❆❘❚❯❘ ❈➱❙❆❘ ❋❆❙❙❖◆■
❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙✿ ❇❆❈■❆❙ ❉❊ ❆❚❘❆➬➹❖ ❊ ❆P▲■❈❆➬Õ❊❙
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❡①✐✲ ❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳
❱■➬❖❙❆
Ficha catalográfica preparada pela Seção de Catalogação e Classificação da Biblioteca Central da UFV
T
Fassoni, Artur César, 1988-
F249s Sistemas dinâmicos: bacias de atração e aplicações / Artur 2012 César Fassoni. – Viçosa, MG, 2012.
vii, 138f. : il. ; 29cm.
Inclui apêndices.
Orientador: Lucy Tiemi Takahashi
Dissertação (mestrado) - Universidade Federal de Viçosa. Referências bibliográficas: f. 136-138
1. Equações diferenciais não-lineares. 2. Sistemas dinâmicos diferenciais. 3. Biomatemática. 4. Dinâmica populacional. 5. Estabilidade. 6. Dinâmica. I. Universidade
❆❘❚❯❘ ❈➱❙❆❘ ❋❆❙❙❖◆■
❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙✿ ❇❆❈■❆❙ ❉❊ ❆❚❘❆➬➹❖ ❊ ❆P▲■❈❆➬Õ❊❙
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❡①✐✲ ❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳
❆P❘❖❱❆❉❆✿ ✷✽ ❞❡ ❋❡✈❡r❡✐r♦ ❞❡ ✷✵✶✷✳
▲á❡r❝✐♦ ❏♦sé ❞♦s ❙❛♥t♦s ▼❡❤r❛♥ ❙❛❜❡t✐
❘✐❝❛r❞♦ ▼❛rt✐♥s ❞❛ ❙✐❧✈❛ ❘♦s❛ ▼❛r❝❡❧♦ ▲♦❜❛t♦ ▼❛rt✐♥s ✭❈♦♦r✐❡♥t❛❞♦r✮
❆●❘❆❉❊❈■▼❊◆❚❖❙
❆ ❉❡✉s✱ ♣❡❧❛ ❡s♣❡r❛♥ç❛ ❞❡ ✈✐❞❛ ❤♦❥❡ ❡ ❛♠❛♥❤ã✳
➚ ♠✐♥❤❛ ❡s♣♦s❛ ❆♥❞ré✐❛✱ q✉❡ t❛♥t❛s ✈❡③❡s ♠❡ ❢❡③ ❝♦♥t✐♥✉❛r ❡ tr❛❜❛❧❤❛r✳ ❖❜r✐❣❛❞♦ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❛❧❡❣r✐❛ ♣❛ss❛❞♦s ❛♦ s❡✉ ❧❛❞♦✳
❆♦s ♠❡✉s ♣❛✐s ❉é❧✐♦ ❡ ❑❧ê♥✐❛✱ ♣♦r t✉❞♦ ♦ q✉❡ ♠❡ ❡♥s✐♥❛r❛♠ ♥❡st❛ ✈✐❞❛✳ ❆♦s ♠❡✉s ✐r♠ã♦s ❆❧✐❝❡ ❡ ❚✐♠ót❡♦✱ ♣♦r q✉❡♠ ❡❧❡s sã♦ ❡ r❡♣r❡s❡♥t❛♠✳
❆ t♦❞♦s ♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s✱ ❛✈ôs ❡ ❛✈ós✱ t✐♦s ❡ t✐❛s✱ ♣r✐♠♦s ❡ ♣r✐♠❛s✱ s♦❣r♦ ❡ s♦❣r❛✱ ❝✉♥❤❛❞♦s ❡ ❝✉♥❤❛❞❛s✳
❆♦s ♠❡✉s ❛♠✐❣♦s✳
➚ ▲✉❝②✱ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ ♣❡❧❛ ❞✐s♣♦s✐çã♦✱ ♣♦r t❡r ❛❝r❡❞✐t❛❞♦ ❡♠ ♠✐♠ ❡ ♠❡ ♦r✐❡♥t❛r ♥ã♦ só ❡♠ ♠❛tér✐❛ ❞❡ ▼❛t❡♠át✐❝❛✳
❆♦ ♠❡✉ ❝♦✲♦r✐❡♥t❛❞♦r ▼❛r❝❡❧♦✳
❆♦s ❝♦❧❡❣❛s ❞❡ ▼❡str❛❞♦✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❆♥❛ P❛✉❧❛✱ ❋❡r♥❛♥❞♦✱ ❋r❡❞✱ ■s❛q✉❡ ❡ ❱✐✲ ♥í❝✐✉s✱ ♣❡❧♦s ♠♦♠❡♥t♦s ❝♦♠♣❛rt✐❧❤❛❞♦s ❥✉♥t♦s✳
❆♦s ♣r♦❢❡ss♦r❡s ❞♦ ♣r♦❣r❛♠❛✱ ♣❡❧❛s ❞✐s❝✐♣❧í♥❛s ❧❡❝✐♦♥❛❞❛s ❡ ❛t❡♥çã♦ ❞✐s♣❡♥s❛❞❛✳ ➚ ❝♦♦r❞❡♥❛❞♦r❛ ❙✐♠♦♥❡✱ ♣❡❧❛ ❞❡❞✐❝❛çã♦ ❡ ❡①❝❡❧ê♥❝✐❛✳
➚ ▼ír✐❛♥✱ s❡✉ ❏❛✐r ❡ t♦❞♦s ♦s ❞❡♠❛✐s ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉▼❆✳
❆♦s ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛✱ ♣♦r ❛❝❡✐t❛r❡♠ ♦ ❝♦♥✈✐t❡ ❡ ❛♣r❡s❡♥t❛r❡♠ s✉❣❡stõ❡s ✈❛❧✐♦s❛s✳ ➚ ❈❛♣❡s✱ ♣❡❧❛ ❜♦❧s❛✳
➚ ❋❆P❊▼■●✱ ❉❡♠❛♥❞❛ ❯♥✐✈❡rs❛❧ ❈❊❳ ❆P◗✵✵✶✹✾✲✵✽✱ ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦✳
❙✉♠ár✐♦
❘❡s✉♠♦ ✈✐
❆❜str❛❝t ✈✐✐
■♥tr♦❞✉çã♦ ✶
✶ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s ✸
✶✳✶ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s ✲ ❉❡✜♥✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❊①✐stê♥❝✐❛ ❡ ❯♥✐❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ❘❡tr❛t♦ ❞❡ ❋❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸✳✶ ❖ Pê♥❞✉❧♦ ✭❙✐♠♣❧❡s ❡ ❆♠♦rt❡❝✐❞♦✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸✳✷ ❈♦♠♣❡t✐çã♦ ❡♥tr❡ ❡s♣é❝✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹ ❖ ❋❧✉①♦ ❞❡ ✉♠❛ ❊❉❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹✳✶ ❙✐❣♥✐✜❝❛❞♦ ❣❡♦♠étr✐❝♦ ❞♦ ✢✉①♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✹✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦ ❋❧✉①♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✺ P♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✻ Ór❜✐t❛s ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ❡ ❝♦♥❥✉♥t♦s ❧✐♠✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✼ ❖ ❚❡♦r❡♠❛ ❞♦ ❋❧✉①♦ ❚✉❜✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✽ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❞❡ ❊❉❖✬s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✽✳✶ ❈❛s♦ n= 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✶✳✽✳✷ ❈❛s♦ A ❞✐❛❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✶✳✽✳✸ ❈❛s♦ A ❞✐❛❣♦♥❛❧✐③á✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✶✳✽✳✺ ❈❛s♦ ●❡r❛❧✿ ❯s♦ ❞❛ ❢♦r♠❛ ❞❡ ❏♦r❞❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✽✳✻ ❙✉❜❡s♣❛ç♦s ❡stá✈❡✐s ❡ ✐♥stá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✾ ❊st❛❜✐❧✐❞❛❞❡ ❞♦s P♦♥t♦s ❞❡ ❊q✉✐❧í❜r✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✶✵ ❖ ❚❡♦r❡♠❛ ❞❡ ❍❛rt♠❛♥✲●r♦❜♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✶✳✶✶ ❆❧❣✉♥s ❝♦♥❝❡✐t♦s s♦❜r❡ ❡st❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✷ ❚❡♦r❡♠❛ ❞❛ ❱❛r✐❡❞❛❞❡ ❊stá✈❡❧ ❡ ❇❛❝✐❛s ❞❡ ❆tr❛çã♦ ✸✶ ✷✳✶ ❖ ❚❡♦r❡♠❛ ❞❛ ❱❛r✐❡❞❛❞❡ ❊stá✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✶✳✶ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❛ ❱❛r✐❡❞❛❞❡ ❊stá✈❡❧ ▲♦❝❛❧ ✳ ✳ ✸✷ ✷✳✶✳✷ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❛ ❱❛r✐❡❞❛❞❡ ❊stá✈❡❧ ●❧♦❜❛❧ ✳ ✳ ✹✷ ✷✳✷ ❇❛❝✐❛s ❞❡ ❆tr❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✷✳✶ ❈❛r❛❝t❡r✐③❛çã♦ ❞❛ ❢r♦♥t❡✐r❛ ❞❛ ❜❛❝✐❛ ❞❡ ❛tr❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✷✳✷ ●❡♥❡r✐❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✷✳✸ ❯♠ ♠ét♦❞♦ ♣❛r❛ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛ ✈❛r✐❡❞❛❞❡ ❡stá✈❡❧ ❞❡ ✉♠ ❡q✉✐✲
❧í❜r✐♦ ❤✐♣❡r❜ó❧✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✷✳✸✳✶ ▼ét♦❞♦ ♣❛r❛ ❞❡t❡r♠✐♥❛çã♦ ❞❡ ❱❛r✐❡❞❛❞❡s ❊stá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✺✸
✸ ❆♣❧✐❝❛çã♦ ❛ ✉♠ s✐st❡♠❛ ✷❉ ✺✻
✸✳✶ ❖ ♠♦❞❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✸✳✷ ❆♥á❧✐s❡ ◗✉❛❧✐t❛t✐✈❛ ❞♦ ▼♦❞❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✸✳✸ ❇✐✲❡st❛❜✐❧✐❞❛❞❡ ❡ ❜❛❝✐❛s ❞❡ ❛tr❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✸✳✹ ❖ ♠ét♦❞♦ ❞❛s ❛♣r♦①✐♠❛çõ❡s s✉❝❡ss✐✈❛s ♣❛r❛ ❡st❡ ♠♦❞❡❧♦ ✳ ✳ ✳ ✳ ✳ ✻✹ ✸✳✹✳✶ ❘❡s✉❧t❛❞♦s ❝♦♠ ♦s ♣❛râ♠❡tr♦s ✜①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✸✳✹✳✷ ❘❡s✉❧t❛❞♦s ❝♦♠α ❡ β ✜①♦s ❡ δ ❧✐✈r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼
✸✳✹✳✸ ❘❡s✉❧t❛❞♦s ❝♦♠α ❡ β ❧✐✈r❡s ❡ δ= 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽
✸✳✺ ❈♦♥❝❧✉sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷
✹ ❆♣❧✐❝❛çã♦ ❛ ✉♠ s✐st❡♠❛ ✸❉ ✼✹
✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✹✳✷ ❖ ▼♦❞❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺
✹✳✸ P♦♥t♦s ❋✐①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✹✳✹ ❆♥á❧✐s❡ ❞❛ ❊st❛❜✐❧✐❞❛❞❡ ❞♦s P♦♥t♦s ❋✐①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✹✳✹✳✶ ❆♥á❧✐s❡ ❞♦ ♣♦♥t♦P5 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✹✳✺ ❆ ✐♥✢✉ê♥❝✐❛ ❞♦s ♣❛râ♠❡tr♦s s♦❜r❡ ❛s ❜❛❝✐❛s ❞❡ ❛tr❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✸ ✹✳✺✳✶ ❖ ♠ét♦❞♦ ♥✉♠ér✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✹ ✹✳✺✳✷ ❖s ❝❡♥ár✐♦s ♥♦ ❡s♣❛ç♦ ❞❡ ♣❛râ♠❡tr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✻ ✹✳✺✳✸ ❆ r❡s♣❡✐t♦ ❞❡ ❜✐❢✉r❝❛çõ❡s ♥♦ s✐st❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾ ✹✳✻ ❈♦♥❝❧✉sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✵
✺ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✶✷✷
❆ ❙✐❣♥✐✜❝❛❞♦ ❞♦s ♣❛râ♠❡tr♦s ❞♦s ❈❛♣ít✉❧♦s ✸ ❡ ✹ ✶✷✸
❆ ❈ó❞✐❣♦ ❞♦ ♣r♦❣r❛♠❛ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ ❱❛r✐❡❞❛❞❡ ❊stá✈❡❧ ✶✷✹
❆ ❈ó❞✐❣♦ ❞♦ ♣r♦❣r❛♠❛ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛s ❇❛❝✐❛s ✶✸✶
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✶✸✽
❘❊❙❯▼❖
❋❆❙❙❖◆■✱ ❆rt✉r ❈és❛r✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❢❡✈❡r❡✐r♦✱ ✷✵✶✷✳ ❙✐st❡♠❛s ❉✐♥â♠✐❝♦s✿ ❇❛❝✐❛s ❞❡ ❆tr❛çã♦ ❡ ❆♣❧✐❝❛çõ❡s✳ ❖r✐❡♥t❛❞♦r❛✿ ▲✉❝② ❚✐❡♠✐ ❚❛❦❛❤❛s❤✐✳ ❈♦♦r✐❡♥t❛❞♦r✿ ▼❛r❝❡❧♦ ▲♦❜❛t♦ ▼❛rt✐♥s✳
❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ♣r♦♣õ❡✲s❡ ❛ ❛♣r❡s❡♥t❛r ✉♠❛ ❞❡s❝r✐çã♦ ❞❛ t❡♦r✐❛ s♦❜r❡ ❛s ❜❛✲ ❝✐❛s ❞❡ ❛tr❛çã♦ ❞❡ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❤✐♣❡r❜ó❧✐❝♦s ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s ❡♠ t❡♠♣♦ ❝♦♥tí♥✉♦✱ ❛ ❞❡s❡♥✈♦❧✈❡r ✉♠ ♠ét♦❞♦ ♣❛r❛ ❛ ❞❡t❡r♠✐♥❛çã♦ ♥✉♠ér✐❝❛ ❞❡ss❛s ❜❛❝✐❛s ❡ ❛ ❡①❛♠✐♥❛r ♦s r❡s✉❧t❛❞♦s ❞❛ ❛♣❧✐❝❛çã♦ ❞❛ t❡♦r✐❛ ❡ ❞♦ ♠ét♦❞♦ ❡♠ ♠♦❞❡❧♦s ❞❡ ❞✐♥â♠✐❝❛ ❞❡ ♣♦♣✉❧❛çõ❡s✳ ❆ ❞❡t❡r♠✐♥❛çã♦ ❞❛s ❜❛❝✐❛s ❞❡ ❛tr❛çã♦ ♣❡r♠✐t❡ ♦ ❡st✉❞♦ ❞❡ ❡str❛té❣✐❛s ❞❡ ❝♦♥tr♦❧❡ s♦❜r❡ ♦s ♣❛râ♠❡tr♦s✱ ❞❡ ♠♦❞♦ ❛ ❛✉♠❡♥t❛r ♦✉ ❞✐♠✐♥✉✐r t❛✐s r❡❣✐õ❡s✱ ❝♦♥❢♦r♠❡ ♦ ✐♥t❡r❡ss❡✳ ❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❡ ❢❡♥ô♠❡♥♦s ❜✐♦❧ó✲ ❣✐❝♦s✱ ❡st❛s ♣r❡✈✐sõ❡s sã♦ ✐♠♣♦rt❛♥t❡s✱ ♣♦✐s✱ s❡ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ r❡♣r❡s❡♥t❛ ❛ ❡①t✐♥çã♦ ❞❡ ✉♠❛ ❡s♣é❝✐❡ q✉❡ ❞❡✈❡ s❡r ♣r❡s❡r✈❛❞❛✱ ❡♥tã♦ ♣r♦❝✉r❛✲s❡ ❣❛r❛♥t✐r q✉❡ ❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ♥❛t✉r❛✐s ♥ã♦ ❡st❡❥❛♠ ♥❛ ❜❛❝✐❛ ❞❡ ❛tr❛çã♦ ❞♦ ♠❡s♠♦✱ ❡st✉❞❛♥❞♦✲s❡ ❡str❛té❣✐❛s ❞❡ ❝♦♥tr♦❧❡ s♦❜r❡ ♦s ♣❛râ♠❡tr♦s ♣❛r❛ q✉❡ ❛ ❜❛❝✐❛ ❞♦ ♣♦♥t♦ ❞✐♠✐♥✉❛ s✉✜❝✐❡♥t❡♠❡♥t❡✳ ❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛ ❛♥á❧✐s❡ ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❞♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ ✉♠ s✐st❡♠❛✱ ❛ t❡♦r✐❛ ❞❡ ❜❛❝✐❛s ❞❡ ❛tr❛çã♦ tr❛③ ❝♦♥s❡q✉ê♥✲ ❝✐❛s t♦♣♦❧ó❣✐❝❛s ❛♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ q✉❡ ♣❡r♠✐t❡♠✱ ❞❡ ❢♦r♠❛ ✐♥❞✐r❡t❛✱ r❡❛❧✐③❛r ✉♠❛ ❛♥á❧✐s❡ ❣❧♦❜❛❧✱ ♥♦ ❡s♣❛ç♦ ❞❡ ♣❛râ♠❡tr♦s✱ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦✱ ❣❛r❛♥t✐♥❞♦ r❡s✉❧t❛❞♦s ♠❛✐s ❛♠♣❧♦s ❞♦s q✉❡ s❡ ♦❜tê♠ ❣❡r❛❧♠❡♥t❡✱ q✉❛♥❞♦ ♥ã♦ s❡ ❢❛③ ✉s♦ ❞❡st❛ t❡♦r✐❛✳
❆❇❙❚❘❆❈❚
❋❆❙❙❖◆■✱ ❆rt✉r ❈és❛r✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❋❡❜r✉❛r②✱ ✷✵✶✷✳ ❉②♥❛♠✐❝❛❧ s②st❡♠s✿ ❇❛s✐♥s ♦❢ ❛ttr❛❝t✐♦♥ ❛♥❞ ❛♣❧✐❝❛t✐♦♥s✳ ❆❞✈✐s❡r✿ ▲✉❝② ❚✐❡♠✐ ❚❛❦❛❤❛s❤✐✳ ❈♦✲❛❞✈✐s❡r✿ ▼❛r❝❡❧♦ ▲♦❜❛t♦ ▼❛rt✐♥s✳
❚❤❡ ♣r❡s❡♥t ✇♦r❦ ♣r♦♣♦s❡s t♦ ♣r❡s❡♥t ❛ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ t❤❡♦r② ♦♥ t❤❡ ❜❛s✐♥s ♦❢ ❛ttr❛❝t✐♦♥ ♦❢ ❤②♣❡r❜♦❧✐❝ ❡q✉✐❧✐❜r✐✉♠ ♣♦✐♥ts ♦❢ ❝♦♥t✐♥✉♦✉s ❞②♥❛♠✐❝❛❧ s②st❡♠s✱ t♦ ❞❡✈❡❧♦♣ ❛ ♠❡t❤♦❞ ❢♦r t❤❡ ♥✉♠❡r✐❝❛❧ ❞❡t❡r♠✐♥❛t✐♦♥ ♦❢ t❤❡s❡ ❜❛s✐♥s ❛♥❞ t♦ ❡①❛♠✐♥❡ t❤❡ r❡s✉❧ts ♦❢ ❛♣♣❧②✐♥❣ t❤❡ t❤❡♦r② ❛♥❞ ♠❡t❤♦❞ ✐♥ ♠♦❞❡❧s ♦❢ ♣♦♣✉❧❛t✐♦♥ ❞②♥❛♠✐❝s✳ ❚❤❡ ❞❡t❡r♠✐♥❛t✐♦♥ ♦❢ t❤❡ ❜❛s✐♥s ♦❢ ❛ttr❛❝t✐♦♥ ❛❧❧♦✇s t❤❡ st✉❞② ♦❢ ❝♦♥tr♦❧ str❛t❡❣✐❡s ♦♥ t❤❡ ♣❛r❛♠❡t❡rs✱ ✐♥ ♦r❞❡r t♦ ✐♥❝r❡❛s❡ ♦r ❞❡❝r❡❛s❡ s✉❝❤ r❡❣✐♦♥s✱ ❛s ✐♥t❡r❡st✳ ❋r♦♠ t❤❡ ❜✐♦❧♦❣✐❝❛❧ ♣❤❡♥♦♠❡♥❛ ✈✐❡✇♣♦✐♥t✱ t❤❡s❡ ♣r❡❞✐❝t✐♦♥s ❛r❡ ✈❡r② ✐♠♣♦rt❛♥t✱ ❜❡❝❛✉s❡ ✐❢ ❛♥ ❡q✉✐❧✐❜r✐✉♠ ♣♦✐♥t r❡♣r❡s❡♥ts t❤❡ ❡①t✐♥❝t✐♦♥ ♦❢ ❛ s♣❡❝✐❡s t❤❛t ♠✉st ❜❡ ♣r❡s❡r✈❡❞✱ t❤❡♥ ♦♥❡ s❡❡❦s t♦ ❣✉❛r❛♥t❡❡ t❤❛t t❤❡ ♥❛t✉r❛❧ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❞♦ ♥♦t ❛r❡ ✐♥ t❤❡ ❜❛s✐♥ ♦❢ ❛ttr❛❝t✐♦♥ ♦❢ t❤❛t ♣♦✐♥t✳ ❚❤✐s ✐s ♠❛❞❡ ❜② st✉❞②✐♥❣ ❝♦♥tr♦❧ str❛t❡❣✐❡s ♦♥ t❤❡ ♣❛r❛♠❡t❡rs✱ ❢♦r t❤❛t t❤❡ ♣♦✐♥t ❜❛s✐♥ ❞❡❝r❡❛s❡s s✉✣❝✐❡♥t❧②✳ ❋r♦♠ t❤❡ ✈✐❡✇♣♦✐♥t ♦❢ st❛❜✐❧✐t② ❛♥❛❧②s✐s ♦❢ ❡q✉✐❧✐❜r✐✉♠ ♣♦✐♥ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s✱ t❤❡ t❤❡♦r② ♦❢ ❜❛s✐♥s ♦❢ ❛ttr❛❝t✐♦♥ ❜r✐♥❣s t♦♣♦❧♦❣✐❝❛❧ ❝♦♥s❡q✉❡♥❝❡s t♦ t❤❡ ♣❤❛s❡ s♣❛❝❡ ✇❤✐❝❤ ❛❧❧♦✇✱ ✐♥❞✐r❡❝t❧②✱ ❝♦♥❞✉❝t ❛ ❣❧♦❜❛❧ ❛♥❛❧②s✐s ✐♥ t❤❡ ♣❛r❛♠❡t❡rs s♣❛❝❡✱ ❛❧❧♦✇✐♥❣ ✇✐❞❡r r❡s✉❧ts ♦❢ ✇❤✐❝❤ ❛r❡ ❣❡♥❡r❛❧❧② ♦❜t❛✐♥❡❞ ✇✐t❤♦✉t t❤✐s t❤❡♦r②✳
■♥tr♦❞✉çã♦
❙✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s sã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ♠♦❞❡❧❛r ❢❡♥ô♠❡♥♦s ❢í✲ s✐❝♦s✱ ❜✐♦❧ó❣✐❝♦s✱ ❡❝♦♥ô♠✐❝♦s✱ ❡t❝ ❬✷✷❪✳ ❊♠❜♦r❛✱ ❡♠ ❣❡r❛❧✱ ♥ã♦ s❡❥❛ ♣♦ss✐✈❡❧ ❞❡✲ t❡r♠✐♥❛r ✉♠❛ ❡①♣r❡ssã♦ ❛♥❛❧ít✐❝❛ ♣❛r❛ ❛ s♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛✱ ♣♦❞❡✲s❡ r❡❛❧✐③❛r ✉♠ ❡st✉❞♦ q✉❛❧✐t❛t✐✈♦ ❞♦ ♠❡s♠♦✱ ❛♥❛❧✐s❛♥❞♦ ❛ r❡❧❛çã♦ ❡♥tr❡ ♦s s❡✉s ♣❛râ♠❡tr♦s ❡ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦s s❡✉s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦✱ q✉❡ r❡♣r❡s❡♥t❛♠ ♦s ❡st❛❞♦s ♣❛r❛ ♦s q✉❛✐s ♦ s✐st❡♠❛ ♣♦❞❡ ❡✈♦❧✉✐r ❡ ♣❡r♠❛♥❡❝❡r✳ ❖❜t❡♠✲s❡✱ ❛ss✐♠✱ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞♦ ❢❡♥ô♠❡♥♦✱ ❛♣❡♥❛s ♣❡❧❛ ❛♥á❧✐s❡ ❞❡ s❡✉s ♣❛râ♠❡tr♦s✳ P♦ré♠✱ ❡♠ ❛❧❣✉♥s ❝❛s♦s ♣♦❞❡♠ ♦❝♦rr❡r ❞♦✐s ♦✉ ♠❛✐s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❛ss✐♥t♦✲ t✐❝❛♠❡♥t❡ ❡stá✈❡✐s✱ ❢❡♥ô♠❡♥♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❜✐❡st❛❜✐❧✐❞❛❞❡ ❬✶❪✳ ◆❡st❡ ❝❛s♦✱ ♥ã♦ só ♦s ♣❛râ♠❡tr♦s t❡rã♦ ✐♥✢✉ê♥❝✐❛ s♦❜r❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s s♦❧✉çõ❡s ♠❛s t❛♠✲ ❜é♠ ❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✱ ♣♦✐s ✉♠❛ ♣❡rt✉r❜❛çã♦ ❡♠ ✉♠❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ♣♦❞❡ ❧❡✈❛r ❛ ✉♠ ❡st❛❞♦ ❛ss✐♥tót✐❝♦ t♦t❛❧♠❡♥t❡ ❞✐❢❡r❡♥t❡ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ♦r✐❣✐♥❛❧✳ ❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❡ ❛♣❧✐❝❛çõ❡s✱ ♣♦❞❡✲s❡ t❡r✱ ♣♦r ❡①❡♠♣❧♦✱ ✉♠ ❡st❛❞♦ ✐♥❞❡s❡❥á✈❡❧ q✉❡ é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧✱ ❞❡ ♠♦❞♦ q✉❡ é ♥❡❝❡ssár✐♦ ❡♥t❡♥❞❡r ♦ q✉❡ ❞❡✈❡ ♦❝♦rr❡r ❝♦♠ ❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ♣❛r❛ ❡✈✐t❛r t❛❧ ❡st❛❞♦✳ P❛r❛ ❝♦♠♣r❡❡♥❞❡r ❝♦♠✲ ♣❧❡t❛♠❡♥t❡ ❡st❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✱ é ❢✉♥❞❛♠❡♥t❛❧ ❝❛r❛❝t❡r✐③❛r ❡ ❞❡t❡r♠✐♥❛r ❛ ❜❛❝✐❛ ❞❡ ❛tr❛çã♦ ❞❡ ❝❛❞❛ ❡q✉✐❧í❜r✐♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧ q✉❡ é✱ ♣♦r ❞❡✜♥✐çã♦✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s q✉❡ sã♦ ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ❞❡ s♦❧✉çõ❡s q✉❡ t❡♥❞❡♠ ♣❛r❛ ❛q✉❡❧❡ ❡q✉✐❧í❜r✐♦ q✉❛♥❞♦ ♦ t❡♠♣♦ t❡♥❞❡ ❛ ✐♥✜♥✐t♦ ❬✻❪✳ ❊st❛ ♥❡❝❡ss✐✲ ❞❛❞❡ ❞♦ ❡st✉❞♦ ❞❛s ❜❛❝✐❛s ❞❡ ❛tr❛çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❛✉tô♥♦♠♦ s✉r❣❡ ❝♦♠♦ ✉♠ ❡stá❣✐♦ ♥❛t✉r❛❧ ❛♣ós ✉♠ ❝✉rs♦ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s q✉❡ ❛❜r❛♥❣❡ ♦ ❚❡♦r❡♠❛ ❞♦ ❋❧✉①♦ ❚✉❜✉❧❛r✱ ♦ ❚❡♦r❡♠❛ ❞❡ ❍❛rt♠❛♥✲●r♦❜♠❛♥ ❡ ♦ ❚❡♦r❡♠❛ ❞❛ ❱❛r✐❡❞❛❞❡ ❊stá✈❡❧ ❬✷✻✱ ✸✸❪✳
■♥✐❝✐❛❧♠❡♥t❡✱ ♥♦ ❈❛♣ít✉❧♦ ✶✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ r❡✈✐sã♦ s♦❜r❡ ♣❛rt❡ ❞❛ t❡♦r✐❛ q✉❛❧✐t❛t✐✈❛ ❞❡ ❊❉❖✬s ❬✷✻✱ ✸✸✱ ✶✻❪✳
◆♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡✱ ❛♣r❡s❡♥t❛♠♦s ❛ t❡♦r✐❛ s♦❜r❡ ❇❛❝✐❛s ❞❡ ❆tr❛çã♦ ❡♠ ❙✐s✲ t❡♠❛s ❉✐♥â♠✐❝♦s ❝♦♥tí♥✉♦s ❬✻✱ ✶❪✳ ❱❡r❡♠♦s q✉❡ t❛✐s r❡s✉❧t❛❞♦s t❡♠ ✉♠ ❝❛rát❡r ❣❧♦❜❛❧ ❡ tr❛③❡♠ ❝♦♥s❡q✉ê♥❝✐❛s t♦♣♦❧ó❣✐❝❛s ❛ ❝♦♥✜❣✉r❛çã♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ❞❡ ✉♠ s✐st❡♠❛ ❛✉tô♥♦♠♦✳ ❊st❛s ❝♦♥s❡q✉ê♥❝✐❛s t❡rã♦ ✉♠ ♣❛♣❡❧ ❢✉♥❞❛♠❡♥t❛❧ ♥❛ ❛♥á❧✐s❡ q✉❛❧✐t❛t✐✈❛ ❞♦ s✐st❡♠❛ ❡st✉❞❛❞♦ ♥♦ ❈❛♣ít✉❧♦ ✹✳ ❈♦♠♦ ❡st❡s r❡s✉❧t❛❞♦s ❛ r❡s♣❡✐t♦ ❞❛s ❜❛❝✐❛s ❞❡ ❛tr❛çã♦ r❡❧❛❝✐♦♥❛♠ ❛ ❢r♦♥t❡✐r❛ ❞❛s ♠❡s♠❛s ❝♦♠♦ ❛ ✉♥✐ã♦ ❞❛s ✈❛✲ r✐❡❞❛❞❡s ❡stá✈❡✐s ❞❡ ❝❡rt♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ s✐st❡♠❛✱ ❛♣r❡s❡♥t❛♠♦s✱ ❛✐♥❞❛ ♥♦ ❈❛♣ít✉❧♦ ✷✱ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❝♦♥str✉t✐✈❛ ❞♦ ❚❡♦r❡♠❛ ❞❛ ❱❛r✐❡❞❛❞❡ ❊stá✲ ✈❡❧✱ q✉❡ ❢♦r♥❡❝❡ ✉♠ ♠ét♦❞♦ ♣❛r❛ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❡st❛s ✈❛r✐❡❞❛❞❡s ❬✽✱ ✶✸❪✳ ❈♦♠
♦s r❡s✉❧t❛❞♦s ❡ ♠ét♦❞♦s ❡♠ ♠ã♦s✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❛❧❣♦rít♠♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♥✉♠❡r✐❝❛♠❡♥t❡ ❛s r❡❣✐õ❡s ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ s✐st❡♠❛ ❛✉tô♥♦♠♦✳
◆♦ ❈❛♣ít✉❧♦ ✸✱ ❛♣❧✐❝❛♠♦s ❛ t❡♦r✐❛ ❡ ❛ r♦t✐♥❛ ❞❡s❡♥✈♦❧✈✐❞❛s ❛ ✉♠ ♠♦❞❡❧♦ ❝❧ás✲ s✐❝♦ ❞❡ ❝♦♠♣❡t✐çã♦ ❡♥tr❡ ❡s♣é❝✐❡s ❡♠ ❞✐♠❡♥sã♦ ❞♦✐s✱ ♦♥❞❡ ♦❝♦rr❡ ❜✐✲❡st❛❜✐❧✐❞❛❞❡ ❬✷✸❪✱ ❛♠♣❧✐❛♥❞♦ ❛ ❛♥á❧✐s❡ q✉❛❧✐t❛t✐✈❛ ❞♦ ♠❡s♠♦✱ ❡st✉❞❛♥❞♦ ❛ ❢♦r♠❛ ❞❛ s❡♣❛r❛tr✐③ ❡♥tr❡ ❛s ❜❛❝✐❛s ❞❡ ❛tr❛çã♦ ❡ ❛ ✐♥✢✉ê♥❝✐❛ ❞❡ ❝❛❞❛ ♣❛râ♠❡tr♦ s♦❜r❡ ♦ t❛♠❛♥❤♦ ❞❛s ♠❡s♠❛s✱ ❞✐s❝✉t✐♥❞♦ ❡str❛té❣✐❛s ♣❡rt✐♥t❡♥t❡s ❛ ❞✐♠✐♥✉✐çã♦ ♦✉ ❛✉♠❡♥t♦ ❞❡ss❛s ❜❛❝✐❛s✳
◆♦ ❈❛♣ít✉❧♦ ✹✱ ♣r♦♣♦♠♦s ✉♠ ♠♦❞❡❧♦ q✉❡ ✈✐s❛ ❞❡s❝r❡✈❡r ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ❞✉❛s ❡s♣é❝✐❡s ❞❡ ♣❧❛♥t❛s✱ ♦♥❞❡ ♦ ❤❛❜✐t❛t ♥❛t✉r❛❧ ❞❡ ✉♠❛ ❞❡❧❛s é ✐♥✈❛❞✐❞♦ ♣❡❧❛ ♦✉tr❛ q✉❡✱ ❛❧é♠ ❞❡ ❝♦♠♣❡t✐r ❝♦♠ ❛ ❡s♣é❝✐❡ ♥❛t✐✈❛ ♣❡❧♦s ❛❧✐♠❡♥t♦s ♣r❡s❡♥t❡s ♥♦ ♠❡✐♦✱ ♣r♦❞✉③ ✉♠❛ ✜t♦t♦①✐♥❛ q✉❡ ✐♥✐❜❡ ♦ ❝r❡s❝✐♠❡♥t♦ ❞❛ ❡s♣é❝✐❡ ♥❛t✐✈❛✱ r❡❝✉rs♦ ❝♦♥❤❡✲ ❝✐❞♦ ❝♦♠♦ s✉♣r❡ssã♦ ❛❧❡❧♦♣át✐❝❛✳ ❖ ♠♦❞❡❧♦ ❝♦♥s✐st❡ ❞❡ ✉♠ s✐st❡♠❛ ❛✉tô♥♦♠♦ ❝♦♠ três ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❡ ♦✐t♦ ♣❛râ♠❡tr♦s✳ ◆❛ ❛♥á❧✐s❡ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦✱ ❞✐❛♥t❡ ❞❛ ♦❝♦rrê♥❝✐❛ s✐♠✉❧tâ♥❡❛ ❞❡ ❞♦✐s ❡q✉✐❧í❜r✐♦s ❛ss✐♥t♦✲ t✐❝❛♠❡♥t❡ ❡stá✈❡✐s✱ ✜③❡♠♦s ✉s♦ ❞❛s ❢❡rr❛♠❡♥t❛s t❡ór✐❝❛s s♦❜r❡ ❜❛❝✐❛s ❞❡ ❛tr❛çã♦ ❛♣r❡s❡♥t❛❞❛s ♥♦ ❈❛♣ít✉❧♦ ✷✳ ❆s ❝♦♥s❡q✉ê♥❝✐❛s ❞❡st❛s ❢❡rr❛♠❡♥t❛s s♦❜r❡ ❛ ❝♦♥✜✲ ❣✉r❛çã♦ t♦♣ó❧♦❣✐❝❛ ❞♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ♣❡r♠✐t❡♠ r❡❛❧✐③❛r ✉♠❛ ❛♥á❧✐s❡ q✉❛❧✐t❛t✐✈❛ ❣❧♦❜❛❧✱ s❡♠ ✜①❛r ♥❡♥❤✉♠ ✈❛❧♦r ♥✉♠ér✐❝♦ ♣❛r❛ ♦s ♣❛râ♠❡tr♦s✱ t❡♥❞♦ ❛ss✐♠✱ ✉♠❛ ❞❡s❝r✐çã♦ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❡♠ t♦❞♦ ❡s♣❛ç♦ ❞❡ ♣❛râ♠❡tr♦s✳ ❚❛❧ ❡st✉❞♦ s❡r✐❛ ✐♥✈✐á✈❡❧ s❡♠ ❛ t❡♦r✐❛ ❡st✉❞❛❞❛ ♥♦ ❈❛♣ít✉❧♦ ✷✳ ❊st✉❞❛♠♦s t❛♠✲ ❜é♠ ❛ ✐♥✢✉ê♥❝✐❛ ❞❡ ❝❛❞❛ ✉♠ ❞♦s ♣❛râ♠❡tr♦s s♦❜ ❛s ❜❛❝✐❛s ❞❡ ❛tr❛çã♦ ❞♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦✱ ♦ q✉❡ ♣♦ss✐❜✐❧✐t❛ ♦ ❞✐r❡❝✐♦♥❛♠❡♥t♦ ❞❡ ❡str❛té❣✐❛s ❞❡ ❝♦♥tr♦❧❡ q✉❡ ✈✐s❡♠ ❛ s♦❜r❡✈✐✈ê♥❝✐❛ ♦✉ ❡①t✐♥çã♦ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ❡s♣é❝✐❡s✳
P♦r ú❧t✐♠♦✱ ♥♦ ❈❛♣ít✉❧♦ ✺ ❛♣r❡s❡♥t❛♠♦s ❛s ❝♦♥❝❧✉sõ❡s✱ r❡ss❛❧t❛♥❞♦ ❛ r❡❧❡✈â♥❝✐❛ ❞❡ s❡ ✉t✐❧✐③❛r ❛ t❡♦r✐❛ s♦❜r❡ ❜❛❝✐❛s ❞❡ ❛tr❛çã♦ ♣❛r❛ ❛ ❛♥á❧✐s❡ q✉❛❧✐t❛t✐✈❛ ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s ♥ã♦✲❧✐♥❡❛r❡s✳
❈❛♣ít✉❧♦ ✶
❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s
◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ r❡✈✐sã♦ ❞❡ ✉♠❛ ♣❛rt❡ ❞❛ t❡♦r✐❛ q✉❛❧✐t❛✲ t✐✈❛ ❞♦s s✐st❡♠❛s ❛✉tô♥♦♠♦s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✳ ❚♦❞♦s ♦s r❡s✉❧t❛❞♦s✱ ❜❡♠ ❝♦♠♦ s✉❛s ❞❡♠♦♥str❛çõ❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬✷✻❪✱ ❬✶✻❪✱ ❬✸✸❪ ❡ ❬✶✶❪✳ ◆ã♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛s ❞❡♠♦♥str❛çõ❡s ❞♦s r❡s✉❧t❛❞♦s✱ ♣♦✐s ❡❧❛s ❢♦❣❡♠ ❞♦ ❡s❝♦♣♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❖ ♦❜❥❡t✐✈♦ ❛q✉✐ é ❛♣❡♥❛s ❛♣r❡s❡♥t❛r ❛s ❢❡rr❛♠❡♥t❛s ❜ás✐❝❛s ♣❛r❛ ♦ ❞❡✲ s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❡st✉❞❛❞❛ ♥♦ ❈❛♣ít✉❧♦ ✷ ❡ ❞♦ ♠ét♦❞♦ ♣r♦♣♦st♦ ♥♦ ♠❡s♠♦ ❝❛♣ít✉❧♦✱ q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ♥♦s ❈❛♣ít✉❧♦s ✸ ❡ ✹✳
✶✳✶ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s ✲ ❉❡✜♥✐çã♦
❯♠ s✐st❡♠❛ ❛✉tô♥♦♠♦ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s é ✉♠ s✐st❡♠❛ ❞❛
❢♦r♠❛✿
dx1
dt = f1(x1, ..., xn)
✳✳✳
dxn
dt = fn(x1, ..., xn)
✭✶✳✶✮
♦♥❞❡ ❛s ✐♥❝ó❣♥✐t❛s sã♦ ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s x1, ..., xn :I → R✱ ❞❡✜♥✐❞❛s ❡♠ ✉♠
✐♥t❡r✈❛❧♦ ❛❜❡rt♦ I ⊂R✱ q✉❡ s❛t✐s❢❛ç❛♠ dxi
dt =fi(x1, ..., xn), i= 1, ..., n. ✭✶✳✷✮
❖ ♥♦♠❡ ❛✉tô♥♦♠♦ ✈❡♠ ❞♦ ❢❛t♦ ❞❛s ❢✉♥çõ❡s fi ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❡♠ ✭✶✳✷✮ ♥ã♦
❞❡♣❡♥❞❡r❡♠ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞♦ t❡♠♣♦✱ ♠❛s ❛♣❡♥❛s ❞♦ ♣♦♥t♦ x = (x1, ..., xn) ∈ U ⊂ Rn✳ ❚♦❞♦ s✐st❡♠❛ ♥ã♦ ❛✉tô♥♦♠♦ ♣♦❞❡ s❡r ❝♦❧♦❝❛❞♦ ♥❛ ❢♦r♠❛ ❛✉tô♥♦♠❛
✐♥tr♦❞✉③✐♥❞♦✲s❡ ✉♠❛ ♥♦✈❛ ✈❛r✐á✈❡❧ xn+1 = t✱ ♣❛r❛ ❛ q✉❛❧ dxdtn+1 = 1✳ ❆ss✐♠✱ ❞❡✜♥❡✲s❡ fn+1(x1, ..., xn+1) = 1❡ ❛❞✐❝✐♦♥❛✲s❡ ❛ ❡q✉❛çã♦ dxdtn+1 =fn+1(x1, ..., xn+1) ❛♦ s✐st❡♠❛✳
❊s❝r❡✈❡♥❞♦ x = (x1, ..., xn) ❡ f = (f1, ..., fn) : U →Rn✱ ♦ s✐st❡♠❛ ✭✶✳✶✮ ♣♦❞❡
s❡r ❡s❝r✐t♦ ♥❛ ❢♦r♠❛
˙
x=f(x) ✭✶✳✸✮
r❛③ã♦ ♣❡❧❛ q✉❛❧✱ ✭✶✳✶✮ ♣♦❞❡ s❡r ❝❤❛♠❛❞♦ ❞❡ ✉♠❛ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧ ❖r❞✐♥ár✐❛ ✭❊❉❖✮ ❛✉tô♥♦♠❛✳
❆ ❛♣❧✐❝❛çã♦ f : U → Rn ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦ ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ q✉❡ ❛ ❝❛❞❛
♣♦♥t♦ x0 ∈ U ❛ss♦❝✐❛ ✉♠ ✈❡t♦r f(x0) ∈ Rn ❝♦♠ ♦r✐❣❡♠ ❡♠ x0 ❡ ❡①tr❡♠✐❞❛❞❡ ❡♠ x0 +f(x0)✳ ❊✱ r❡❝✐♣r♦❝❛♠❡♥t❡✱ ❞❛❞♦ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s g : U → Rn✱ ❡❧❡ ❞á ♦r✐❣❡♠ ❛ ❊❉❖ x˙ = g(x)✳ P♦rt❛♥t♦✱ ❊❉❖✬s ❡ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❡stã♦ ❡♠
❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛✳ ❆ss✐♠✱ ♠✉✐t❛s ✈❡③❡s ♥♦s r❡❢❡r✐r❡♠♦s ❛ ✭✶✳✶✮✱ ♦✉ ❛ ✭✶✳✸✮✱ ❝♦♠♦ ❊❉❖✬s ❞❡✜♥✐❞❛s ♣❡❧♦ ❝❛♠♣♦f ❡✱ ♥♦s r❡❢❡r✐r❡♠♦s ❛f ❝♦♠♦ ❝❛♠♣♦ ❞❛
❊❉❖ ✭✶✳✸✮✳
◆❛ ♥♦t❛çã♦ ✈❡t♦r✐❛❧✱ ❛ s♦❧✉çã♦ q✉❡ ♣r♦❝✉r❛♠♦s ♣❛r❛ ✭✶✳✸✮ é ✉♠❛ ❛♣❧✐❝❛çã♦
x : I ⊂ R → Rn t❛❧ q✉❡ dxi
dt = fi(x1, ..., xn)✱ i = 1, ..., n✳ ●❡♦♠❡tr✐❝❛♠❡♥t❡✱
♣r♦❝✉r❛♠♦s ✉♠❛ ❝✉r✈❛ ❞✐❢❡r❡♥❝✐á✈❡❧ x(t) ❝♦♥t✐❞❛ ❡♠ U t❛❧ q✉❡✱ ❡♠ ❝❛❞❛ ♣♦♥t♦ x(t)✱ ♦ s❡✉ ✈❡t♦r t❛♥❣❡♥t❡ x′(t) ❝♦✐♥❝✐❞❛ ❝♦♠ ♦ ❝❛♠♣♦ ♥❛q✉❡❧❡ ♣♦♥t♦✱ ✐st♦ é✱
q✉❡r❡♠♦s q✉❡ x(t)s❛t✐s❢❛ç❛
x′(t) = f(x(t)).
❙❡ ❡s❝r❡✈❡r♠♦s ♦ s✐st❡♠❛ ✭✶✳✸✮ ❛❞✐❝✐♦♥❛♥❞♦ ✉♠❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧✱ t❡♠♦s ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ ✭P❱■✮✿
˙
x=f(x)
x(0) =x0 .
✭✶✳✹✮
❯♠❛ s♦❧✉çã♦ ❞❡st❡ P❱■ s❡rá ✉♠❛ ❝✉r✈❛x(t)q✉❡ ♣❛ss❡ ♣♦rx0 ∈U ♥♦ t❡♠♣♦t=
0❡ q✉❡ s❡❥❛ t❛♥❣❡♥t❡ ❛♦ ❝❛♠♣♦f ❡♠ ❝❛❞❛ ♣♦♥t♦✳ ❱❡❥❛ ❛ ❋✐❣✉r❛ ✶✳✶✳ ❋♦r♠❛❧♠❡♥t❡✱
t❡♠♦s ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿
❉❡✜♥✐çã♦ ✶✳✶ ❯♠❛ s♦❧✉çã♦ ❞♦ P❱■ ✭✶✳✹✮ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ I ⊂R é ✉♠❛
❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ x:I →Rn t❛❧ q✉❡
• x(t)∈U, ∀t∈I✳
• 0∈I ❡ x(0) =x0✳ • x′(t) = f(x(t)), ∀t∈I✳
➱ ✐♠♣♦rt❛♥t❡ r❡❢♦rç❛r q✉❡ ❡st❛ ❞❡✜♥✐çã♦ ❞✐③ q✉❛♥❞♦ ✉♠❛ ❝✉r✈❛ é ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✶✳✹✮ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❡s♣❡❝í✜❝♦ I✳ ◆♦t❡ t❛♠❜é♠ q✉❡ ♥❡❝❡ss❛r✐❛♠❡♥t❡ 0∈ I✳
➚s ✈❡③❡s✱ q✉❛♥❞♦ ♥ã♦ ❢♦r ✐♠♣♦rt❛♥t❡ ❡s♣❡❝✐✜❝❛r ♦ ✐♥t❡r✈❛❧♦✱ ♥♦s r❡❢❡r✐r❡♠♦s ❛ ✉♠❛ t❛❧ s♦❧✉çã♦ ❞❡ ✭✶✳✹✮ ❝♦♠♦ ✉♠❛ s♦❧✉çã♦ ❞❛ ❊❉❖ ✭✶✳✸✮ ♣❛ss❛♥❞♦ ♣♦r x0✳ ❆✐♥❞❛✱
❝❤❛♠❛♠♦s x(t) ❞❡ ❝✉r✈❛ ✐♥t❡❣r❛❧ ❞❡ f✱ ♦✉ ❝✉r✈❛ ✐♥t❡❣r❛❧ ❞❡ f ♣❛ss❛♥❞♦
♣♦r x0✱ q✉❛♥❞♦ ❛❞✐❝✐♦♥❛r♠♦s ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ❡♠ ✭✶✳✹✮ ♦✉ ❛✐♥❞❛✱ tr❛❥❡tór✐❛
❞♦ ❝❛♠♣♦ f✳
❋✐❣✉r❛ ✶✳✶✿ ❈❛♠♣♦ ♥✉♠ ❛❜❡rt♦U ⊂R2❡ s♦❧✉çã♦ ❞❡ ✉♠ P❱■ ❛ss♦❝✐❛❞♦ ❛♦ ❝❛♠♣♦✳
❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❡ ❬✸❪✳
✶✳✷ ❊①✐stê♥❝✐❛ ❡ ❯♥✐❝✐❞❛❞❡
❱❡r❡♠♦s ❛❣♦r❛ ❛❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ ❣❛r❛♥t❡♠ ❛ ❡①✐stê♥❝✐❛ ❡ ❛ ✉♥✐❝✐❞❛❞❡ ❞❛s s♦❧✉çõ❡s ❞❡ ✉♠❛ ❊❉❖ ❛✉tô♥♦♠❛ ❡ t❛♠❜é♠ ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❚r❛❜❛❧❤❛r❡♠♦s s❡♠♣r❡ ❝♦♠ ❛ ❤✐♣ót❡s❡ ❞❡ q✉❡ f :U →Rn é ✉♠ ❝❛♠♣♦ ❞❡ ❝❧❛ss❡ C1 ❞❡✜♥✐❞♦ ♥✉♠ ❛❜❡rt♦ U ⊆Rn✳ P❛r❛ ❞❡♥♦t❛r ❡st❡ ❢❛t♦✱ ❡♠ t♦❞❛ ❛ ❞✐ss❡rt❛çã♦✱
✉s❛r❡♠♦s ❛ ♥♦t❛çã♦ f ∈C1(U)✱ U ❛❜⊂Rn✳
❚❡♦r❡♠❛ ✶✳✷ ✭❊①✐stê♥❝✐❛ ❡ ❯♥✐❝✐❞❛❞❡ ❉❡ P✐❝❛r❞✮ ❙❡ f : U → Rn é ✉♠
❝❛♠♣♦ ❞❡ ❝❧❛ss❡ C1 ♥♦ ❛❜❡rt♦U ❡ x
0 ∈U ❡♥tã♦ ❡①✐st❡ a >0 t❛❧ q✉❡ ♦ P❱■
˙
x=f(x)
x(0) =x0
✭✶✳✺✮
t❡♠ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ x(t) ♥♦ ✐♥t❡r✈❛❧♦ I = (−a, a)✳
❉❡♠♦♥str❛çã♦✿ ❆ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷✻❪✱ ♣♣✳ ✼✵✲✼✻✳
❖❜s❡r✈❡ q✉❡ ♦ t❡♦r❡♠❛ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ✐♥t❡r✈❛❧♦ ♥♦ q✉❛❧ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ❞♦ P❱■ ✭✶✳✺✮✳ ❆ss✐♠✱ ♣❛r❛ ✉♠ ✐♥t❡r✈❛❧♦ ❞❛❞♦✱ ♥ã♦ s❛❜❡♠♦s ❛✐♥❞❛ s❡ ❡①✐st❡ ♦✉ ♥ã♦ s♦❧✉çã♦ ♥♦ ♠❡s♠♦✱ ♠✉✐t♦ ♠❡♥♦s s❡ ❡❧❛ é ú♥✐❝❛✳ ❆✐♥❞❛✱ ♥ã♦ t❡♠♦s ❝♦♠♦ ♣r❡❝✐s❛r ♦ r❛✐♦a❞♦ ✐♥t❡r✈❛❧♦ ❞❛❞♦ ♣❡❧♦ t❡♦r❡♠❛✳ ❈♦♥t✉❞♦ ♦ ♣ró①✐♠♦
t❡♦r❡♠❛ ♥♦s ❞á ✉♠ ✐♥t❡r✈❛❧♦ ♠❛①✐♠❛❧✳
❚❡♦r❡♠❛ ✶✳✸ ✭❙♦❧✉çã♦ ▼❛①✐♠❛❧✮ ❙❡❥❛ f ∈ C1(U)✳ ❊♥tã♦✱ ♣❛r❛ ❝❛❞❛ x 0 ∈ U ❡①✐st❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ I(x0) ❡ x :I(x0) → Rn ✉♠❛ s♦❧✉çã♦ ú♥✐❝❛ ❞❡ ✭✶✳✺✮ ❡♠ I(x0)✱ ❝♦♠ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿
❙❡ y:J →Rn é s♦❧✉çã♦ ❞❡ ✭✶✳✺✮ ❡♠ J ⊂❛❜ R✱ ❡♥tã♦ J ⊆I(x
0) ❡ x|J =y✳
❉❡♠♦♥str❛çã♦✿ ❆ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷✻❪✱ ♣♣✳ ✽✼✲✽✾✳
❉❡✜♥✐çã♦ ✶✳✹ ❆ s♦❧✉çã♦ x : I(x0) → Rn ❞❡ ✭✶✳✺✮ ❞❛❞❛ ♣❡❧♦ ❚❡♦r❡♠❛ ✭✶✳✸✮ é ❝❤❛♠❛❞❛ s♦❧✉çã♦ ♠❛①✐♠❛❧ ❞❡ ✭✶✳✺✮ ♣❛ss❛♥❞♦ ♣♦r x0✳
❈♦♠♦ I(x0) é ❛❜❡rt♦ ❡ 0 ∈ I(x0)✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r I(x0) = (−α, β) ❝♦♠
α, β > 0✱ ♥ã♦ ❡①❝❧✉✐♥❞♦ ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s α, β = ∞✳ ❈♦♠ ❛❧❣✉♠❛s ❤✐♣ót❡s❡s ❛
♠❛✐s✱ é ♣♦ss✐✈❡❧ ❞❡t❡r♠✐♥❛r q✉❛♥❞♦ ♦ ✐♥t❡r✈❛❧♦ ♠❛①✐♠❛❧ é ❛ r❡t❛ t♦❞❛✳ ❚❡♦r❡♠❛ ✶✳✺ ❙❡❥❛♠ f ∈ C1(U)✱ x
0 ∈ U✱ ❡ x : I(x0) → Rn ❛ s♦❧✉çã♦ ♠❛①✐♠❛❧ ❞❡ ✭✶✳✺✮ ❝♦♠ I(x0) = (−α, β)✳ ❊♥tã♦
✭✐✮ ❙❡ β ∈R✱ ❡♥tã♦x(t)→∂U q✉❛♥❞♦ t→β−✱ ✐st♦ é✱ ❞❛❞♦ q✉❛❧q✉❡r ❝♦♠♣❛❝t♦ K ⊂U✱ ❡①✐st❡ t ∈(0, β) t❛❧ q✉❡ x(t)∈/K✳
✭✐✐✮ ❙❡ β ∈R ❡ ❡①✐st❡ ♦ ❧✐♠✐t❡ lim
t→β−x(t) ❡♥tã♦ tlim→β−x(t)∈∂U
✭✐✐✐✮ ❙❡ ❛ tr❛❥❡tór✐❛ x(I(x0)) é ❧✐♠✐t❛❞❛ ❡ U =Rn✱ ❡♥tã♦ β =∞✳
❉❡♠♦♥str❛çã♦✿ ❆ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷✻❪✱ ♣♣✳ ✾✵✲✾✶✳
P♦❞❡♠♦s ❛♣❧✐❝❛r ♦ ❚❡♦r❡♠❛ ✶✳✺ ❛♦ ❝❛♠♣♦ −f ♣❛r❛ ♦❜t❡r ❝♦♥❝❧✉sõ❡s ❛♥á❧♦❣❛s
♣❛r❛ ♦ ❡①tr❡♠♦−α ❞♦ ✐♥t❡r✈❛❧♦ ♠❛①✐♠❛❧ I(x0)✳
✶✳✸ ❘❡tr❛t♦ ❞❡ ❋❛s❡
❖ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ❞♦ s✐st❡♠❛ ✭✶✳✸✮ é ♦ ❛❜❡rt♦U ⊆Rn ♦♥❞❡ ♦ ❝❛♠♣♦f ❛t✉❛✳
❊❧❡ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❡st❛❞♦s ♣♦ssí✈❡✐s ❞♦ s✐st❡♠❛✳ ❖ r❡tr❛t♦ ❞❡ ❢❛s❡ ❞♦ s✐st❡♠❛ ❝♦♥s✐st❡ ❞♦ ❡s❜♦ç♦ ❞❛s ❝✉r✈❛s ✐♥t❡❣r❛✐s ♣❛ss❛♥❞♦ ♣♦r ❞❡t❡r♠✐♥❛❞♦s ♣♦♥t♦s ❞❡ U ❞❡ ♠❛♥❡✐r❛ q✉❡ s❡❥❛♠ ❡✈✐❞❡♥❝✐❛❞♦s ♦s ❛s♣❡❝t♦s
♠❛✐s ✐♠♣♦rt❛♥t❡s ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ q✉❛❧✐t❛t✐✈♦ ❞♦ s✐st❡♠❛✳ ❈♦♠♦ ✐❧✉str❛çã♦✱ ❝♦❧♦❝❛♠♦s✱ s❡♠ ❝♦♥t❛s ♦✉ ❥✉st✐✜❝❛t✐✈❛s✱ ♦ ❡s❜♦ç♦ ❞♦ r❡tr❛t♦ ❞❡ ❢❛s❡s ❞❡ ❛❧❣✉♥s s✐st❡♠❛s ❛✉tô♥♦♠♦s✳
✶✳✸✳✶ ❖ Pê♥❞✉❧♦ ✭❙✐♠♣❧❡s ❡ ❆♠♦rt❡❝✐❞♦✮
❈♦♥s✐❞❡r❡ ✉♠ ♣ê♥❞✉❧♦ ❞❡ ♠❛ss❛m✱ ❝♦♠♣r✐♠❡♥t♦l✱ s♦❜ ❛ ❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡g✳
❙✉♣♦♥❤❛ q✉❡ ❡❧❡ ❡st❡❥❛ s✉❥❡✐t♦ ❛ ✉♠❛ ❢♦rç❛ ❞❡ ❛♠♦rt❡❝✐♠❡♥t♦ ♣r♦♣♦r❝✐♦♥❛❧ ❛ s✉❛ ✈❡❧♦❝✐❞❛❞❡✳ ❙❡❥❛θ ♦ â♥❣✉❧♦ q✉❡ ❞❡s❝r❡✈❡ ❛ s✉❛ ✐♥❝❧✐♥❛çã♦ ❡♠ r❡❧❛çã♦ ❛♦ r❡♣♦✉s♦✱
❝♦♥❢♦r♠❡ ✐❧✉str❛❞♦ ♥❛ ❋✐❣✉r❛ ✶✳✷✳
❈♦♥s✐❞❡r❛♥❞♦µ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛♠♦rt❡❝✐♠❡♥t♦ ❞♦ ♣ê♥❞✉❧♦✱ ♣❡❧❛s ❧❡✐s ❞❛ ❢ís✐❝❛✱
t❡♠♦s q✉❡
ml2θ¨+ml2µ θ˙=−mgls❡♥θ ✭✶✳✻✮
❋✐❣✉r❛ ✶✳✷✿ ❉❡❝♦♠♣♦s✐çã♦ ❞❛s ❢♦rç❛s ❛t✉❛♥❞♦ s♦❜r❡ ✉♠ ♣ê♥❞✉❧♦ ❞❡ ♠❛ss❛ m✱ ❝♦♠♣r✐♠❡♥t♦ l✱ s✉❥❡✐t♦ ❛ ❣r❛✈✐❞❛❞❡ g✳ ❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❡
❤tt♣✿✴✴❞♠♣❡❧✐✳♠❝♠❛st❡r✳❝❛✴▼❛t❧❛❜✴❈▲▲s♦❢t✇❛r❡✴P❡♥❞✉❧✉♠✴P❡♥❞✉❧✉♠✳❤t♠❧✱ ✷✶ ❞❡ ❞❡③❡♠❜r♦ ❞❡ ✷✵✶✶✳
❊st❛ é ✉♠❛ ❞❛s ❡q✉❛çõ❡s ❝❧áss✐❝❛s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ q✉❡ sã♦ ❡st✉❞❛❞❛s ♥✉♠ ♣r✐♠❡✐r♦ ❝✉rs♦ ❞❡ ❊❉❖✳ ▼✉✐t❛s ❞❛q✉❡❧❛s ❡q✉❛çõ❡s ♣♦❞❡♠ s❡r tr❛♥s❢♦r♠❛❞❛s ❡♠ s✐st❡♠❛s ❛✉tô♥♦♠♦s✱ ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣♦ ❛ q✉❡ ❢❛r❡♠♦s ❝♦♠ ❛ ❡q✉❛çã♦ ❞♦ ♣ê♥❞✉❧♦✳ ❊s❝r❡✈❡♠♦s
x=θ, y= ˙x.
❉❛í✱ ♣♦♥❞♦ g
l =k✱ ✭✶✳✻✮ é ❡q✉✐✈❛❧❡♥t❡ ❛♦ s✐st❡♠❛ ❛✉tô♥♦♠♦
˙ x=y
˙
y=−µy−ks❡♥x . ✭✶✳✼✮
◗✉❛♥❞♦ µ= 0✱ ♦ ♠♦✈✐♠❡♥t♦ ♥ã♦ t❡♠ ❛♠♦rt❡❝✐♠❡♥t♦✱ ♣♦rt❛♥t♦✱ ❞❡♣❡♥❞❡♥❞♦
❞❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥✐❝✐❛❧✱ ♦ ♣❡♥❞✉❧♦ ✐rá ♦s❝✐❧❛r ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ ❞❡ ❡q✉✐❧í❜r✐♦ ♦✉ ✐rá ❣✐r❛r ✐♥❞❡✜♥✐❞❛♠❡♥t❡ ♥♦ s❡♥t✐❞♦ ❞♦ ♠♦✈✐♠❡♥t♦ ✐♥✐❝✐❛❧✳ ❖ r❡tr❛t♦ ❞❡ ❢❛s❡ ❡stá ❡s❜♦ç❛❞♦ ♥❛ ❋✐❣✉r❛ ✶✳✸✳
❋✐❣✉r❛ ✶✳✸✿ ❘❡tr❛t♦ ❞❡ ❢❛s❡ ❞♦ s✐st❡♠❛ ✭✶✳✼✮ q✉❛♥❞♦ µ = 0✳ ❖ ❝✐❝❧♦s r❡♣r❡✲
s❡♥t❛♠ ❛ ♦s❝✐❧❛çã♦ ❞♦ ♣ê♥❞✉❧♦ ❡♠ t♦r♥♦ ❞❛ ♦r✐❣❡♠✳ ❆s ❞❡♠❛✐s tr❛❥❡tór✐❛s r❡♣r❡s❡♥t❛♠ ♦ ♣❡♥❞✉❧♦ ❣✐r❛♥❞♦ ❡♠ ✉♠ ú♥✐❝♦ s❡♥t✐❞♦✳ ❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❡ ❤tt♣✿✴✴❞♠♣❡❧✐✳♠❝♠❛st❡r✳❝❛✴▼❛t❧❛❜✴❈▲▲s♦❢t✇❛r❡✴P❡♥❞✉❧✉♠✴P❡♥❞✉❧✉♠✳❤t♠❧✱ ✷✶ ❞❡ ❞❡③❡♠❜r♦ ❞❡ ✷✵✶✶✳
◗✉❛♥❞♦ µ >0✱ ♦ ♠♦✈✐♠❡♥t♦ é ❛♠♦rt❡❝✐❞♦✳ ❖ ♣❡♥❞✉❧♦ ✐rá ♦s❝✐❧❛r ❡♠ t♦r♥♦ ❞♦
❡q✉✐❧í❜r✐♦ ❛té ❛❧❝❛♥ç❛r ♦ r❡♣♦✉s♦✱ ♦✉ ✐rá ❞❛r ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ✈♦❧t❛s ❡ ❞❡♣♦✐s ♦s❝✐❧❛rá ❛té ♦ r❡♣♦✉s♦✳ P♦❞❡♠♦s ♦❜s❡r✈❛r ✉♠❛ ♠✉❞❛♥ç❛ s✐❣♥✐✜❝❛t✐✈❛ ♥♦ r❡tr❛t♦ ❞❡ ❢❛s❡✱ ❡s❜♦ç❛❞♦ ♥❛ ❋✐❣✉r❛ ✶✳✹✳
❋✐❣✉r❛ ✶✳✹✿ ❘❡tr❛t♦ ❞❡ ❢❛s❡ ❞♦ s✐st❡♠❛ ✭✶✳✼✮ q✉❛♥❞♦ µ > 0✳ ❈❛❞❛ ❡s♣✐r❛❧
❡♠ t♦r♥♦ ❞♦ ♣♦♥t♦ (2kπ,0) r❡♣r❡s❡♥t❛ ♦ ♠♦✈✐♠❡♥t♦ ❞♦ ♣ê♥❞✉❧♦ ❞❛♥❞♦ k
✈♦❧t❛s ❡ ❞❡♣♦✐s ♦s❝✐❧❛♥❞♦ ❡♠ t♦r♥♦ ❞♦ ❡q✉✐❧í❜r✐♦ ❛té ❝❤❡❣❛r ❛♦ r❡♣♦✉s♦ ✭♦ q✉❡✱ ♠❛t❡♠❛t✐❝❛♠❡♥t❡✱ ❛❝♦♥t❡❝❡ ❞❡ ❢❛t♦ s♦♠❡♥t❡ q✉❛♥❞♦ t → ∞✳ ❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❡ ❤tt♣✿✴✴❞♠♣❡❧✐✳♠❝♠❛st❡r✳❝❛✴▼❛t❧❛❜✴❈▲▲s♦❢t✇❛r❡✴P❡♥❞✉❧✉♠✴P❡♥❞✉❧✉♠✳❤t♠❧✱ ✷✶ ❞❡ ❞❡③❡♠❜r♦ ❞❡ ✷✵✶✶✳
✶✳✸✳✷ ❈♦♠♣❡t✐çã♦ ❡♥tr❡ ❡s♣é❝✐❡s
❱❡r❡♠♦s ♥♦ ❈❛♣ít✉❧♦ ✸✱ q✉❡ ♦ s✐st❡♠❛
˙
N =N(1−N −aI) ˙
I =rI(1−I −bN) ✭✶✳✽✮
♦♥❞❡ a ❡ b sã♦ ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s✱ ❞❡s❝r❡✈❡ ❛ ✐♥t❡r❛çã♦ ❞❡ ❞✉❛s ❡s♣é❝✐❡s N ❡ I ✈✐✈❡♥❞♦ ♥✉♠ ♠❡s♠♦ ❛♠❜✐❡♥t❡ ❝♦♠ r❡❝✉rs♦s ❧✐♠✐t❛❞♦s ❡ ❝♦♠♣❡t✐♥❞♦ ❡♥tr❡ s✐✱
♣♦r ❡st❡s r❡❝✉rs♦s✳ ❖ r❡tr❛t♦ ❞❡ ❢❛s❡ ❞❡ ✭✶✳✽✮ ❡stá ❡s❜♦ç❛❞♦ ♥❛ ❋✐❣✉r❛ ✶✳✺ ♣❛r❛ ❞✐❢❡r❡♥t❡s ❝♦♥✜❣✉r❛çõ❡s ❞♦s ♣❛râ♠❡tr♦s✳
✶✳✹ ❖ ❋❧✉①♦ ❞❡ ✉♠❛ ❊❉❖
❱✐♠♦s ❛té ♦ ♠♦♠❡♥t♦ q✉❡ ❞❛❞♦s ✉♠ ❝❛♠♣♦ f ❞❡ ❝❧❛ss❡ C1 ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ U ❛❜⊂ Rn ❡ ✉♠ ♣♦♥t♦ x
0 ∈ U✱ ❡①✐st❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ ♠❛①✐♠❛❧ I(x0) ✭q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r Ix0✮ ❞❡♣❡♥❞❡♥t❡ ❞❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ x:Ix0 →Rn ❞❡x˙ =f(x) ❡♠ Ix0✳ ❚❛❧ s♦❧✉çã♦x(t)t❛♠❜é♠ ❞❡♣❡♥❞❡ ❞❛ ❝♦♥❞✐çã♦
✐♥✐❝✐❛❧✱ ♣♦✐s s❡ ♠✉❞❛r♠♦s ♦ ♣♦♥t♦ x0✱ ♦ ❚❡♦r❡♠❛ ✶✳✸ ♥♦s ❞❛rá ♦✉tr❛ ❛♣❧✐❝❛çã♦ y ❞❡✜♥✐❞❛ ❡♠ ♦✉tr♦ ✐♥t❡r✈❛❧♦✳ P♦r ❡st❡ ♠♦t✐✈♦✱ ♠✉❞❛r❡♠♦s ❛ ♥♦t❛çã♦ ♣❛r❛ ❛s
s♦❧✉çõ❡s✱ ❡s❝r❡✈❡♥❞♦ φ ❡♠ ✈❡③ ❞❡ x✱ ❡ ❡①♣❧✐❝✐t❛r❡♠♦s q✉❡ ❛ s♦❧✉çã♦ φ t❛♠❜é♠
❋✐❣✉r❛ ✶✳✺✿ ❊s♣❛ç♦ ❞❡ ❢❛s❡ ❞♦ s✐st❡♠❛ ✭✶✳✽✮ ♣❛r❛ ❞✐❢❡r❡♥t❡s ❝♦♥✜❣✉r❛çõ❡s ❞♦s ♣❛râ♠❡tr♦s✳ ◆❛ ❋✐❣✉r❛ ✭❛✮✱ t❡♠♦s ✉♠ ❝❡♥ár✐♦ ❞❡ ❝♦❡①✐stê♥❝✐❛ ❞❛s ❞✉❛s ❡s♣é❝✐❡s✳ ◆❛ ❋✐❣✉r❛ ✭❜✮✱ ✉♠ ❝❡♥ár✐♦ ❞❡ ❡①❝❧✉sã♦ ❝♦♠♣❡t✐t✐✈❛✿ ♦✉N s♦❜r❡✈✐✈❡ ❡I é ❡①t✐♥t❛✱
♦✉ ♦ ♦❝♦rr❡ ❝♦♥trár✐♦✱ ❛ ❞❡♣❡♥❞❡r ❞❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✳ ◆❛ ❋✐❣✉r❛ ✭❝✮✱ ❛♣❡♥❛s ❛ ❡s♣é❝✐❡ N s♦❜r❡✈✐✈❡ ❡ ♥❛ ❋✐❣✉r❛ ✭❞✮ ❛♣❡♥❛s ❛ ❡s♣é❝✐❡ I s♦❜r❡✈✐✈❡✳ ❋✐❣✉r❛
❛❞❛♣t❛❞❛ ❞❡ ❬✷✸❪✳
❞❡♣❡♥❞❡ ❞❡ x0✳ ❆ss✐♠ ❛ s♦❧✉çã♦ ❞❛❞❛ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✺ s❡rá ❞❡♥♦t❛❞❛ ♣♦r φ(·, x0) :Ix0 −→ Rn
t 7→ φ(t, x0). ✭✶✳✾✮
◆♦t❡ ❛❣♦r❛ q✉❡✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ s♦❧✉çã♦ ✭❉❡✜♥✐çã♦ ✶✳✶✮✱ t❡♠♦s q✉❡ ❛ ✐♠❛❣❡♠ ❞✐r❡t❛ φ(Ix0, x0) ❡stá ❝♦♥t✐❞❛ ❡♠ U✳ ❆ss✐♠✱ ❝♦❧♦❝❛♥❞♦ Ω ={(t, x)∈R×U :t ∈
Ix} ⊂Rn✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❛♣❧✐❝❛çã♦
φ : Ω −→ U
(t, x) 7→ φ(t, x). ✭✶✳✶✵✮
❆ ❡st❛ ❛♣❧✐❝❛çã♦ φ : Ω → U ❞❛♠♦s ♦ ♥♦♠❡ ❞❡ ✢✉①♦ ❞❛ ❊❉❖ ✭✶✳✶✮ ♦✉ ✢✉①♦
❣❡r❛❞♦ ♣❡❧♦ ❝❛♠♣♦ f✳ P♦r ✉♠❛ r❛③ã♦ ❞❡ ♥♦t❛çã♦✱ q✉❡ ✜❝❛rá ♠❛✐s ❝❧❛r❛ ❛❞✐❛♥t❡✱
♠✉✐t❛s ✈❡③❡s ❡s❝r❡✈❡r❡♠♦s φ(t, x) = φt(x)✱ ♣❛r❛ (t, x) ∈ Ω✳ ❆ s❡❣✉✐r✱ ✈❡❥❛♠♦s ♦
s✐❣♥✐✜❝❛❞♦ ❣❡♦♠étr✐❝♦ ❞♦ ✢✉①♦✳
✶✳✹✳✶ ❙✐❣♥✐✜❝❛❞♦ ❣❡♦♠étr✐❝♦ ❞♦ ✢✉①♦
❙❡ ✜①❛r♠♦s ✉♠ ♣♦♥t♦ x0 ❞♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ U ❡ ❝♦♥s✐❞❡r❛r♠♦s ❛ ❛♣❧✐❝❛çã♦
t7→φt(x0), t∈Ix0
♦❜t❡♠♦s ✉♠❛ tr❛❥❡tór✐❛ Γ ❡♠ U q✉❡ é ❛ ❝✉r✈❛ ✐♥t❡❣r❛❧ ❞❡ x˙ = f(x) ♣❛ss❛♥❞♦
♣♦r x0✳ ❚❛❧ tr❛❥❡tór✐❛ ❞❡s❝r❡✈❡ ♦ ♠♦✈✐♠❡♥t♦ ❞♦ ♣♦♥t♦ x0 s✉❥❡✐t♦ ❛♦ ❝❛♠♣♦ ❞❡ ✈❡❧♦❝✐❞❛❞❡s f✳ ◆❛ ❋✐❣✉r❛ ✶✳✻✱ à ❡sq✉❡r❞❛✱ ✈❡♠♦s ❡st❛ tr❛❥❡tór✐❛ ❞❡s❝r✐t❛ ♣♦r x0✳
❙❡ ✜①❛r♠♦s t∈R ❡ t♦♠❛r♠♦s ✉♠ ❝♦♥❥✉♥t♦ E ⊂U✱ ❛ ❛♣❧✐❝❛çã♦ x7→φt(x), x∈E
❞❡s❝r❡✈❡ ♦ ❡st❛❞♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ E ♥♦ t❡♠♣♦ t✳ ❆♦ ❢❛③❡r♠♦s ✉♠ ❡s❜♦ç♦
❞❡ E ❡ φt(E) ❡♠ U✱ ✈❡♠♦s ♦ ❡❢❡✐t♦ ❞♦ ✢✉①♦ φ s♦❜r❡ ❛ r❡❣✐ã♦ E ❛♣ós ✉♠ t❡♠♣♦ t✳ ❯♠❛ ✐❧✉str❛çã♦ ❞❡st❡ ❢❛t♦ ♣♦❞❡ s❡r ✈✐st❛ ♥❛ ❋✐❣✉r❛ ✶✳✻✱ ❛♦ ❝❡♥tr♦ ❡ à ❞✐r❡✐t❛✳
❋✐❣✉r❛ ✶✳✻✿ ◆❛ ✜❣✉r❛ ❞❛ ❡sq✉❡r❞❛✱ ✈❡♠♦s ❛ ❛♣❧✐❝❛çã♦ t 7→ φt(x0)✱ q✉❡ ❞❡s❝r❡✈❡ ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠ ú♥✐❝♦ ♣♦♥t♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✳ ❆♦ ♠❡✐♦✱ ✈❡♠♦s ♦ ❡❢❡✐t♦ ❞♦ ✢✉①♦ ❛♣ós ✉♠ t❡♠♣♦ts♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦E ❝♦♥t✐❞♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✳ ➚ ❞✐r❡✐t❛✱
♦✉tr♦ ❡①❡♠♣❧♦ ❞♦ ❡❢❡✐t♦ ❞♦ ✢✉①♦ s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ E✳ ❋✐❣✉r❛s r❡t✐r❛❞❛s ❞❡ ❬✷✻❪
❡ ❬✸❪✳
❆ss✐♠✱ s❡ ♣❡♥s❛r♠♦s q✉❡ ♦ ❝❛♠♣♦ f ❞❡s❝r❡✈❡ ❛s ✈❡❧♦❝✐❞❛❞❡s ❞❡ ✉♠ ✢✉✐❞♦✱
❝❛❞❛ tr❛❥❡tór✐❛ t 7→ φt(x0) ❞❡s❝r❡✈❡ ♦ ♠♦✈✐♠❡♥t♦ ✐♥❞✐✈✐❞✉❛❧ ❞❡ ✉♠ ♣♦♥t♦ x0 ❞♦ ✢✉✐❞♦✱ ❡♥q✉❛♥t♦ x7→φt(x) ❞❡s❝r❡✈❡ ♦ ❞❡s❧♦❝❛♠❡♥t♦ ❡ ❛ ❝♦♥tr❛çã♦✱ ♦✉ ❡①♣❛♥sã♦✱
❞❡ ✉♠❛ ❢❛t✐❛ ❞♦ ✢✉✐❞♦ ♦✉ ❞❡ t♦❞♦ ❡❧❡✳
P♦rt❛♥t♦✱ ♦ ❡st✉❞♦ ❞♦ ✢✉①♦ ❞❡ ✉♠❛ ❊❉❖ é ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛ ♣r❡✈✐sã♦ q✉❛❧✐t❛t✐✈❛ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s ♣♦♥t♦s ❡ tr❛❥❡tór✐❛s ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✳ ❱❡❥❛♠♦s ❛❣♦r❛ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ✢✉①♦✳
✶✳✹✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦ ❋❧✉①♦
❯♠❛ ✐♠♣♦rt❛♥t❡ ♣r♦♣r✐❡❞❛❞❡ é q✉❡ s❡ ♦ ❝❛♠♣♦ f ❢♦r ❞❡ ❝❧❛ss❡ C1 ❡♥tã♦ ♦ ✢✉①♦ ❞❡♣❡♥❞❡rá ❞❡ ❢♦r♠❛ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞♦ t❡♠♣♦✱ ❞♦s ♣❛râ♠❡tr♦s ❡ ❞❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✳ ❆s ❞❡♠♦♥str❛çõ❡s ❞♦s ❚❡♦r❡♠❛s ✶✳✻✱ ✶✳✼✱ ✶✳✽ ❡ ✶✳✾ q✉❡ s❡❣✉❡♠ ❛❜❛✐①♦ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✷✻❪✱ ♣á❣✐♥❛s ✾✺✲✾✾✳
❚❡♦r❡♠❛ ✶✳✻ ◆❛s ♥♦t❛çõ❡s ❛❝✐♠❛✱ s❡❥❛♠ f ∈C1(U)✱ Ω = {(t, x)∈R×U :t ∈
Ix} ❡ φ : Ω→U ♦ ✢✉①♦ ❞❡ f✳ ❊♥tã♦ Ω é ❛❜❡rt♦ ❡♠ Rn+1 ❡ φ é ❞❡ ❝❧❛ss❡ C1 ❡♠ Ω✳
❖✉tr❛ ♣r♦♣r✐❡❞❛❞❡ ✐♠♣♦rt❛♥t❡ ❞♦ ✢✉①♦ é q✉❡ ❡❧❡ t❡♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❣r✉♣♦ ✭s♦❜ ❛❧❣✉♠❛s ❤✐♣ót❡s❡s✮ ❡♠ r❡❧❛çã♦ ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ❛♣❧✐❝❛çõ❡s✳
❚❡♦r❡♠❛ ✶✳✼ ❙❡❥❛φ: Ω→U ♦ ✢✉①♦ ❞❡ ✉♠ ❝❛♠♣♦f ∈C1(U)❞❡✜♥✐❞♦ ♥♦ ❛❜❡rt♦
U✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ x∈U ❡ ♣❛r❛ t♦❞♦s t∈Ix ❡ s∈Iφt(x) t❡♠✲s❡ s+t∈Ix ❡ φs+t(x) =φs(φt(x)).
❚❡♦r❡♠❛ ✶✳✽ ❙❡❥❛φ: Ω→U ♦ ✢✉①♦ ❞❡ ✉♠ ❝❛♠♣♦f ∈C1(U)❞❡✜♥✐❞♦ ♥♦ ❛❜❡rt♦
U✳ ❉❛❞♦ (t, x0)∈Ω❡①✐st❡ ✉♠ ❛❜❡rt♦A⊂U ❝♦♥t❡♥❞♦x0 t❛❧ q✉❡ t×A⊂Ω✳ ❉❛í✱
B =φt(A) é ❛❜❡rt♦✱
φ−t(φt(x)) = x ♣❛r❛ t♦❞♦x∈A
❡
φt(φ−t(y)) =y ♣❛r❛ t♦❞♦y∈B.
❚❡♦r❡♠❛ ✶✳✾ ❙❡❥❛ φ : Ω → U ♦ ✢✉①♦ ❞❡ ✉♠ ❝❛♠♣♦ f ∈ C1(U) ❞❡✜♥✐❞♦ ♥♦ ❛❜❡rt♦ U✳ ❙❡ Ix =R ♣❛r❛ t♦❞♦ x∈U✱ ❡♥tã♦✱ ♣❛r❛ t♦❞♦ t∈R ❛ ❛♣❧✐❝❛çã♦
φt: U −→ U
x 7→ φt(x) ✭✶✳✶✶✮
é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❡♠ U ⊂Rn ❝♦♠ ✐♥✈❡rs❛ φ
−t :U →U ❡ ♠❛✐s✱ ❝♦♠♦ φs+t = φs◦φt✱ ♦ ❝♦♥❥✉♥t♦ {φt}t∈R é ✉♠ ❣r✉♣♦ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦s ❡♠ U ❡ ❛ ❛♣❧✐❝❛çã♦
R −→ {φt}t∈R
t 7→ φt
é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✳
✶✳✺ P♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦
❉❡✜♥✐çã♦ ✶✳✶✵ ❯♠ ♣♦♥t♦x0 ∈U é ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ♦✉ ♣♦♥t♦ s✐♥❣✉❧❛r ❞♦ ❝❛♠♣♦ f s❡ f(x0) = 0✳ ❙❡ f(x0)6= 0✱ x0 é ✉♠ ♣♦♥t♦ r❡❣✉❧❛r ❞❡ f✳
❙❡❥❛♠ x0 ∈ U ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ f ❡ φ ♦ ✢✉①♦ ❞❡ f✳ ❊♥tã♦ φ(., x0) é ❛ ❝✉r✈❛ ✐♥t❡❣r❛❧ ❞❡ f ♣❛ss❛♥❞♦ ♣♦r x0 ❡♠ t = 0✳ ▲♦❣♦✱ φ(0, x0) = x0✳ ❈♦♠♦
φt+s(x0) = φt(φs(x0))✱ ♣❡❧❛ ❘❡❣r❛ ❞❛ ❈❛❞❡✐❛✱ ♣❛r❛ t♦❞♦ t∈Ix0✱ t❡♠♦s q✉❡ d
dtφt(x0) = d
dsφt+s(x0)s= 0 = d
ds(φt(φs(x0))) s = 0 =
d(φt)x0 ·
d
dsφs(x0) s= 0
=d(φt)x0 ·(f(φ0(x0))) =d(φt)x0 ·0 = 0.
▲♦❣♦✱ ❡①✐st❡ c ∈ U t❛❧ q✉❡ φ(t, x0) = c ♣❛r❛ t♦❞♦ t ∈ Ix0✳ ❈♦♠♦ φ(0, x0) = x0✱ s❡❣✉❡ q✉❡
φ(t, x0) =x0 ♣❛r❛ t♦❞♦t∈Ix0.
❖✉ s❡❥❛✱ ♦ ✢✉①♦ ❡♠x0 é ❝♦♥st❛♥t❡ ❡ ✐❣✉❛❧ ❛x0✳ P♦rt❛♥t♦✱ t♦❞♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ f é ✉♠ ♣♦♥t♦ ✜①♦ ❞❡ φt✱ ♣❛r❛ t♦❞♦ t✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ x0 é ✉♠ ♣♦♥t♦ ✜①♦ ❞❡φt ❡♥tã♦φ(t, x0) =x0 ♣❛r❛ t♦❞♦t∈Ix0✳ ❉❛í✱
d
dtφ(t, x0) = 0♣❛r❛ t♦❞♦ t∈Ix0✳
▲♦❣♦✱
0 = dφ(t, x0)
dt t=0 =f(φ(t, x0))t=0 =f(x0)
♦✉ s❡❥❛✱ x0 é ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ f✳ P♦rt❛♥t♦✱ t♦❞♦ ♣♦♥t♦ ✜①♦ ❞♦ ✢✉①♦ é ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡f✳
❙❡♥❞♦ ❛ss✐♠✱ ✐r❡♠♦s ♥♦s r❡❢❡r✐r ❛ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❝♦♠♦ ♣♦♥t♦s ✜①♦s ❡ ✈✐❝❡✲✈❡rs❛✳ ❖❜s❡r✈❛♠♦s t❛♠❜é♠ q✉❡✱ s❡ ✉♠❛ s♦❧✉çã♦ ♣❛ss❛r ♦✉ ❝♦♠❡ç❛r ❡♠ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ x0 ❡❧❛ ❞❡✈❡rá s❡r ❝♦♥st❛♥t❡ ❡ ✐❣✉❛❧ ❛ x0✳
✶✳✻ Ór❜✐t❛s ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ❡ ❝♦♥❥✉♥t♦s ❧✐♠✐t❡
❉❡✜♥✐çã♦ ✶✳✶✶ ❉❛❞♦ ✉♠ ❝❛♠♣♦ f ∈C1(U) ❡ s❡✉ ✢✉①♦ φ : Ω→U✱ ❛ ór❜✐t❛ ❞❡
f ♣♦r ✉♠ ♣♦♥t♦ x∈U é ♦ ❝♦♥❥✉♥t♦
γx ={φ(t, x) :t∈Ix}✳
❆ s❡♠✐✲ór❜✐t❛ ♣♦s✐t✐✈❛ ❞❡ ❢ ♣♦r x é ♦ ❝♦♥❥✉♥t♦ γ+
x = {φ(t, x) : t ∈ Ix ∩R+} ❡ ❛
s❡♠✐✲ór❜✐t❛ ♥❡❣❛t✐✈❛ ❞❡ ❢ ♣♦r x é ❞❡✜♥✐❞❛ ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✳
Pr♦♣♦s✐çã♦ ✶✳✶✷ ❙❡❥❛♠ f ∈ C1(U)✱ x
0 ∈ U ❡ φ(. , x0) : Ix0 → U ❛ ❝✉r✈❛
✐♥t❡❣r❛❧ ❞❡ f ♣❛ss❛♥❞♦ ♣♦r x0✳ ❙❡ φ(. , x0) ♥ã♦ é ✐♥❥❡t♦r❛ ❡♥tã♦
• φ(. , x0) é ❝♦♥st❛♥t❡ ❡ γx0 ={x0} ♦✉
• φ(. , x0) é ♣❡r✐ó❞✐❝❛ ❡ γx0 é ✉♠❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ ✭♣❡r✐ó❞✐❝❛✮✳
❊st❛ ♣r♦♣♦s✐çã♦ ♥♦s ✐♥❢♦r♠❛ ❝♦♠♦ sã♦ ❛s tr❛❥❡tór✐❛s ❞❡ q✉❛❧q✉❡r s✐st❡♠❛ ♦♥❞❡ ♦ ❝❛♠♣♦ é ❞❡ ❝❧❛ss❡ C1✳ ❖✉ ❡❧❛s sã♦✱ ❝✉r✈❛s ♣❛r❛♠❡tr✐③❛❞❛s s✐♠♣❧❡s ✭♥♦ ❝❛s♦ ❡♠ q✉❡ φ(., x0) é ✐♥❥❡t♦r❛✮ ♦✉ ♣♦♥t♦s ✜①♦s ♦✉ ór❜✐t❛s ♣❡r✐ó❞✐❝❛s✳ ❱❡❥❛ s✉❛ ❞❡♠♦♥s✲ tr❛çã♦ ❡♠ ❬✸✸❪✱ ♣á❣✐♥❛s ✷✶✼✲✷✶✽✳
❖s ❝♦♥❥✉♥t♦s ω✲❧✐♠✐t❡ ❡ α✲❧✐♠✐t❡ t❡♠ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♥❛ t❡♦r✐❛ q✉❛✲
❧✐t❛t✐✈❛✳ ❆ ❣r♦ss♦ ♠♦❞♦✱ ❡❧❡s ❝❛r❛❝t❡r✐③❛♠ ♦✭s✮ ♣♦♥t♦✭s✮ ✜♥❛❧✭✐s✮ ❡ ✐♥✐❝✐❛❧✭✐s✮ ❞❡ ✉♠❛ tr❛❥❡tór✐❛ ❞♦ ❝❛♠♣♦✳
❉❡✜♥✐çã♦ ✶✳✶✸ ❙❡❥❛♠ f ∈ C1(U) ✉♠ ❝❛♠♣♦ ♥♦ ❛❜❡rt♦ U ⊂ Rn✱ x
0 ∈ U ❡
φ(t, x0)❛ s♦❧✉çã♦ ❞❡ x˙ =f(x)♣❛ss❛♥❞♦ ♣♦rx0✱ ❞❡✜♥✐❞❛ ♥♦ s❡✉ ✐♥t❡r✈❛❧♦ ♠❛①✐♠❛❧
I(x0) = (α, β)✳ ❙❡ β = +∞✱ ❞❡✜♥✐♠♦s ♦ ❝♦♥❥✉♥t♦ ω✲❧✐♠✐t❡ ❞❡ x0 ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♥❥✉♥t♦
ω(x0) = {y∈U :∃(tn), ❝♦♠ I(x0)∋tn → ∞ ❡ φtn(x0)→y, q✉❛♥❞♦ n→ ∞}.
❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ α =−∞✱ ❞❡✜♥✐♠♦s ♦ ❝♦♥❥✉♥t♦ α✲❧✐♠✐t❡ ❞❡ x0 ❝♦♠♦
α(x0) = {y∈U :∃(tn), ❝♦♠ I(x0)∋tn → −∞❡ φtn(x0)→y, q✉❛♥❞♦ n→ ∞}.
❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♥❥✉♥t♦ ✐♥✈❛r✐❛♥t❡✱ q✉❡ t❛♠❜é♠ é ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛ ♥♦ ❡st✉❞♦ q✉❛❧✐t❛t✐✈♦✳
❉❡✜♥✐çã♦ ✶✳✶✹ ❙❡❥❛ f ∈ C1(U) ✉♠ ❝❛♠♣♦ ♥♦ ❛❜❡rt♦ U ⊂ Rn✳ ❯♠ ❝♦♥❥✉♥t♦ S ⊂U é ❝❤❛♠❛❞♦ ✐♥✈❛r✐❛♥t❡ q✉❛♥❞♦ x∈S ✐♠♣❧✐❝❛ φ(t, x)∈S ♣❛r❛ t♦❞♦ t∈R✳
❖ ♣ró①✐♠♦ ❚❡♦r❡♠❛ ♥♦s ❞á ✐♥❢♦r♠❛çõ❡s t♦♣♦❧ó❣✐❝❛s ❛ r❡s♣❡✐t♦ ❞♦s ❝♦♥❥✉♥t♦s ❧✐♠✐t❡✳ ❙✉❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✸✸❪✱ ♣á❣✐♥❛s ✷✹✺✲✷✹✼✳
❚❡♦r❡♠❛ ✶✳✶✺ ❙❡❥❛♠f ✉♠ ❝❛♠♣♦ ❞❡ ❝❧❛ss❡C1 ♥♦ ❛❜❡rt♦U ⊂Rn ❡ γ+
p ✭r❡s♣❡❝✲
t✐✈❛♠❡♥t❡ γ−
p ✮ ❛ s❡♠✐✲ór❜✐t❛ ♣♦s✐t✐✈❛ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ ❛ s❡♠✐✲ór❜✐t❛ ♥❡❣❛t✐✈❛✮ ❞♦
❝❛♠♣♦f ♣❡❧♦ ♣♦♥t♦ p✳ ❙❡ γ+
p ✭r❡s♣❡❝t✐✈❛♠❡♥t❡γp−✮ ❡stá ❝♦♥t✐❞❛ ♥✉♠ s✉❜❝♦♥❥✉♥t♦
❝♦♠♣❛❝t♦ K ⊂U✱ ❡♥tã♦
✭✐✮ ω(p)6=∅ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ α(p)✮✳
✭✐✐✮ ω(p) é ❝♦♠♣❛❝t♦ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ α(p)✮✳
✭✐✐✐✮ ω(p) é ✐♥✈❛r✐❛♥t❡ ♣♦r φt ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ α(p)✮✳
✭✐✈✮ ω(p) é ❝♦♥❡①♦ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ α(p)✮✳
❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡st❡ ❚❡♦r❡♠❛✱ s❡U =Rn✱ ♦✉ ✉♠❛ s♦❧✉çã♦ ✏❡①♣❧♦❞❡✑✱ ✐st♦
é✱ |φ(t, x)| → ∞ q✉❛♥❞♦ t → ∞✱ ♦✉ ❡❧❛ s❡ ❛♣r♦①✐♠❛ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦✲✈❛③✐♦✱
❝♦♠♣❛❝t♦✱ ❝♦♥❡①♦ ❡ ✐♥✈❛r✐❛♥t❡✳
❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛♠♦s ♦ ❡♥✉♥❝✐❛❞♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ P♦✐♥❝❛ré✲❇❡♥❞✐①♦♥✱ ✉♠ ✐♠♣♦rt❛♥t❡ r❡s✉❧t❛❞♦ ♣❛r❛ ❝❛♠♣♦s ❡♠R2 q✉❡ ❝❛r❛❝t❡r✐③❛ ❝♦♠♣❧❡t❛♠❡♥t❡ ♦s ❝♦♥✲
❥✉♥t♦s ❧✐♠✐t❡ ❞❛s ❝✉r✈❛s ✐♥t❡❣r❛✐s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❡❧❡ ❛✜r♠❛ q✉❡ ♥ã♦ ♣♦❞❡ ❡①✐st✐r ✏❝❛♦s✑ ❡♠ ❞✐♠❡♥sã♦ ❞♦✐s ✭♥ã♦ ❞❛r❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ❝❛♦s ❛q✉✐✮✳ ❆ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ P♦✐♥❝❛ré✲❇❡♥❞✐①♦♥ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✸✸❪✱ ♣á❣✐♥❛s ✷✹✽✲✷✺✸✳ ❚❡♦r❡♠❛ ✶✳✶✻ ✭❚❡♦r❡♠❛ ❞❡ P♦✐♥❝❛ré✲❇❡♥❞✐①♦♥✮ ❙❡❥❛♠f ∈C1(U)✉♠ ❝❛♠♣♦ ♥♦ ❛❜❡rt♦ U ⊂ R2 ❡ φ(t, p) ❛ ❝✉r✈❛ ✐♥t❡❣r❛❧ ❞❡ f ♣❛ss❛♥❞♦ ♣♦r p✳ ❙✉♣♦♥❤❛ q✉❡ φ(t, p)❡stá ❞❡✜♥✐❞❛ ♣❛r❛ t♦❞♦ t≥0✱ q✉❡ γ+
p ❡stá ❝♦♥t✐❞❛ ❡♠ ✉♠ ❝♦♠♣❛❝t♦K ⊂U
❡ q✉❡ f ♣♦ss✉❛ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡s ❡♠ω(p)✳ ❊♥tã♦✱ ✉♠❛ ❞❛s s❡✲
❣✉✐♥t❡s ♣♦ss✐❜✐❧✐❞❛❞❡s ♦❝♦rr❡✿
✭✐✮ ❙❡ ω(p) ❝♦♥tê♠ s♦♠❡♥t❡ ♣♦♥t♦s r❡❣✉❧❛r❡s✱ ❡♥tã♦ ω(p) é ✉♠❛ ór❜✐t❛ ♣❡r✐ó✲
❞✐❝❛✳
✭✐✐✮ ❙❡ ω(p) ❝♦♥tê♠ ♣♦♥t♦s r❡❣✉❧❛r❡s ❡ s✐♥❣✉❧❛r❡s✱ ❡♥tã♦ ω(p) ❝♦♥s✐st❡ ❞❡ ✉♠
❝♦♥❥✉♥t♦ ❞❡ ór❜✐t❛s✱ ❝❛❞❛ ✉♠❛ ❞❛s q✉❛✐s t❡♥❞❡ ❛ ✉♠ ❞❡ss❡s ♣♦♥t♦s s✐♥❣✉❧❛r❡s q✉❛♥t♦ t → ±∞✳
✭✐✐✐✮ ❙❡ ω(p) ♥ã♦ ❝♦♥tê♠ ♣♦♥t♦s r❡❣✉❧❛r❡s✱ ❡♥tã♦ ω(p) é ✉♠ ♣♦♥t♦ s✐♥❣✉❧❛r✳
❆ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ P♦✐♥❝❛ré✲❇❡♥❞✐①♦♥ ❡ ♥♦s ❢♦r♥❡❝❡ ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ór❜✐t❛ ❢❡❝❤❛❞❛✳ P♦❞❡♠♦s ✉sá✲❧❛ ♣❛r❛ ♣r♦✈❛r ❛ ♥ã♦✲❡①✐stê♥❝✐❛ ❞❡ t❛✐s ór❜✐t❛s✳ ❙✉❛ ❞❡♠♦♥str❛çã♦ t❛♠❜é♠ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✸✸❪✱ ♥❛ ♣á❣✐♥❛ ✷✺✹✳
Pr♦♣♦s✐çã♦ ✶✳✶✼ ❙❡❥❛ f ✉♠ ❝❛♠♣♦ ❞❡ ❝❧❛ss❡ C1 ♥♦ ❛❜❡rt♦ U ⊂R2✳ ❙❡γ é ✉♠❛ ór❜✐t❛ ❢❡❝❤❛❞❛ ❞❡ f ❝✉❥♦ ✐♥t❡r✐♦r ❡st❡❥❛ ❝♦♥t✐❞♦ ❡♠ U✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ♣♦♥t♦
s✐♥❣✉❧❛r ❞❡ f ♥♦ ✐♥t❡r✐♦r ❞❡ γ✳
❆ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r ♥♦s ❢♦r♥❡❝❡ ✉♠❛ ♠❛♥❡✐r❛ ❞❡ ❞❡t❡r♠✐♥❛r ❝♦♥❥✉♥t♦s ❧✐♠✐t❡✳ ❯♠❛ ✐❞é✐❛ ❞❡ s✉❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✸✸❪✱ ♣á❣✐♥❛ ✷✽✵✳
Pr♦♣♦s✐çã♦ ✶✳✶✽ ❙❡❥❛ f ✉♠ ❝❛♠♣♦ ❞❡ ❝❧❛ss❡ C1 ♥♦ ❛❜❡rt♦ U ⊂ Rn✳ ❙❡ ❡①✐st❡
✉♠❛ ❢✉♥çã♦ V :U →R t❛❧ q✉❡
h∇V(x), f(x)i ≤0 ♣❛r❛ t♦❞♦x∈U
❡♥tã♦✱ ♣❛r❛ t♦❞♦ p ∈ U✱ ♦ ❝♦♥❥✉♥t♦ ω✲❧✐♠✐t❡ ❞❡ p ❡stá ❝♦♥t✐❞♦ ♥♦ ❝♦♥❥✉♥t♦ Σ =
{x∈U :h∇V(x), f(x)i= 0}✳
❆♣r❡s❡♥t❛♠♦s ❛ s❡❣✉✐r✱ ✉♠ r❡s✉❧t❛❞♦ q✉❡ s❡❣✉❡ ❝♦♠♦ ❝♦r♦❧ár✐♦ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✶✽ ❡ s❡rá ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ♥♦s ❈❛♣ít✉❧♦s ✸ ❡ ✹✳
