimensional. \u00C9 poss\u00EDvel prever que a medida que o tempo varia, o ponto agora dado por e que descreve o estado da part\u00EDcula oscilat\u00F3ria ir\u00E1 se mover ao longo de um caminho de fase no plano descrito pelo caminho de fase. Entretanto, se as condi\u00E7\u00F5es iniciais do sistema \u00E9 diferente de um oscilador ent\u00E3o o movimento ser\u00E1 descrito por diferentes caminhos de fase. Onde qualquer caminho fornecido representa o hist\u00F3rico temporal completo do oscilador para uma determinada condi\u00E7\u00E3o inicial. Assim, a totalidade de todos os caminhos de fase poss\u00EDveis formam o que chamamos de diagrama de fase de um oscilador. Como j\u00E1 descrito neste trabalho as equa\u00E7\u00F5es de posi\u00E7\u00E3o e velocidade para um oscilador harm\u00F4nico simples s\u00E3o dados por: x(t)=A sen (\u03C9 t- \u03B4) (1-a) \u1E8B(t)=A\u03C9 co s(\u03C9 t- \u03B4) (1-b) Se o tempo (t) for elimidado de ambas as equa\u00E7\u00F5es tem-se uma express\u00E3o que representa uma fam\u00EDlia de el\u00EDpses, e \u00E9 dada por: ( x\u00B2)/A\u00B2 + \u1E8B\u00B2/(A\u00B2\u03C9 \u00B2)=1 (2) A partir da equa\u00E7\u00E3o (9) e da rela\u00E7\u00E3o pode-se substituir na equa\u00E7\u00E3o (2) para obter: ( x\u00B2)/(2E\u27CBk) + \u1E8B\u00B2/(2E\u27CBm)=1 (3) Como estamos considerando sistemas conservativos, chega-se, ent\u00E3o, a conclus\u00E3o que cada caminho de fase corresponde \u00E1 energia total definida do oscilador. Nesta representa\u00E7\u00E3o (velocidade por espa\u00E7o) os eixos de coordenadas do plano de fase foram escolhidos de modo que o movimento representativo p(x,) ser\u00E1 invariavelmente em sentido hor\u00E1rio, pois para x > 0 a velocidade ser\u00E1 ser\u00E1 sempre descrescente e para x < 0 a velocidade ser\u00E1 sempre crescente. Neste tipo de sistema dois caminhos de fase de um oscilador n\u00E3o podem se cruzar, pois isto implicaria que para um determinado conjunto de condi\u00E7\u00F5es iniciais x(t) e \u1E8B(t) o movimento poderia ocorrer ao longo de caminhos de fase diferentes, entretanto isto n\u00E3o \u00E9 verdade j\u00E1 que a solu\u00E7\u00E3o da equa\u00E7\u00E3o diferencial \u00E9 \u00FAnica. Para obter o diagrama de faseintegrasse a equa\u00E7\u00E3o de elipse: (d\u00B2x)/dt\u00B2 +\u03C9\u00B2x=0 (4) Como um dos eixos cartesianos \u00E9 \u1E8B, pode-se simplificar a solu\u00E7\u00E3o da equa\u00E7\u00E3o (4) substituindo apenas: (dx)/dt=\u1E8B e tamb\u00E9m, ( d\u1E8B)/dt=-\u03C9\u00B2x (5) Realizando mais uma manipula\u00E7\u00E3o matem\u00E1tica, dividi-se a equa\u00E7\u00E3o (d\u1E8B)/dt=-\u03C9)\u00B2x por \u1E8B e assim obt\u00E9m-se: (d\u1E8B)/dt=-\u03C9\u00B2 x/\u1E8B (6) A solu\u00E7\u00E3o da equa\u00E7\u00E3o (6) \u00E9 dada pela equa\u00E7\u00E3o (2) descriminda acima (pg. 3). No caso do oscilador harm\u00F4nico a simplifica\u00E7\u00E3o feita aqui \u00E9 bastante \u00FAtil, por\u00E9m para sistemas mais complexos pode ser mais simples a resolu\u00E7\u00E3o da fun\u00E7\u00E3o \u1E8B(x) diretamente. Osciladores Amortecidos e Diagrama de Fase Antes de apresentar diagramas de fase para osciladores amortecidos e subamortecidos ser\u00E1 feira uma breve introdu\u00E7\u00E3o acerca destes tipos de osciladores. No caso descrito anteriormente, os osciladores harm\u00F4nicos n\u00E3o apresentam perda de energia, ou seja, uma vez oscilantes permanecer\u00E1 sempre neste movimento. Esta \u00E9 uma simplifica\u00E7\u00E3o bastante \u00FAtil e valiosa para estudar fen\u00F4menos f\u00EDsicos oscilantes, por\u00E9m na natureza n\u00E3o vemos osciladores harm\u00F4nicos, mas sim osciladores amortecidos. Osciladores amortecidos s\u00E3o aqueles que apresentam for\u00E7as de dissipa\u00E7\u00E3o, como por exemplo, o atrito que ir\u00E1 atuar no freamento da oscila\u00E7\u00E3o at\u00E9 sua completa paraliza\u00E7\u00E3o. Neste trabalho adotaremos a for\u00E7a dissipativa como uma fun\u00E7\u00E3o linear da velocidade que ser\u00E1 dada por F=\u03B1v tamb\u00E9m iremos considerar movimentos unidimensionais de forma que pode-se apresentar o termo de amortecimento por -b \u1E8B. Neste caso pode-se pensar que o no sistema amortecido, teremos o termo da equa\u00E7\u00E3o de um sistema harm\u00F4nico acrescido de um termo que represente a for\u00E7a de amortecimento, da\u00ED surge o termo de amortecimento dado acima. Onde o par\u00E2metro b deve ser positivo para que a for\u00E7a seja resistiva, uma vez que devemos ter em mente que a for\u00E7a aqui considerada deve diminuir a velocidade do oscilador. Por isso, se consider\u00E1ssemos o par\u00E2metro b um valor negativo ent\u00E3o a velocidade do oscilador deveria aumentar. Sendo assim, pode-se considerar uma part\u00EDcula de massa m que se move sob influencia de uma combina\u00E7\u00E3o de uma for\u00E7a restauradora \u2013kx e uma for\u00E7a resistiva -b \u1E8B, portanto a equa\u00E7\u00E3o diferencial para este sistema \u00E9 dada por: m\u1E8D +b\u1E8B +kx=0 (7) Como definimos anteriormente \u03C9_0\u2261\u221Akm, que \u00E9 a frequ\u00EAncia angular caracter\u00EDstica na aus\u00EAncia do amortecimento. E definido \u03B2\u2261b\u27CB2m como sendo o par\u00E2metro de amortecimento, tem-se portanto substituindo na equa\u00E7\u00E3o acima: \u1E8D +2\u03B2\u1E8B +\u03C9_0 \u00B2x=0 (8) Temos que as ra\u00EDzes da equa\u00E7\u00E3o s\u00E3o dadas por: R1=-\u03B2 +\u221A(\u03B2\u00B2+\u03C9 \u00B2) (9) R2 = -\u03B2-\u221A(\u03B2\u00B2-\u03C9\u00B2) Desta forma, a solu\u00E7\u00E3o para a equa\u00E7\u00E3o (8) \u00E9 dada por: x(t)= e^(-\u03B2t) [A_1 exp\u2061(\u221A(\u03B2\u00B2-\u03C9\u00B2) t)+ A_2 exp\u2061(-\u221A(\u03B2^2-\u03C9^2 ) t)] (10) A partir da equa\u00E7\u00E3o (10) pode-se concluir que dependendo da rela\u00E7\u00E3o entre \u03C9\u00B2 e \u03B2\u00B2 tem-se resultados diferentes, a estes casos gerais especiais temos Sistemas de Oscila\u00E7\u00E3o Especiais Subamortecimento \u03C9 \u00B2 > \u03B2\u00B2 Amortecimento cr\u00EDtico \u03C9\u00B2 = \u03B2\u00B2 Sobreamortecimento \u03C9 \u00B2 < \u03B2\u00B2"@pt . . . . . "Un ritratto di fase (talvolta chiamato con il nome inglese phase portrait) \u00E8 una rappresentazione geometrica delle traiettorie di un sistema dinamico nello spazio delle fasi. Ogni insieme di condizioni iniziali \u00E8 rappresentato da una differente curva o punto. I ritratti di fase sono uno strumento fondamentale nello studio dei sistemi dinamici. Costituiti dalla rappresentazione grafica delle tipiche traiettorie del sistema nello spazio di stato, rivelano informazioni riguardanti la presenza di attrattori, orbite periodiche e punti di equilibrio. Il concetto di \u00E8 importante per classificare i diversi comportamenti dei sistemi studiati, in quanto \u00E8 necessario per capire se due differenti ritratti di fase rappresentano qualitativamente lo stesso comportamento dinamico. In un ritratto di fase di un sistema dinamico vengono rappresentate le traiettorie del sistema (con delle frecce), gli stati di stabilit\u00E0 (con dei punti) e gli stati di instabilit\u00E0 (con dei cerchi) nello spazio di stato. Gli assi sono costituiti dalle variabili di stato."@it . . . "Un portrait de phase est une repr\u00E9sentation g\u00E9om\u00E9trique des trajectoires d'un syst\u00E8me dynamique dans l'espace des phases : \u00E0 chaque ensemble de conditions initiales correspond une courbe ou un point."@fr . . "Un retrato de fase es una representaci\u00F3n geom\u00E9trica de todas las trayectorias de un sistema din\u00E1mico en el plano. Cada curva representa una condici\u00F3n inicial diferente. Un retrato de fase es una herramienta valiosa en el estudio de los sistemas din\u00E1micos aut\u00F3nomos de segundo orden. La configuraci\u00F3n de las curvas en el espacio de fase revela informaci\u00F3n sobre la existencia de atractores, , .\u200B El concepto de desempe\u00F1a un papel importante en la clasificaci\u00F3n de los sistemas din\u00E1micos para especificar cuando dos sistemas diferentes muestran el mismo comportamiento cualitativo. Un gr\u00E1fico de un retrato de fases de un sistema din\u00E1mico representa las trayectorias del sistema con flechas y sus estados de equilibrio estable e inestable con puntos.\u200B El p\u00E9ndulo simple es un sistema f\u00EDsico que ejemplifica un retrato de fase, as\u00ED como el oscilador arm\u00F3nico, donde el retrato de fase se compone de formas el\u00EDpticas centradas en el origen.\u200B"@es . "Retrat de fase"@ca . . . . . . . "\u76F8\u5716 (\u52D5\u614B\u7CFB\u7D71)"@zh . "\u76F8\u5716\u662F\u5728\u7528\u7E6A\u5716\u7684\u65B9\u5F0F\u5728\u76F8\u5E73\u9762\u4E0A\u8868\u793A\u52D5\u614B\u7CFB\u7D71\u7684\u8ECC\u8DE1\u3002\u6BCF\u4E00\u500B\u4E0D\u540C\u7684\u521D\u59CB\u689D\u4EF6\u90FD\u7528\u4E00\u689D\u66F2\u7DDA\uFF08\u6216\u662F\u4E00\u500B\u9EDE\uFF09\u8868\u793A\u3002 \u5728\u7814\u7A76\u52D5\u614B\u7CFB\u7D71\u6642\uFF0C\u76F8\u5716\u662F\u5F88\u91CD\u8981\u7684\u5DE5\u5177\u3002\u76F8\u5716\u662F\u7531\u5728\u76F8\u7A7A\u9593\u4E2D\u5404\u9EDE\u8ECC\u8DE1\u7684\u7D44\u6210\u3002\u76F8\u5716\u53EF\u4EE5\u770B\u51FA\u52D5\u614B\u7CFB\u7D71\u5728\u7D66\u5B9A\u7684\u53C3\u6578\u4E0B\uFF0C\u662F\u5426\u6709\u5438\u5F15\u5B50\u3001\u6392\u65A5\u5B50\u6216\u662F\u6781\u9650\u73AF\u3002\u7684\u6982\u5FF5\u5728\u70BA\u7CFB\u7D71\u884C\u70BA\u5206\u985E\u6642\u975E\u5E38\u91CD\u8981\uFF0C\u4F8B\u5982\u4E8C\u500B\u4E0D\u540C\u7684\u76F8\u5716\u53EF\u80FD\u6703\u51FA\u73FE\u76F8\u540C\u7684\u672C\u8CEA\u6027\u52D5\u614B\u7279\u6027\u3002 \u5728\u76F8\u5716\u4E2D\u6703\u63CF\u7E6A\u7CFB\u7D71\u7684\u8ECC\u8DE1\uFF08\u4EE5\u7BAD\u982D\u8868\u793A\uFF09\u3001\u7A69\u5B9A\u7A69\u614B\uFF08\u4EE5\u9ED1\u9EDE\u8868\u793A\uFF09\u53CA\u4E0D\u7A69\u5B9A\u7A69\u614B\uFF08\u4EE5\u5713\u5708\u9EDE\u8868\u793A\uFF09\uFF0C\u76F8\u5716\u7684\u8EF8\u5C0D\u61C9\u72C0\u614B\u8B8A\u6578\u3002"@zh . . . "1065975713"^^ . "Retrato de fase"@pt . . . . . . . "Un ritratto di fase (talvolta chiamato con il nome inglese phase portrait) \u00E8 una rappresentazione geometrica delle traiettorie di un sistema dinamico nello spazio delle fasi. Ogni insieme di condizioni iniziali \u00E8 rappresentato da una differente curva o punto. In un ritratto di fase di un sistema dinamico vengono rappresentate le traiettorie del sistema (con delle frecce), gli stati di stabilit\u00E0 (con dei punti) e gli stati di instabilit\u00E0 (con dei cerchi) nello spazio di stato. Gli assi sono costituiti dalle variabili di stato."@it . "Diagram fazowy, portret fazowy \u2013 zbi\u00F3r punkt\u00F3w w przestrzeni fazowej reprezentuj\u0105cy mo\u017Cliwe ruchy dla zadanego hamiltonianu lub lagran\u017Cjanu. W pierwszym wypadku przestrze\u0144 fazowa zawiera wymiar\u00F3w przestrzennych i wymiar\u00F3w p\u0119dowych, gdzie jest stopniem swobody uk\u0142adu opisywanego hamiltonianem. W przypadku mechaniki Lagrange'a wymiary p\u0119dowe s\u0105 zast\u0105pione przez wymiary pr\u0119dko\u015Bciowe. Wobec tego ka\u017Cdy punkt na diagramie fazowym odpowiada pewnemu po\u0142o\u017Ceniu w przestrzeni sk\u0142adowych uk\u0142adu (wsp\u00F3\u0142rz\u0119dne przestrzenne punktu) i odpowiadaj\u0105cym im p\u0119dom (pr\u0119dko\u015Bciom). Dla uk\u0142adu, w kt\u00F3rym po\u0142o\u017Cenia i p\u0119dy (pr\u0119dko\u015Bci) mog\u0105 przyjmowa\u0107 warto\u015Bci ci\u0105g\u0142e, diagram fazowy sk\u0142ada si\u0119 zwykle z krzywych (zwanych krzywymi fazowymi). Niekiedy krzywe tworz\u0105 krzyw\u0105 prze\u0142\u0105cze\u0144, czyli krzyw\u0105 wzgl\u0119dem kt\u00F3rej rozpatrywane jest badane zagadnienie. Diagram fazowy mo\u017Ce jednak zawiera\u0107 r\u00F3wnie\u017C odseparowane punkty. Z diagramu fazowego mo\u017Cna \u0142atwo odczyta\u0107 charakter ruchu uk\u0142adu: \n* ruch nieograniczony \u2013 krzywa fazowa \u201Eucieka\u201D do niesko\u0144czono\u015Bci; \n* ruch ograniczony \u2013 krzywa fazowa jest ograniczona w pewnym sko\u0144czonym obszarze, istniej\u0105 dwa jego podtypy: \n* ruch okresowy \u2013 krzywe fazowe s\u0105 krzywymi zamkni\u0119tymi (np. drgania harmoniczne); \n* ruch nieokresowy \u2013 krzywe fazowe s\u0105 krzywymi otwartymi (np. drgania t\u0142umione). Podprzestrze\u0144 przestrzeni fazowej zawieraj\u0105ca tylko wsp\u00F3\u0142rz\u0119dne przestrzenne jest nazywana przestrzeni\u0105 konfiguracyjn\u0105, krzywe w tej przestrzeni to trajektorie."@pl . . . . "Diagram fazowy, portret fazowy \u2013 zbi\u00F3r punkt\u00F3w w przestrzeni fazowej reprezentuj\u0105cy mo\u017Cliwe ruchy dla zadanego hamiltonianu lub lagran\u017Cjanu. W pierwszym wypadku przestrze\u0144 fazowa zawiera wymiar\u00F3w przestrzennych i wymiar\u00F3w p\u0119dowych, gdzie jest stopniem swobody uk\u0142adu opisywanego hamiltonianem. W przypadku mechaniki Lagrange'a wymiary p\u0119dowe s\u0105 zast\u0105pione przez wymiary pr\u0119dko\u015Bciowe. Wobec tego ka\u017Cdy punkt na diagramie fazowym odpowiada pewnemu po\u0142o\u017Ceniu w przestrzeni sk\u0142adowych uk\u0142adu (wsp\u00F3\u0142rz\u0119dne przestrzenne punktu) i odpowiadaj\u0105cym im p\u0119dom (pr\u0119dko\u015Bciom)."@pl . "In der Mathematik dient das Phasenportr\u00E4t (auch Phasenportrait) der Veranschaulichung einer autonomen Differentialgleichung. Das Phasenraumportr\u00E4t gibt eine M\u00F6glichkeit, die zeitlichen Entwicklungen dynamischer Systeme graphisch zu analysieren. Dazu werden nur die dynamischen Gleichungen des Systems ben\u00F6tigt, eine explizite Darstellung der Zeitentwicklung, etwa durch analytisches L\u00F6sen einer Differentialgleichung, ist nicht n\u00F6tig. Das Phasenportr\u00E4t besteht aus der Gesamtheit aller Orbits des dynamischen Systems, zusammen mit Pfeilen, die die zeitliche Entwicklung entlang der Orbits angeben. Da die Gesamtheit aller Orbits der gesamte Phasenraum des dynamischen Systems ist, zeichnet man nur einige charakteristische Orbits. Aus dem Phasenportr\u00E4t eines dynamischen Systems l\u00E4sst sich ein erster Eindruck \u00FCber sein globales Verhalten gewinnen, beispielsweise die Existenz und Stabilit\u00E4t von Fixpunkten und periodischen Orbits. Aus Gr\u00FCnden der \u00DCbersichtlichkeit ist meist nur das Zeichnen von Phasenportr\u00E4ts in und sinnvoll. Man betrachtet also eine Differentialgleichung erster Ordnung: mit f\u00FCr eine Teilmenge .Die einzige Information, die wir \u00FCber die gesuchte Bahn haben, ist ihreAbleitung , die an der Stelle durch gegeben ist. Die Funktion ordnet also jedem Element aus dem Definitionsbereich eine Steigung oder auch Richtung zu. Tr\u00E4gt man diese Richtungen in Form von Geradenst\u00FCcken an den zugeh\u00F6rigen Punkten ein, wird ein Muster sichtbar. Die L\u00F6sungen der Differentialgleichung sind Kurven, die tangential zu diesen Geradenst\u00FCcken stehen und als Bahnkurven oder Trajektorien bezeichnet werden. Die Menge aller Bahnkurven, bzw. Trajektorien, gibt das Phasenportr\u00E4t. F\u00FCr ein Raster von Punkten wird die Richtung der Bewegung im Phasenraum durch Pfeile dargestellt; so wird ein Vektorfeld eingezeichnet. Folgt man nun ausgehend von einem bestimmten Startpunkt dem Pfeil, kommt man zu einem neuen Punkt, wo man dieses Vorgehen wiederholen kann. So kann man anhand des Vektorfelds zus\u00E4tzlich typische Trajektorien in das Phasenraumportr\u00E4t einzeichnen, die das qualitative Verhalten der zeitlichen Entwicklung einzusch\u00E4tzen helfen. F\u00FCr einfache dynamische Systeme kann man Vektorfeld und Beispieltrajektorien oft mit der Hand einzeichnen, bei komplexeren Systemen kann dies durch Computerprogramme geschehen."@de . "Phase portrait"@en . . "Un retrat de fase \u00E9s una representaci\u00F3 geom\u00E8trica de les traject\u00F2ries d'un sistema din\u00E0mic en el pla de fase. Cada conjunt de condicions inicials \u00E9s representat per una corba diferent, o per un punt. Els retrats de fase s\u00F3n una eina molt \u00FAtil en l'estudi de sistemes din\u00E0mics. Consisteixen en una representaci\u00F3 gr\u00E0fica de les traject\u00F2ries t\u00EDpiques en un espai d'estats. Aix\u00F2 revela informaci\u00F3 com ara si un punt d'equilibri \u00E9s atractor, repel\u00B7lent o cicle l\u00EDmit segons el valor que s'escolleixi d'un par\u00E0metre. El concepte d'equival\u00E8ncia topol\u00F2gica \u00E9s important en la classificaci\u00F3 del comportament dels sistemes per especificar quan dos retrats de fase diferents representen el mateix comportament din\u00E0mic qualitativament. Un atractor \u00E9s un punt estable que \u00E9s tamb\u00E9 anomenat \"embornal\" (en angl\u00E8s, \"sink\"). Un repel\u00B7lent \u00E9s considerat com un punt inestable i tamb\u00E9 \u00E9s conegut com a \"font\" (en angl\u00E8s, \"source\"). Una gr\u00E0fica del retrat de fases d'un sistema din\u00E0mic descriu les traject\u00F2ries del sistema (amb fletxes) i estats estacionaris estables (amb punts) i estats estacionaris inestables (amb cercles) en un espai d'estats. Els eixos representen les variables d'estat."@ca . . . . . . . "Un retrato de fase es una representaci\u00F3n geom\u00E9trica de todas las trayectorias de un sistema din\u00E1mico en el plano. Cada curva representa una condici\u00F3n inicial diferente. Un retrato de fase es una herramienta valiosa en el estudio de los sistemas din\u00E1micos aut\u00F3nomos de segundo orden. La configuraci\u00F3n de las curvas en el espacio de fase revela informaci\u00F3n sobre la existencia de atractores, , .\u200B El concepto de desempe\u00F1a un papel importante en la clasificaci\u00F3n de los sistemas din\u00E1micos para especificar cuando dos sistemas diferentes muestran el mismo comportamiento cualitativo. Un gr\u00E1fico de un retrato de fases de un sistema din\u00E1mico representa las trayectorias del sistema con flechas y sus estados de equilibrio estable e inestable con puntos.\u200B"@es . . "\u0424\u0430\u0437\u043E\u0432\u0438\u0439 \u043F\u043E\u0440\u0442\u0440\u0435\u0442 - \u0437\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u043D\u044F \u0442\u0440\u0430\u0454\u043A\u0442\u043E\u0440\u0456\u0439 \u0434\u0438\u043D\u0430\u043C\u0456\u0447\u043D\u043E\u0457 \u0441\u0438\u0441\u0442\u0435\u043C\u0438 \u0443 \u0444\u0430\u0437\u043E\u0432\u043E\u043C\u0443 \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0456. \u041A\u043E\u0436\u0435\u043D \u0441\u0442\u0430\u043D \u0441\u0438\u0441\u0442\u0435\u043C\u0438 \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u0430\u0454 \u043F\u0435\u0432\u043D\u0456\u0439 \u0442\u043E\u0447\u0446\u0456 \u043D\u0430 \u0444\u0430\u0437\u043E\u0432\u043E\u043C\u0443 \u043F\u043E\u0440\u0442\u0440\u0435\u0442\u0456. \u0424\u0430\u0437\u043E\u0432\u0456 \u043F\u043E\u0440\u0442\u0440\u0435\u0442\u0438 \u0441\u043B\u0443\u0436\u0430\u0442\u044C \u0434\u043B\u044F \u043D\u0430\u043E\u0447\u043D\u043E\u0433\u043E \u0432\u0456\u0434\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u043D\u044F \u043E\u0441\u043E\u0431\u043B\u0438\u0432\u043E\u0441\u0442\u0435\u0439 \u0435\u0432\u043E\u043B\u044E\u0446\u0456\u0457 \u0434\u0438\u043D\u0430\u043C\u0456\u0447\u043D\u043E\u0457 \u0441\u0438\u0441\u0442\u0435\u043C\u0438: \u0441\u0442\u0430\u0446\u0456\u043E\u043D\u0430\u0440\u043D\u0438\u0445 \u0442\u043E\u0447\u043E\u043A, \u0446\u0438\u043A\u043B\u0456\u0432, \u0431\u0430\u0441\u0435\u0439\u043D\u0456\u0432 \u043F\u0440\u0438\u0442\u044F\u0433\u0430\u043D\u043D\u044F. \u0414\u043B\u044F \u0434\u0432\u043E\u0432\u0438\u043C\u0456\u0440\u043D\u043E\u0457 \u0441\u0438\u0441\u0442\u0435\u043C\u0438 \u0444\u0430\u0437\u043E\u0432\u0438\u0439 \u043F\u043E\u0440\u0442\u0440\u0435\u0442 \u043F\u043E\u0432\u043D\u0456\u0441\u0442\u044E \u0432\u0456\u0434\u043E\u0431\u0440\u0430\u0436\u0430\u0454 \u0442\u0438\u043F\u0438 \u0442\u0440\u0430\u0454\u043A\u0442\u043E\u0440\u0456\u0439, \u044F\u043A\u0456 \u043C\u043E\u0436\u0443\u0442\u044C \u0440\u0435\u0430\u043B\u0456\u0437\u0443\u0432\u0430\u0442\u0438\u0441\u044F. \u0414\u043B\u044F \u0441\u0438\u0441\u0442\u0435\u043C\u0438 \u0431\u0456\u043B\u044C\u0448\u043E\u0457 \u0432\u0438\u043C\u0456\u0440\u043D\u043E\u0441\u0442\u0456 \u0431\u0443\u0434\u0443\u044E\u0442\u044C\u0441\u044F \u043F\u0440\u043E\u0454\u043A\u0446\u0456\u0457 \u0444\u0430\u0437\u043E\u0432\u0438\u0445 \u0442\u0440\u0430\u0454\u043A\u0442\u043E\u0440\u0456\u0439 \u043D\u0430 \u0432\u0438\u0431\u0440\u0430\u043D\u0443 \u043F\u043B\u043E\u0449\u0438\u043D\u0443 \u0444\u0430\u0437\u043E\u0432\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0443."@uk . "Un portrait de phase est une repr\u00E9sentation g\u00E9om\u00E9trique des trajectoires d'un syst\u00E8me dynamique dans l'espace des phases : \u00E0 chaque ensemble de conditions initiales correspond une courbe ou un point."@fr . "\u0424\u0430\u0437\u043E\u0432\u0438\u0439 \u043F\u043E\u0440\u0442\u0440\u0435\u0442"@uk . "\u0635\u0648\u0631\u0629 \u0627\u0644\u0637\u0648\u0631"@ar . . . . . . "4519252"^^ . "\u0424\u0430\u0437\u043E\u0432\u0438\u0439 \u043F\u043E\u0440\u0442\u0440\u0435\u0442 - \u0437\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u043D\u044F \u0442\u0440\u0430\u0454\u043A\u0442\u043E\u0440\u0456\u0439 \u0434\u0438\u043D\u0430\u043C\u0456\u0447\u043D\u043E\u0457 \u0441\u0438\u0441\u0442\u0435\u043C\u0438 \u0443 \u0444\u0430\u0437\u043E\u0432\u043E\u043C\u0443 \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0456. \u041A\u043E\u0436\u0435\u043D \u0441\u0442\u0430\u043D \u0441\u0438\u0441\u0442\u0435\u043C\u0438 \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u0430\u0454 \u043F\u0435\u0432\u043D\u0456\u0439 \u0442\u043E\u0447\u0446\u0456 \u043D\u0430 \u0444\u0430\u0437\u043E\u0432\u043E\u043C\u0443 \u043F\u043E\u0440\u0442\u0440\u0435\u0442\u0456. \u0424\u0430\u0437\u043E\u0432\u0456 \u043F\u043E\u0440\u0442\u0440\u0435\u0442\u0438 \u0441\u043B\u0443\u0436\u0430\u0442\u044C \u0434\u043B\u044F \u043D\u0430\u043E\u0447\u043D\u043E\u0433\u043E \u0432\u0456\u0434\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u043D\u044F \u043E\u0441\u043E\u0431\u043B\u0438\u0432\u043E\u0441\u0442\u0435\u0439 \u0435\u0432\u043E\u043B\u044E\u0446\u0456\u0457 \u0434\u0438\u043D\u0430\u043C\u0456\u0447\u043D\u043E\u0457 \u0441\u0438\u0441\u0442\u0435\u043C\u0438: \u0441\u0442\u0430\u0446\u0456\u043E\u043D\u0430\u0440\u043D\u0438\u0445 \u0442\u043E\u0447\u043E\u043A, \u0446\u0438\u043A\u043B\u0456\u0432, \u0431\u0430\u0441\u0435\u0439\u043D\u0456\u0432 \u043F\u0440\u0438\u0442\u044F\u0433\u0430\u043D\u043D\u044F. \u0414\u043B\u044F \u0434\u0432\u043E\u0432\u0438\u043C\u0456\u0440\u043D\u043E\u0457 \u0441\u0438\u0441\u0442\u0435\u043C\u0438 \u0444\u0430\u0437\u043E\u0432\u0438\u0439 \u043F\u043E\u0440\u0442\u0440\u0435\u0442 \u043F\u043E\u0432\u043D\u0456\u0441\u0442\u044E \u0432\u0456\u0434\u043E\u0431\u0440\u0430\u0436\u0430\u0454 \u0442\u0438\u043F\u0438 \u0442\u0440\u0430\u0454\u043A\u0442\u043E\u0440\u0456\u0439, \u044F\u043A\u0456 \u043C\u043E\u0436\u0443\u0442\u044C \u0440\u0435\u0430\u043B\u0456\u0437\u0443\u0432\u0430\u0442\u0438\u0441\u044F. \u0414\u043B\u044F \u0441\u0438\u0441\u0442\u0435\u043C\u0438 \u0431\u0456\u043B\u044C\u0448\u043E\u0457 \u0432\u0438\u043C\u0456\u0440\u043D\u043E\u0441\u0442\u0456 \u0431\u0443\u0434\u0443\u044E\u0442\u044C\u0441\u044F \u043F\u0440\u043E\u0454\u043A\u0446\u0456\u0457 \u0444\u0430\u0437\u043E\u0432\u0438\u0445 \u0442\u0440\u0430\u0454\u043A\u0442\u043E\u0440\u0456\u0439 \u043D\u0430 \u0432\u0438\u0431\u0440\u0430\u043D\u0443 \u043F\u043B\u043E\u0449\u0438\u043D\u0443 \u0444\u0430\u0437\u043E\u0432\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0443."@uk . . "Retrato de fase"@es . "A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each set of initial conditions is represented by a different curve, or point. Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the state space. This reveals information such as whether an attractor, a repellor or limit cycle is present for the chosen parameter value. The concept of topological equivalence is important in classifying the behaviour of systems by specifying when two different phase portraits represent the same qualitative dynamic behavior. An attractor is a stable point which is also called \"sink\". The repeller is considered as an unstable point, which is also known as \"source\"."@en . "A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each set of initial conditions is represented by a different curve, or point. Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the state space. This reveals information such as whether an attractor, a repellor or limit cycle is present for the chosen parameter value. The concept of topological equivalence is important in classifying the behaviour of systems by specifying when two different phase portraits represent the same qualitative dynamic behavior. An attractor is a stable point which is also called \"sink\". The repeller is considered as an unstable point, which is also known as \"source\". A phase portrait graph of a dynamical system depicts the system's trajectories (with arrows) and stable steady states (with dots) and unstable steady states (with circles) in a state space. The axes are of state variables."@en . . . "Portrait de phase"@fr .