Estabilidade de pontos de equilíbrio e existência de soluções periódicas em alguns modelos bidimensionais

❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❈â♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦

❊st❛❜✐❧✐❞❛❞❡ ❞❡ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❡

❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s ❡♠ ❛❧❣✉♥s

♠♦❞❡❧♦s ❜✐❞✐♠❡♥s✐♦♥❛✐s

❙❛❧✈❛❞♦r ❚❛✈❛r❡s ❞❡ ❖❧✐✈❡✐r❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ✲ ▼❡str❛❞♦ Pr♦✜s✲ s✐♦♥❛❧✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡

❖r✐❡♥t❛❞♦r❛

Pr♦❢❛✳ ❉r❛✳ ❘❡♥❛t❛ ❩♦t✐♥ ●♦♠❡s ❞❡ ❖❧✐✈❡✐r❛

✺✶✼✳✸✽ ❖✹✽❡

❖❧✐✈❡✐r❛✱ ❙❛❧✈❛❞♦r ❚❛✈❛r❡s ❞❡

❊st❛❜✐❧✐❞❛❞❡ ❞❡ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❡ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❡✲ r✐ó❞✐❝❛s ❡♠ ❛❧❣✉♥s ♠♦❞❡❧♦s ❜✐❞✐♠❡♥s✐♦♥❛✐s✴ ❙❛❧✈❛❞♦r ❚❛✈❛r❡s ❞❡ ❖❧✐✈❡✐r❛✲ ❘✐♦ ❈❧❛r♦✿ ❬s✳♥✳❪✱ ✷✵✶✺✳

✼✼ ❢✳✿✜❣✳

❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛✱ ■♥st✐✲ t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✳

❖r✐❡♥t❛❞♦r❛✿ ❘❡♥❛t❛ ❩♦t✐♥ ●♦♠❡s ❞❡ ❖❧✐✈❡✐r❛

✶✳ ❙✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s✳ ✷✳ P♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✳ ✸✳ ❊st❛❜✐❧✐❞❛❞❡✳ ✹✳ ❋✉♥çã♦ ❞❡ ▲②❛♣✉♥♦✈✳ ✺✳ ❙♦❧✉çã♦ ♣❡r✐ó❞✐❝❛✳ ■✳ ❚ít✉❧♦

❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖

❙❛❧✈❛❞♦r ❚❛✈❛r❡s ❞❡ ❖❧✐✈❡✐r❛

❊st❛❜✐❧✐❞❛❞❡ ❞❡ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❡ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s

♣❡r✐ó❞✐❝❛s ❡♠ ❛❧❣✉♥s ♠♦❞❡❧♦s ❜✐❞✐♠❡♥s✐♦♥❛✐s

❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❯♥✐✈❡rs✐tár✐❛ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿

Pr♦❢❛✳ ❉r❛✳ ❘❡♥❛t❛ ❩♦t✐♥ ●♦♠❡s ❞❡ ❖❧✐✈❡✐r❛ ❖r✐❡♥t❛❞♦r❛

Pr♦❢✳ ❉r❛✳ ❙✉③✐♥❡✐ ❆♣ ❙✐q✉❡✐r❛ ▼❛r❝♦♥❛t♦ ■●❈❊✴❯◆❊❙P✴❘✐♦ ❈❧❛r♦ ✭❙P✮

Pr♦❢✳ ❉r✳ ❆❞✐❧s♦♥ ❏♦sé ❱✐❡✐r❛ ❇r❛♥❞ã♦

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s✴❆r❛r❛s ✭❙P✮

❆❣r❛❞❡❝✐♠❡♥t♦s

❙❡ ❆ é ♦ s✉❝❡ss♦✱ ❡♥tã♦ ❆ é ✐❣✉❛❧ ❛ ❳ ♠❛✐s ❨ ♠❛✐s ❩✳ ❖ tr❛❜❛❧❤♦ é ❳❀ ❨ é ♦ ❧❛③❡r❀ ❡ ❩ é ♠❛♥t❡r ❛ ❜♦❝❛ ❢❡❝❤❛❞❛✳

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ❝r✐tér✐♦s ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ♣❛r❛ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s ♥ã♦ ❧✐♥❡❛r❡s✱ ❡♠ ♣❛rt✐❝✉❧❛r ♦s ♠ét♦❞♦s ❞❡ ▲②❛♣✉♥♦✈ ✭❞✐r❡t♦ ❡ ✐♥❞✐r❡t♦✮✳ ❆♥❛❧✐s❛♠♦s t❛♠❜é♠ ❛❧❣✉♥s ❝r✐tér✐♦s q✉❡ ♥♦s ♣♦ss✐❜✐❧✐t❛♠✱ às ✈❡③❡s✱ ❞❡t❡r♠✐♥❛r ❛ ❡①✐stê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ s♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ✉♠❛ ✈❛r✐❛çã♦ ❞♦ ♠♦❞❡❧♦ ♣r❡s❛✲♣r❡❞❛❞♦r ❝❧áss✐❝♦ é ❛♥❛❧✐s❛❞❛ q✉❛♥t♦ à ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❡r✐ó❞✐❝❛✳

❆❜str❛❝t

❚❤✐s ✇♦r❦ ♣r❡s❡♥ts st❛❜✐❧✐t② ❝r✐t❡r✐❛ ❢♦r ❡q✉✐❧✐❜r✐✉♠ ♣♦✐♥ts ♦❢ ♥♦♥❧✐♥❡❛r s②st❡♠s ♦❢ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ✐♥ ♣❛rt✐❝✉❧❛r t❤❡ ▲②❛♣✉♥♦✈ ♠❡t❤♦❞s ✭❞✐r❡❝t ❛♥❞ ✐♥❞✐r❡❝t✮✳ ❲❡ ❛❧s♦ ❧♦♦❦ ❛t s♦♠❡ ❝r✐t❡r✐❛ t❤❛t ❡♥❛❜❧❡ ✉s s♦♠❡t✐♠❡s ❞❡t❡r♠✐♥❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♣❡r✐♦❞✐❝ s♦❧✉t✐♦♥s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❛ ✈❛r✐❛t✐♦♥ ♦❢ t❤❡ ❝❧❛ss✐❝ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ✐s ❛♥❛❧②③❡❞ ❢♦r t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♣❡r✐♦❞✐❝ s♦❧✉t✐♦♥✳

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✸✳✶ ❈❛♠♣♦ ❞❡ ❞✐r❡çõ❡s ♣❛r❛ ♦ s✐st❡♠❛ ✭✸✳✷✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✷ ❖s q✉❛tr♦ ❝❛s♦s ♣❛r❛ ❞✉❛s ❡s♣é❝✐❡s ❡♠ ❝♦♠♣❡t✐çã♦ ✭✸✳✶✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✸ P♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❡ ❝❛♠♣♦ ❞❡ ❞✐r❡çõ❡s ♣❛r❛ ♦ s✐st❡♠❛ ✭✸✳✶✹✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✹ ❘❡tr❛t♦ ❞❡ ❢❛s❡ ♣❛r❛ ♦ s✐st❡♠❛ ✭✸✳✶✹✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✺ ❱❛r✐❛çõ❡s ♥❛s ♣♦♣✉❧❛çõ❡s ❞❡ ♣r❡s❛s ❡ ❞❡ ♣r❡❞❛❞♦r❡s ❡♠ r❡❧❛çã♦ ❛♦ t❡♠♣♦

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶✼

✷ ❙✐st❡♠❛s q✉❛s❡ ❧✐♥❡❛r❡s ✶✾

✷✳✶ ❆❧❣✉♠❛s ❞❡✜♥✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷ ❖ s❡❣✉♥❞♦ ♠ét♦❞♦ ❞❡ ▲②❛♣✉♥♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✸ ▼♦❞❡❧♦s P♦♣✉❧❛❝✐♦♥❛✐s ✸✺

✸✳✶ ❊s♣é❝✐❡s ❡♠ ❝♦♠♣❡t✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷ ▼♦❞❡❧♦ ♣r❡s❛✲♣r❡❞❛❞♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹ ❆❧❣✉♥s ❝r✐tér✐♦s ♣❛r❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s ✹✼

✺ ❱❛r✐❛çõ❡s ❞♦ ♠♦❞❡❧♦ ❝❧áss✐❝♦ ♣r❡s❛✲♣r❡❞❛❞♦r ✺✼

✻ ❈♦♥❝❧✉sã♦ ✻✺

❘❡❢❡rê♥❝✐❛s ✻✼

❆ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❍♦♠♦❣ê♥❡♦s ❝♦♠ ❈♦❡✜❝✐❡♥t❡s ❈♦♥st❛♥t❡s ✻✾ ❆✳✶ ❆✉t♦✈❛❧♦r❡s r❡❛✐s ❡ ❞✐st✐♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ❆✳✷ ❆✉t♦✈❛❧♦r❡s ❝♦♠♣❧❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ❆✳✸ ❆✉t♦✈❛❧♦r❡s ❘❡♣❡t✐❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸

❇ ❋ór♠✉❧❛ ❞❛ ❱❛r✐❛çã♦ ❞❛s ❈♦♥st❛♥t❡s ✼✺

✶ ■♥tr♦❞✉çã♦

P❡r✐♦❞✐❝✐❞❛❞❡ é ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ✐♠♣♦rt❛♥t❡ q✉❡ ❛♣❛r❡❝❡ ❡♠ ♠✉✐t♦s ❢❡♥ô♠❡♥♦s ❢í✲ s✐❝♦s ❡ ❜✐♦❧ó❣✐❝♦s✳ ❉❡♣♦✐s ❞❡ s♦❧✉çõ❡s ❝♦♥st❛♥t❡s ✭♦r✐✉♥❞❛s ❞❡ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦s✮✱ ❛s s♦❧✉çõ❡s ♠❛✐s ✐♠♣♦rt❛♥t❡s sã♦ ❛s ór❜✐t❛s ♣❡r✐ó❞✐❝❛s✱ ❝✉❥❛s tr❛❥❡tór✐❛s sã♦ ❝✉r✈❛s ❢❡❝❤❛❞❛s ♥♦ ♣❧❛♥♦ ❞❡ ❢❛s❡✳ ❆ ♣r❡s❡♥ç❛ ❞❡ ✉♠ ú♥✐❝♦ ♠♦✈✐♠❡♥t♦ ♣❡r✐ó❞✐❝♦ q✉❡ ❛tr❛✐ t♦❞❛s ❛s s♦✲ ❧✉çõ❡s ✭♣ró①✐♠❛s✮✱ ✐st♦ é✱ ❞❡ ✉♠ ❝✐❝❧♦ ❧✐♠✐t❡ ❡stá✈❡❧✱ é ✉♠ ❞♦s ❢❡♥ô♠❡♥♦s ❝❛r❛❝t❡ríst✐❝♦s ❛ss♦❝✐❛❞♦s à ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♥ã♦✲❧✐♥❡❛r❡s ❬✶❪✳

P❡r✐♦❞✐❝✐❞❛❞❡ ❡ ❝♦♠♣♦rt❛♠❡♥t♦s ♦s❝✐❧❛tór✐♦s ❡stã♦ ♣r❡s❡♥t❡s ❡♠ ♠♦❞❡❧♦s ♣♦♣✉❧❛❝✐♦♥❛✐s ♦♥❞❡ ❛❧❣✉♠ ♣❛râ♠❡tr♦ ❡♥✈♦❧✈✐❞♦ ✈❛r✐❛ ♣❡r✐♦❞✐❝❛♠❡♥t❡✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ s✉♣♦rt❡ ❞♦ ♠❡✐♦✳ ❊♠ ♠♦❞❡❧♦s ❡♣✐❞❡♠✐♦❧ó❣✐❝♦s ❛ ♣❡r✐♦❞✐❝✐❞❛❞❡ ♣♦❞❡ ❡st❛r r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ❛ s❛③♦♥❛❧✐❞❛❞❡ ❞❡ ❛❧❣✉♥s ❢❛t♦r❡s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❛ t❛①❛ ❞❡ ❝♦♥t❛t♦ ❡♥tr❡ ✐♥❞✐✈í❞✉♦s s✉s❝❡tí✈❡✐s ❡ ✐♥❢❡❝t❛❞♦s ♦✉ ❛té ♠❡s♠♦✱ ❞❡✈✐❞♦ à ♣ró♣r✐❛ ❡str✉t✉r❛ ❞♦ ♠♦❞❡❧♦✳ ❬✷❪

◆❡ss❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛♠♦s ❝r✐tér✐♦s ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ♣❛r❛ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ s✐st❡♠❛s ♥ã♦✲❧✐♥❡❛r❡s✱ ❡♠ ♣❛rt✐❝✉❧❛r ♦s ♠ét♦❞♦s ❞❡ ▲②❛♣✉♥♦✈ ✭❞✐r❡t♦ ❡ ✐♥❞✐r❡t♦✮✳ ▼♦t✐✲ ✈❛❞♦s ♣❡❧❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❡r✐ó❞✐❝❛ ♥♦ ♠♦❞❡❧♦ ❝❧áss✐❝♦ ♣r❡s❛✲♣r❡❞❛❞♦r✱ ❛♥❛❧✐s❛♠♦s ❛❧❣✉♥s ❝r✐tér✐♦s q✉❡ ♥♦s ♣♦ss✐❜✐❧✐t❛♠✱ às ✈❡③❡s✱ ❞❡t❡r♠✐♥❛r ❛ ❡①✐stê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ s♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛♥❛❧✐s❛♠♦s ✉♠❛ ✈❛r✐❛çã♦ ❞♦ ♠♦❞❡❧♦ ♣r❡s❛✲♣r❡❞❛❞♦r ♣r♦♣♦st♦ ❡♠ ❬✸❪ q✉❡✱ ❞✐❢❡r❡♥t❡♠❡♥t❡ ❞♦ ♠♦❞❡❧♦ ❝❧áss✐❝♦✱ ♥ã♦ ♣♦ss✉✐ s♦❧✉çã♦ ♣❡r✐ó❞✐❝❛✳

◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛s ♣r✐♥❝✐♣❛✐s ❞❡✜♥✐çõ❡s ✉t✐❧✐③❛❞❛s ♥❡st❡ tr❛❜❛❧❤♦ ❜❡♠ ❝♦♠♦ ✉♠ t❡♦r❡♠❛ q✉❡ ♥♦s ❢♦r♥❡❝❡ ✉♠❛ ❝♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ✉♠ s✐st❡♠❛ s❡r q✉❛s❡ ❧✐♥❡❛r✳ ❆♣r❡s❡♥t❛♠♦s ✉♠ t❡♦r❡♠❛ q✉❡ ❝❛r❛❝t❡r✐③❛✱ ♣♦r ♠❡✐♦ ❞❡ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ❝♦rr❡s♣♦♥❞❡♥t❡✱ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❛ss✐♥tót✐❝❛ ♦✉ ❛ ✐♥st❛❜✐❧✐❞❛❞❡ ♣❛r❛ s✐st❡♠❛s q✉❛s❡ ❧✐♥❡❛r❡s✳ ❆♣r❡s❡♥t❛♠♦s t❛♠❜é♠ ✉♠ ❡st✉❞♦ q✉❛❧✐t❛t✐✈♦ ❞❡ ✉♠ s✐st❡♠❛ ✈✐❛ ✉♠❛ ❢✉♥çã♦ ❞❡ ▲②❛♣✉♥♦✈✱ q✉❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❙❡❣✉♥❞♦ ▼ét♦❞♦ ❞❡ ▲②❛♣✉♥♦✈✳

◆♦ ❝❛♣ít✉❧♦ ✸ ❛♥❛❧✐s❛♠♦s ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦s ♠♦❞❡❧♦s ❝❧áss✐❝♦s ❞❡ ❝♦♠♣❡t✐çã♦ ❡♥tr❡ ❞✉❛s ❡s♣é❝✐❡s ❡ ♦ ♠♦❞❡❧♦ ♣r❡s❛✲♣r❡❞❛❞♦r✳ ◆♦ ❝❛♣ít✉❧♦ ✹ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s ❝r✐tér✐♦s ♣❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ❛ ❡①✐stê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ tr❛❥❡tór✐❛s ❢❡❝❤❛❞❛s ✭s♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s✮✳

◆♦ ❝❛♣ít✉❧♦ ✺ ❛♣r❡s❡♥t❛♠♦s ♦ ♠♦❞❡❧♦ ♣r♦♣♦st♦ ❡♠ ❬✸❪ q✉❡ ❛♣r❡s❡♥t❛ ✉♠❛ ✈❛r✐❛çã♦ ❞♦ ♠♦❞❡❧♦ ♣r❡s❛✲♣r❡❞❛❞♦r ❞❡ ▲♦t❦❛✲❱♦❧t❡rr❛ ❡ ✉t✐❧✐③❛ ♦ ❝r✐tér✐♦ ❞❡ ❉✉❧❛❝ ♣❛r❛ ♠♦str❛r ❛ ♥ã♦ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s✳

✶✵

P♦r ✜♠ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❝♦♥❝❧✉sã♦ ❡ ✉♠ ❛♣ê♥❞✐❝❡ ❝♦♠ ❛❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ ❢♦r❛♠ ✉t✐❧✐③❛❞♦s ♥♦ t❡①t♦✳

P❛r❛ ❛ ❝♦♥❢❡❝çã♦ ❞❛s ✜❣✉r❛s ♣r❡s❡♥t❡s ♥❡st❡ tr❛❜❛❧❤♦ ✉t✐❧✐③❛♠♦s ♦s s♦❢t✇❛r❡s ❲✐♥♣❧♦t

✷ ❙✐st❡♠❛s q✉❛s❡ ❧✐♥❡❛r❡s

❆♣r❡s❡♥t❛♠♦s ♥❡st❡ ❝❛♣ít✉❧♦ ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s s♦❜r❡ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ♣❛r❛ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s q✉❛s❡ ❧✐♥❡❛r❡s✱ q✉❡ sã♦ ♦s ♠❛✐s ❝♦♠✉♥s ❡♠ t❡r♠♦s ❞❡ ❛♣❧✐❝❛çõ❡s às ♠❛✐s ❞✐✈❡rs❛s ár❡❛s✳ ❆s ❞❡✜♥✐çõ❡s ❛♣r❡s❡♥t❛❞❛s ♥❡st❡ ❝❛♣ít✉❧♦ sã♦ ❜❛s❡❛❞❛s ❡♠ ❬✶❪ ❡ ❬✹❪✳

✷✳✶ ❆❧❣✉♠❛s ❞❡✜♥✐çõ❡s

❙❡❥❛ t ✉♠ ❡s❝❛❧❛r r❡❛❧✱ D ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❞❡ Rn+1 ❝♦♠ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ _D

❞❡s❝r✐t♦ ♣♦r (t, x) ❡ f : D _→ Rn ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✱ ❝♦♠ _x′

= dx/dt✳ ❯♠❛ ❡q✉❛çã♦

❞✐❢❡r❡♥❝✐❛❧ é ✉♠❛ r❡❧❛çã♦ ❞❛ ❢♦r♠❛

x′

(t) =f(t, x(t)) ♦✉✱ s✐♠♣❧❡s♠❡♥t❡ x′

=f(t, x). ✭✷✳✶✮ ❉❡✜♥✐çã♦ ✷✳✶✳ ❉✐③❡♠♦s q✉❡ x é ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✷✳✶✮ ♥♦ ✐♥t❡r✈❛❧♦ I _⊂ R s❡ _x é ✉♠❛

❢✉♥çã♦ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡✜♥✐❞❛ ❡♠ I✱ (t, x(t)) _∈D✱ t _∈I ❡ x s❛t✐s❢❛③ ✭✷✳✶✮ ❡♠ I✳ ◆♦s r❡❢❡r✐♠♦s ❛ f ❝♦♠♦ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ❞❡ D✳

❉❡✜♥✐çã♦ ✷✳✷✳ ❙✉♣♦♥❤❛ (t0, x0) ∈ D ❞❛❞♦✳ ❯♠ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ ♣❛r❛ ❛ ❡q✉❛✲ çã♦ ✭✷✳✶✮ ❝♦♥s✐st❡ ❡♠ ❡♥❝♦♥tr❛r ✉♠ ✐♥t❡r✈❛❧♦ I ❝♦♥t❡♥❞♦ t0 ❡ ✉♠❛ s♦❧✉çã♦ x ❞❡ ✭✷✳✶✮ s❛t✐s❢❛③❡♥❞♦ x(t0) =x0✳ ❊s❝r❡✈❡♠♦s ❡st❡ ♣r♦❜❧❡♠❛ s✐♠❜♦❧✐❝❛♠❡♥t❡ ❝♦♠♦

x′

=f(t, x), x(t0) =x0, t∈I. ✭✷✳✷✮ ❙❡ ❡①✐st❡ ✉♠ ✐♥t❡r✈❛❧♦ I ❝♦♥t❡♥❞♦ t0 ❡ ✉♠❛ ❢✉♥çã♦ x(t) s❛t✐s❢❛③❡♥❞♦ ✭✷✳✷✮✱ ❞✐③❡♠♦s q✉❡x é ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✷✳✶✮ ♣❛ss❛♥❞♦ ♣♦r (t0, x0)✳

❙❡f(t, x)é ❝♦♥tí♥✉❛ ❡♠ ✉♠ ❞♦♠í♥✐♦ D✱ ❡♥tã♦ ♦ t❡♦r❡♠❛ ❞❡ ❡①✐stê♥❝✐❛ ♣❛r❛ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ✐♠♣❧✐❝❛ ♥❛ ❡①✐stê♥❝✐❛✱ ❞❡ ♥♦ ♠í♥✐♠♦✱ ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✷✳✶✮ ♣❛ss❛♥❞♦ ♣♦r ✉♠ ❞❛❞♦ ♣♦♥t♦(t0, x0)❡♠ D✳ P❛r❛ q✉❛❧q✉❡r (t0, x0)∈D✱ s❡❥❛ (a(t0, x0), b(t0, x0))♦ ✐♥t❡r✈❛❧♦ ♠á①✐♠♦ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ x(t, t0, x0) ❡ s❡❥❛E ⊂Rn+2 ❞❡✜♥✐❞♦ ❝♦♠♦

E =_{(t, t0, x0) :a(t0, x0)< t < b(t0, x0),(t0, x0)∈D}.

❆❧❣✉♠❛s ❞❡✜♥✐çõ❡s ✶✷

❉❡✜♥✐çã♦ ✷✳✸✳ ❯♠❛ tr❛❥❡tór✐❛ ❛tr❛✈és ❞❡(t0, x0) é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❡♠Rn+1 ❞❛❞♦ ♣♦r (t, x(t, t0, x0)) ♣❛r❛ t ✈❛r✐❛♥❞♦ ❡♥tr❡ t♦❞♦s ♦s ✈❛❧♦r❡s ♣♦ssí✈❡✐s ♥♦s q✉❛✐s (t, t0, x0) ♣❡rt❡♥❝❡ ❛ E✳ ❖ ❝♦♥❥✉♥t♦ E é ❝❤❛♠❛❞♦ ❞❡ ❞♦♠í♥✐♦ ❞❡ ❞❡✜♥✐çã♦ ❞❡ x(t, t0, x0)✳

❉❡✜♥✐çã♦ ✷✳✹✳ Ór❜✐t❛ ❞❡ ✉♠❛ tr❛❥❡tór✐❛ é ❛ ♣r♦❥❡çã♦ ❞❡ ✉♠❛ tr❛❥❡tór✐❛ ❡♠Rn✱ ♦ ❡s♣❛ç♦

❞❛s ✈❛r✐á✈❡✐s ❞❡♣❡♥❞❡♥t❡s ❡♠ ✭✷✳✶✮✳ ❖ ❡s♣❛ç♦ ❞❛s ✈❛r✐á✈❡✐s ❞❡♣❡♥❞❡♥t❡s é ♥♦r♠❛❧♠❡♥t❡ ❝❤❛♠❛❞♦ ❞❡ ❡s♣❛ç♦ ❞❡ ❡st❛❞♦ ♦✉ ❡s♣❛ç♦ ❞❡ ❢❛s❡✳ ❆s ❝♦♦❞❡r♥❛❞❛s ❞❡ ❢❛s❡ ♣❛r❛ ✉♠❛ ❡q✉❛çã♦ ❡s❝❛❧❛r ❞❡ ♦r❞❡♠n ❡♠ x é ♦ ✈❡t♦r (x1, x2, . . . , xn)✳

❉❡✜♥✐çã♦ ✷✳✺✳ ❯♠ s✐st❡♠❛ ❞❡n ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ é ❝❤❛♠❛❞♦ ❞❡ ❛✉tô♥♦♠♦ q✉❛♥❞♦ ❛s ❢✉♥çõ❡s fi✱ i = 1, . . . , n ♥ã♦ ❞❡♣❡♥❞❡♠ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞❛ ✈❛r✐á✈❡❧

✐♥❞❡♣❡♥❞❡♥t❡ t✱ ♠❛s ❛♣❡♥❛s ❞❛s ✈❛r✐á✈❡✐s x1, . . . , xn✱ ✐st♦ é✱



    

    

x′

1 =f1(x1, . . . , xn)

x′

2 =f2(x1, . . . , xn)

✳✳✳ x′

n =fn(x1, . . . , xn)

✭✷✳✸✮

◆❡♠ s❡♠♣r❡ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r ✉♠❛ s♦❧✉çã♦ ❡①♣❧í❝✐t❛ ♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧✳ ❉❡ss❛ ❢♦r♠❛✱ ♣r♦❝✉r❛✲s❡ ❞❡t❡r♠✐♥❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ s✐st❡♠❛✱ ❛♥❛❧✐s❛♥❞♦ s❡ ❛s s♦❧✉çõ❡s s❡ ❛♣r♦①✐♠❛♠ ♦✉ ♥ã♦ ❞❡ s♦❧✉çõ❡s ❝♦♥st❛♥t❡s✳

❉❡✜♥✐çã♦ ✷✳✻✳ ❖s ♣♦♥t♦s ♦♥❞❡f(x) = 0✱ f = (f1, f2, . . . , fn)s❡ ❡①✐st✐r❡♠✱ sã♦ ❝❤❛♠❛❞♦s

❞❡ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ✭♦✉ ♣♦♥t♦s ❝rít✐❝♦s✮ ❞♦ s✐st❡♠❛ ❛✉tô♥♦♠♦ ✭✷✳✸✮✳

❖❜s❡r✈❡ q✉❡ ♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❞ã♦ ♦r✐❣❡♠ às s♦❧✉çõ❡s ❝♦♥st❛♥t❡s ❞❡ ✉♠ s✐st❡♠❛ ❞♦ t✐♣♦ ✭✷✳✸✮✳

❉❡✜♥✐çã♦ ✷✳✼✳ ❯♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦x ❞♦ s✐st❡♠❛ ✭✷✳✸✮ é ❞✐t♦ ❡stá✈❡❧ s❡✱ ❞❛❞♦ q✉❛❧q✉❡r ε >0✱ ❡①✐st❡ δ >0 t❛❧ q✉❡ t♦❞❛ s♦❧✉çã♦ x=ϕ(t) ❞♦ s✐st❡♠❛✱ q✉❡ s❛t✐s❢❛③✱ ❡♠ t= 0✱

||ϕ(0)₋x_||< δ, ✭✷✳✹✮ ❡①✐st❡ ♣❛r❛ t♦❞♦ t ♣♦s✐t✐✈♦ ❡ s❛t✐s❢❛③

||ϕ(t)₋x_||< ε ✭✷✳✺✮ ♣❛r❛ t♦❞♦t _≥0✳ ❆q✉✐ _{|| ||} ❞❡♥♦t❛ ❛ ♥♦r♠❛ ❡✉❝❧✐❞✐❛♥❛ ❡♠ Rn_✳

❱❛♠♦s ❝♦♥s✐❞❡r❛r ♦ t❡♠♣♦ ✐♥✐❝✐❛❧ ❝♦♠♦ t0 = 0 ♣♦✐s ♦ s✐st❡♠❛ é ❛✉tô♥♦♠♦✳ ❬✶❪

❉❡✜♥✐çã♦ ✷✳✽✳ ❯♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦x é ❞✐t♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧ s❡ é ❡stá✈❡❧ ❡ s❡ ❡①✐st❡δ0(δ0 >0)t❛❧ q✉❡✱ s❡ ✉♠❛ s♦❧✉çã♦ x=ϕ(t) s❛t✐s❢❛③❡♥❞♦

❆❧❣✉♠❛s ❞❡✜♥✐çõ❡s ✶✸

❡♥tã♦

lim

t→∞ϕ(t) =x. ✭✷✳✼✮

❙❡❥❛ ♦ s✐st❡♠❛ ❞❡ n ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❧✐♥❡❛r❡s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ x′

1 =p11(t)x1+. . .+p1n(t)xn+g1(t), ✳✳✳

x′

n =pn1(t)x1+. . .+pnn(t)xn+gn(t)

✭✷✳✽✮

❱❛♠♦s ❝♦♥s✐❞❡r❛rx1(t), . . . , xn(t) ❝♦♠♦ ❝♦♠♣♦♥❡♥t❡s ❞❡ ✉♠ ✈❡t♦r x=x(t)✳ ❆♥❛❧♦❣❛✲

♠❡♥t❡✱g1(t), . . . , gn(t)sã♦ ❝♦♠♣♦♥❡♥t❡s ❞❡ ✉♠ ✈❡t♦rg(t)❡p11(t), . . . , pnn(t)sã♦ ❡❧❡♠❡♥t♦s

❞❡ ✉♠❛ ♠❛tr✐③ n_×n, P(t)✳ ❆ss✐♠✱ ♦ s✐st❡♠❛ ✭✷✳✽✮ é ❡s❝r✐t♦ ❡♠ ♥♦t❛çã♦ ♠❛tr✐❝✐❛❧ ❝♦♠♦

x′

=P(t)x+g(t). ✭✷✳✾✮

❉❡✜♥✐çã♦ ✷✳✾✳ ❙❡ t♦❞❛s ❛s ❢✉♥çõ❡sg1(t), . . . , gn(t)❢♦r❡♠ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛s ♥♦ ✐♥t❡r✈❛❧♦

I = _{t _∈R_{, α < t < β}}✱ ❞✐③❡♠♦s q✉❡ ♦ s✐st❡♠❛ ✭✷✳✽✮ é ❤♦♠♦❣ê♥❡♦❀ ❝❛s♦ ❝♦♥trár✐♦✱ ❡❧❡ é

♥ã♦✲❤♦♠♦❣ê♥❡♦✳

❉❡✜♥✐çã♦ ✷✳✶✵✳ ❉✐③❡♠♦s q✉❡ ❛s ❢✉♥çõ❡s

x1 =ϕ1(t), . . . , xn=ϕn(t) ✭✷✳✶✵✮

❢♦r♠❛♠ ✉♠❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ✭✷✳✽✮ ♥♦ ✐♥t❡r✈❛❧♦ I s❡ ❡❧❛s✿ ✭✐✮ sã♦ ❞✐❢❡r❡♥❝✐á✈❡✐s ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❞♦ ✐♥t❡r✈❛❧♦ I ❡ ✭✐✐✮ s❛t✐s❢❛③❡♠ ♦ s✐st❡♠❛ ✭✷✳✽✮ ❡♠ t♦❞♦ t _∈I✳

❯♠ ❞♦s s✐st❡♠❛s ♠❛✐s s✐♠♣❧❡s✱ ❛ s❛❜❡r✱ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ❤♦♠♦❣ê♥❡♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♥st❛♥t❡s✱ ❞❡ ❞✐♠❡♥sã♦ ❞♦✐s✱ t❡♠ ❛ ❢♦r♠❛

x′

=Ax, ✭✷✳✶✶✮

♦♥❞❡A é ✉♠❛ ♠❛tr✐③ ❝♦♥st❛♥t❡2_×2 ❡x é ✉♠ ✈❡t♦r 2_×1✳

◆♦ ❆♣ê♥❞✐❝❡ ❆ ♠♦str❛♠♦s q✉❡ ❛s s♦❧✉çõ❡s ♣❛r❛ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❧✐♥❡❛r❡s ❤♦♠♦❣ê♥❡♦s sã♦ ❞❛ ❢♦r♠❛x =ξert_{✱ ♦♥❞❡ r é ✉♠ ❛✉t♦✈❛❧♦r ❞❡ A ❡ ξ ♦ ❛✉t♦✈❡t♦r}

❛ss♦❝✐❛❞♦✳ ❊♥tã♦✱ s✉❜st✐t✉✐♥❞♦x=ξert _{♥❛ ❡q✉❛çã♦ ✭✷✳✶✶✮✱ ♦❜t❡♠♦s}

(A₋rI)ξ= 0. ✭✷✳✶✷✮

❖s ❛✉t♦✈❛❧♦r❡s sã♦ ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧

❆❧❣✉♠❛s ❞❡✜♥✐çõ❡s ✶✹

❡ ♦s ❛✉t♦✈❡t♦r❡s sã♦ ❞❡t❡r♠✐♥❛❞♦s ♣❡❧❛ ❡q✉❛çã♦ ✭✷✳✶✷✮✱ ❛ ♠❡♥♦s ❞❡ ✉♠❛ ❝♦♥st❛♥t❡ ♠✉❧t✐✲ ♣❧✐❝❛t✐✈❛✳

❈♦♠♦ ❥á ❞❡✜♥✐❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ♦s ♣♦♥t♦s t❛✐s q✉❡ Ax = 0 sã♦ ❝❤❛♠❛❞♦s ❞❡ ♣♦♥t♦s

❞❡ ❡q✉✐❧í❜r✐♦✳ ❱❛♠♦s s✉♣♦r q✉❡ A s❡❥❛ ✐♥✈❡rtí✈❡❧ ✭detA ₆= 0✮ ❡ ♣♦rt❛♥t♦ x = 0 é ♦ ú♥✐❝♦

♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ s✐st❡♠❛ ✭✷✳✶✶✮✳

❆s s♦❧✉çõ❡s✱ q✉❡ sã♦ ❢✉♥çõ❡s ✈❡t♦r✐❛✐s q✉❡ s❛t✐s❢❛③❡♠ ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧✱ ♣♦❞❡♠ s❡r ✈✐st❛s ❝♦♠♦ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❛r❛♠étr✐❝❛ ❞❡ ✉♠❛ ❝✉r✈❛ ♥♦ ♣❧❛♥♦ x1x2✳ ❖❜s❡r✈❛♠♦s ❡ss❛ ❝✉r✈❛ ❝♦♠♦ ✉♠❛ tr❛❥❡tór✐❛ ♦✉ ✉♠ ❝❛♠✐♥❤♦ ♣❡r❝♦rr✐❞♦ ♣♦r ✉♠ ♦❜❥❡t♦ ❝✉❥❛ ✈❡❧♦❝✐❞❛❞❡ dx/dt é ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧✳ ❖ ♣❧❛♥♦ x1x2 r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ ♣❧❛♥♦ ❞❡ ❢❛s❡ ❡ ♦ ❝♦♥❥✉♥t♦ r❡♣r❡s❡♥t❛t✐✈♦ ❞❡ tr❛❥❡tór✐❛s é ❝❤❛♠❛❞♦ ❞❡ r❡tr❛t♦ ❞❡ ❢❛s❡✳

P❛r❛ ✉♠❛ ❛♥á❧✐s❡ ❝♦♠♣❧❡t❛ ❞❛s s♦❧✉çõ❡s ❡ ♣❧❛♥♦ ❞❡ ❢❛s❡ ❞♦ s✐st❡♠❛ ✭✷✳✶✶✮ ❝♦♥s✉❧t❡ ❆♣ê♥❞✐❝❡ ❆✳

❱❛♠♦s ❛❣♦r❛ ❝♦♥s✐❞❡r❛r ✉♠ s✐st❡♠❛ ❜✐❞✐♠❡♥s✐♦♥❛❧ ♥ã♦ ❧✐♥❡❛r

x′

=f(x). ✭✷✳✶✹✮

◆♦ss♦ ♦❜❥❡t✐✈♦ é ✐♥✈❡st✐❣❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s tr❛❥❡tór✐❛s ❞♦ s✐st❡♠❛ ✭✷✳✶✹✮ ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦x✳ P❛r❛ ✐ss♦ ✈❛♠♦s ❛♥❛❧✐s❛r q✉❛♥❞♦ é ♣♦ssí✈❡❧ ❛♣r♦①✐♠❛r ♦ s✐st❡♠❛ ♥ã♦ ❧✐♥❡❛r ✭✷✳✶✹✮ ♣♦r ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ❛♣r♦♣r✐❛❞♦ ❝✉❥❛s tr❛❥❡tór✐❛s s❡❥❛♠ ❢á❝❡✐s ❞❡ ❞❡s❝r❡✈❡r✳

❆❧❣✉♠❛s ♣❡r❣✉♥t❛s ♣♦❞❡♠ s✉r❣✐r ❝♦♠♦✿ ❛s tr❛❥❡tór✐❛s ❞♦ s✐st❡♠❛ ❧✐♥❡❛r sã♦ ❜♦❛s ❛♣r♦①✐♠❛çõ❡s ❞❛s tr❛❥❡tór✐❛s ❞♦ s✐st❡♠❛ ♥ã♦ ❧✐♥❡❛r❄ ❈♦♠♦ ❡♥❝♦♥tr❛r ♦ s✐st❡♠❛ ❧✐♥❡❛r ❛♣r♦♣r✐❛❞♦❄ P❛r❛ r❡s♣♦♥❞❡r ❛ ❡ss❛s ♣❡r❣✉♥t❛s ✈❛♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ ❞❡✜♥✐r ♦ q✉❡ é ❡st❛r ♣ró①✐♠♦ ❡♠ ✉♠ s❡♥t✐❞♦ ❛♣r♦♣r✐❛❞♦✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ✈❛♠♦s ❝♦♥✈❡♥✐❡♥t❡♠❡♥t❡ ❡s❝♦❧❤❡r ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❝♦♠♦ s❡♥❞♦ ❛ ♦r✐❣❡♠✱ ♣♦✐s s❡ x₆= 0✱ s❡♠♣r❡ ♣♦❞❡✲s❡ ❢❛③❡r

❛ s✉❜st✐t✉✐çã♦✱ u=x₋x ♥❛ ❡q✉❛çã♦ ✭✷✳✶✹✮✳ ❙❡❥❛ ♦ s✐st❡♠❛ ♥ã♦ ❧✐♥❡❛r

x′

=Ax+g(x) ✭✷✳✶✺✮

♦♥❞❡Aé ✉♠❛ ♠❛tr✐③ r❡❛❧2_×2❡g(x)❝♦♥tí♥✉❛ é ❞❛❞❛ ♣♦r ✉♠ ✈❡t♦r ❝♦❧✉♥❛2_×1✳ ❙✉♣♦♥❤❛

q✉❡x= 0é ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ✐s♦❧❛❞♦✱ ♦✉ s❡❥❛✱ ❡①✐st❡ ✉♠ ❝ír❝✉❧♦ ❡♠ t♦r♥♦ ❞❛ ♦r✐❣❡♠

❞❡♥tr♦ ❞♦ q✉❛❧ ♥ã♦ ❡①✐st❡ q✉❛❧q✉❡r ♦✉tr♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✳ ❆❞♠✐t✐♥❞♦ q✉❡det(A)₆= 0

❡ ❛ss✐♠x= 0 é ♦ ú♥✐❝♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ✐s♦❧❛❞♦ ❞♦ s✐st❡♠❛ ❧✐♥❡❛r x′

=Ax✳

P❛r❛ q✉❡ ♦ s✐st❡♠❛ ♥ã♦ ❧✐♥❡❛r ✭✷✳✶✺✮ s❡❥❛ ♣ró①✐♠♦ ❛♦ s✐st❡♠❛ ❧✐♥❡❛r ✭✷✳✶✶✮ é ♣r❡❝✐s♦ q✉❡g s❛t✐s❢❛ç❛ ❛ ❝♦♥❞✐çã♦

||g(x)_||

||x_|| →0q✉❛♥❞♦ x→0, ✭✷✳✶✻✮

♦✉ s❡❥❛✱ ||g(x)_|| é ♣❡q✉❡♥♦ ❝♦♠♣❛r❛❞♦ ❛♦ _||x_|| ♣ró①✐♠♦ à ♦r✐❣❡♠✳

❆❧❣✉♠❛s ❞❡✜♥✐çõ❡s ✶✺

é ❝❤❛♠❛❞♦ ❞❡ s✐st❡♠❛ q✉❛s❡ ❧✐♥❡❛r ♥❛ ✈✐③✐♥❤❛♥ç❛ ❞♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ x= 0✳

❊①❡♠♣❧♦ ✷✳✶✳ ❱❛♠♦s ♠♦str❛r q✉❡ ♦ s✐st❡♠❛ ❛❜❛✐①♦ é q✉❛s❡ ❧✐♥❡❛r ♥❛ ✈✐③✐♥❤❛♥ç❛ ❞♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦x= (0,0)✳ ❈♦♥s✐❞❡r❡

(

dx1/dt = x1+x22 dx2/dt = x1+x2

✭✷✳✶✼✮

q✉❡ ♥❛ ❢♦r♠❛ ❞❛ ❡q✉❛çã♦ ✭✷✳✶✺✮ ♥♦s ❞á

x1 x2

!′

= 1 0 1 1

!

x1 x2

!

+ x

2 2

0

!

❊♥❝♦♥tr❛♥❞♦ ♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦✿

x1+x2 = 0 ⇒x1 =−x2 ✭✷✳✶✽✮

x1+x22 = 0 ✭✷✳✶✾✮

❙✉❜st✐t✉✐♥❞♦ ✭✷✳✶✽✮ ❡♠ ✭✷✳✶✾✮ t❡♠♦s

x2₂₋x2 = 0 ⇒x2(x2−1) = 0⇒x2 = 0 ♦✉x2 = 1.

❊♥tã♦✱ s❡ x2 = 0 t❡♠♦s x1 = 0✳ ❆♥❛❧♦❣❛♠❡♥t❡ s❡ x2 = 1 ❡♥tã♦ x1 =−1✳ ❖s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ s✐st❡♠❛ ✭✷✳✶✼✮ sã♦✿ (₋1,1) ❡ (0,0)✳ ❈♦♠♦ det(A) = 1₆= 0, x = (0,0)é ✉♠

♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ✐s♦❧❛❞♦✳

◆♦t❡♠♦s q✉❡ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡gtê♠ ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❝♦♥tí♥✉❛s ❡

||g(x)_||

||x_|| =

p

(0)2_{+ (x}2 2)2

p

(x1)2+ (x2)2

= x

2 2

p

x2 1 +x22

.

❋❛③❡♥❞♦ ✉♠❛ ♠✉❞❛♥ç❛ ♣❛r❛ ❝♦♦r❞❡♥❛❞❛s ♣♦❧❛r❡s✱

(

x1 =rcosθ x2 =rsenθ

✭✷✳✷✵✮

♦❜t❡♠♦s

||g(x)_||

||x_|| =

r2_sen2_(θ) r

lim

r→0rsen

2_{θ= 0.}

▲♦❣♦✱ ♦ s✐st❡♠❛ ✭✷✳✶✼✮ é q✉❛s❡ ❧✐♥❡❛r ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❛ ♦r✐❣❡♠✳

❆❧❣✉♠❛s ❞❡✜♥✐çõ❡s ✶✻

P❛r❛ ❢❛❝✐❧✐t❛r ♦s ❝á❧❝✉❧♦s✱ ✈❛♠♦s ❡s❝r❡✈❡r ♦ s✐st❡♠❛ ✭✷✳✶✺✮ ❡♠ ❢♦r♠❛ ❡s❝❛❧❛r✱ r❡s✉❧t❛♥❞♦ ❡♠

x′

=F(x, y), y′

=G(x, y). ✭✷✳✷✶✮

❉❡✜♥✐çã♦ ✷✳✶✷✳ ❬✺❪❉✐③❡♠♦s q✉❡ f : I _→ R é ✉♠❛ ❢✉♥çã♦ ❞❡ ❝❧❛ss❡ _Cn✱ ❡ ❡s❝r❡✈❡♠♦s

f _∈ Cn_{✱ q✉❛♥❞♦ f é n ✈❡③❡s ❞❡r✐✈á✈❡❧ ❡✱ ❛❧é♠ ❞✐ss♦✱ ❛ ❢✉♥çã♦ f}(n) _{: I → R é ❝♦♥tí♥✉❛✳} ◗✉❛♥❞♦f _∈Cn _{♣❛r❛ t♦❞♦ n ∈N✱ ❞✐③❡♠♦s q✉❡f é ❞❡ ❝❧❛ss❡ C}∞ ❡ ❡s❝r❡✈❡♠♦s

f _∈C∞✳ ➱

❝♦♥✈❡♥✐❡♥t❡ ❝♦♥s✐❞❡r❛rf ❝♦♠♦ s✉❛ ♣ró♣r✐❛ ✏❞❡r✐✈❛❞❛ ❞❡ ♦r❞❡♠ ③❡r♦✑ ❡ ❡s❝r❡✈❡r f(0) _=f✳ ❆ss✐♠✱ f _∈C0 _{s✐❣♥✐✜❝❛ q✉❡ f é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳}

❚❡♦r❡♠❛ ✷✳✶✳ ❬✶❪❖ s✐st❡♠❛ ✭✷✳✷✶✮ s❡rá q✉❛s❡ ❧✐♥❡❛r ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ x= (x0, y0) s❡♠♣r❡ q✉❡ ❛s ❢✉♥çõ❡s F ❡ G t✐✈❡r❡♠ ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s ❝♦♥tí♥✉❛s ❛té ❛ s❡❣✉♥❞❛ ♦r❞❡♠✳

❉❡♠♦♥str❛çã♦✳ P❛r❛ ❞❡♠♦♥str❛r ❡ss❛ ❛✜r♠❛çã♦✱ ✉s❛♠♦s ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❚❛②❧♦r ❡♠ t♦r♥♦ ❞♦ ♣♦♥t♦(x0, y0) ♣❛r❛ ❡s❝r❡✈❡rF(x, y)❡ G(x, y)♥❛ ❢♦r♠❛✿

F(x, y) =F(x0, y0) +Fx(x0, y0)(x−x0) +Fy(x0, y0)(y−y0) +η1(x, y) G(x, y) =G(x0, y0) +Gx(x0, y0)(x−x0) +Gy(x0, y0)(y−y0) +η2(x, y), ♦♥❞❡ η1(x, y)/[(x−x0)2 + (y−y0)2]

1

2 → 0 q✉❛♥❞♦ (x, y) → (x₀, y₀). ❆♥❛❧♦❣❛♠❡♥t❡ ♣❛r❛

η2✳

❖❜s❡r✈❡♠♦s q✉❡ F(x0, y0) = G(x0, y0) = 0✱ ❡ q✉❡ dx/dt = d(x − x0)/dt ❡ dy/dt = d(y₋y0)/dt✳ ❊♥tã♦ ♦ s✐st❡♠❛ ✭✷✳✷✶✮ s❡ r❡❞✉③ ❛

d dt

x₋x0 y₋y0

!

= Fx(x0, y0) Fy(x0, y0)

Gx(x0, y0) Gy(x0, y0)

!

x₋x0 y₋y0

!

+ η1(x, y)

η2(x, y)

!

✭✷✳✷✷✮

♦✉✱ ❡♠ ♥♦t❛çã♦ ✈❡t♦r✐❛❧✱

du dt =

Df

dx (x0, y0)u+η(x), ✭✷✳✷✸✮ ♦♥❞❡ u = (x₋x0, y −y0)T ❡ η = (η1, η2)T✳ ❊♥tã♦✱ ♦ s✐st❡♠❛ ✭✷✳✷✶✮ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✭✷✳✶✻✮✳

❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡st❡ r❡s✉❧t❛❞♦✱ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡ s❡ ❛s ❢✉♥çõ❡s F ❡G ❢♦r❡♠ ❞❡ ❝❧❛ss❡ C2_{✱ ❡♥tã♦ ♦ s✐st❡♠❛ ✭✷✳✷✶✮ é q✉❛s❡ ❧✐♥❡❛r✱ ♦✉ s❡❥❛✱ ♥ã♦ é ♥❡❝❡ssár✐♦} ❝❛❧❝✉❧❛r ♦ ❧✐♠✐t❡ ❝♦♠♦ ♥♦ ❡①❡♠♣❧♦ ✭✷✳✶✮✳ ❚❛♠❜é♠ ♦❜s❡r✈❛♠♦s q✉❡ ♦ s✐st❡♠❛ ❧✐♥❡❛r q✉❡ ❛♣r♦①✐♠❛ ♦ s✐st❡♠❛ ♥ã♦ ❧✐♥❡❛r ✭✷✳✷✶✮ ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦(x0, y0)é ❞❛❞♦ ♣❡❧❛ ♣❛rt❡ ❧✐♥❡❛r ❞❛s ❡q✉❛çõ❡s ✭✷✳✷✷✮ ♦✉ ✭✷✳✷✸✮✳

d dt

u1 u2

!

= Fx(x0, y0) Fy(x0, y0)

Gx(x0, y0) Gy(x0, y0)

!

u1 u2

!

, ✭✷✳✷✹✮

❆❧❣✉♠❛s ❞❡✜♥✐çõ❡s ✶✼

❡q✉❛çã♦ ✭✷✳✷✹✮✳ ❆ ♠❛tr✐③ Df(x0, y0) é ❝❤❛♠❛❞❛ ♠❛tr✐③ ❥❛❝♦❜✐❛♥❛ ❞❡ f ♥♦ ♣♦♥t♦ (x0, y0) ♦♥❞❡f(x, y) = (F(x, y), G(x, y)).

❚❡♦r❡♠❛ ✷✳✷✳ ❙❡ t♦❞♦s ♦s ❛✉t♦✈❛❧♦r❡s ❞❛ ♠❛tr✐③ ❞❡ ❝♦❡✜❝✐❡♥t❡s A ♥♦ s✐st❡♠❛ ❧✐♥❡❛r x′

=Ax t❡♠ ♣❛rt❡s r❡❛✐s ♥❡❣❛t✐✈❛s✱ ❡♥tã♦ s❡✉ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦x= 0 é ❛ss✐♥t♦t✐❝❛♠❡♥t❡

❡stá✈❡❧✳ ❆✐♥❞❛ ♠❛✐s✱ ❡①✐st❡♠ ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s k ❡ α t❛✐s q✉❡

||eAt_x0

|| ≤ke−αt

||x0_|| ♣❛r❛ t♦❞♦t _≥0, x0 _∈R2_. ✭✷✳✷✺✮

❙❡ ✉♠ ❞♦s ❛✉t♦✈❛❧♦r❡s ❞❛ ♠❛tr✐③ ❞❡ ❝♦❡✜❝✐❡♥t❡s A t❡♠ ♣❛rt❡ r❡❛❧ ♣♦s✐t✐✈❛✱ ❡♥tã♦ ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ x= 0 é ✐♥stá✈❡❧✳

❆ ♠❛tr✐③eAt _{é ❛♣r❡s❡♥t❛❞❛ ♥♦ ❆♣ê♥❞✐❝❡ ❬❇❪✳}

❚❡♦r❡♠❛ ✷✳✸✳ ▲✐♥❡❛r✐③❛çã♦✿ ❬✻❪ ❙❡❥❛ x = 0 ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦s s✐st❡♠❛s q✉❛s❡

❧✐♥❡❛r❡s ✭✷✳✷✶✮ ❡ ❞♦ s✐st❡♠❛ ❧✐♥❡❛r ✭✷✳✷✹✮ ❝♦rr❡s♣♦♥❞❡♥t❡✱ ♦♥❞❡f é ✉♠❛ ❢✉♥çã♦C1_{✳ ❊♥tã♦✿} ✭✶✮ ❙❡ t♦❞♦s ♦s ❛✉t♦✈❛❧♦r❡s ❞❛ ♠❛tr✐③ ❥❛❝♦❜✐❛♥❛ Df(x) t❡♠ ♣❛rt❡s r❡❛✐s ♥❡❣❛t✐✈❛s✱ ❡♥tã♦

♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ x ❞❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ x′

=f(x) é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧❀

✭✷✮ ❙❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ❛✉t♦✈❛❧♦r❡s ❞❛ ♠❛tr✐③ ❏❛❝♦❜✐❛♥❛Df(x)t❡♠ ♣❛rt❡ r❡❛❧ ♣♦s✐t✐✈❛✱

❡♥tã♦ ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ x ❞❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ x′

=f(x) é ✐♥stá✈❡❧✳

❉❡♠♦♥str❛çã♦✳ ✭✶✮ P❛r❛ ❛♥❛❧✐s❛r♠♦s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❞❡x✈❛♠♦s ♣r✐♠❡✐✲ r❛♠❡♥t❡ ❢❛③❡r ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ y(t) = x₋x(t)✱ ❞❡ ♠♦❞♦ q✉❡ ♦ ♣♦♥t♦ ❞❡

❡q✉✐❧í❜r✐♦x ❞❡ x′

=f(x)❝♦rr❡s♣♦♥❞❛ ❛♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ y= (0,0)❞♦ s✐st❡♠❛

y′

=f(y(t) +x). ✭✷✳✷✻✮

❈♦♠♦ ♦ s✐st❡♠❛ ✭✷✳✷✶✮ é q✉❛s❡ ❧✐♥❡❛r✱ t❡♠♦s q✉❡ f é ❞❡ ❝❧❛ss❡ C1_{✳ ❆♣❧✐❝❛♥❞♦ ❛} ❡①♣❛♥sã♦ ❞❡ ❚❛②❧♦r ♥❛ ❢✉♥çã♦f(y(t) +x)❡♠ t♦r♥♦ ❞♦ ♣♦♥t♦ x♦❜t❡♠♦s✿

f(y(t) +x) = f(x) +Df(x)y+g(y) ✭✷✳✷✼✮

♦♥❞❡ g(y)s❛t✐s❢❛③

g(0,0) = 0❡ Dg(0,0) = 0. ✭✷✳✷✽✮ ❊♥tã♦✱ ❝♦♠♦ f(x) = 0✱ ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ y′

= f(y(t) +x) ♣♦❞❡ s❡r ❡s❝r✐t❛ ♥❛

❢♦r♠❛

y′

❆❧❣✉♠❛s ❞❡✜♥✐çõ❡s ✶✽

◆♦t❡♠♦s q✉❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ✭✷✳✷✽✮ ❞❡ g(y) ✐♠♣❧✐❝❛♠ q✉❡ ♣ró①✐♠♦ à ♦r✐❣❡♠ g(y)

é ✏♣❡q✉❡♥♦✑ ❝♦♠♣❛r❛❞♦ ❛ y✳ ❙❡❣✉❡ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ q✉❡ ♣❛r❛ ❛❧❣✉♠ m >0✱ ❡①✐st❡ ✉♠ ε >0t❛❧ q✉❡

||g(y)_{|| ≤}m_||y_|| s❡ _||y_||< ε. ✭✷✳✸✵✮ ❘❡t♦r♥❛♥❞♦ à ❡q✉❛çã♦ ✭✷✳✷✾✮✱ s✉♣♦♥❤❛ q✉❡ y s❡❥❛ ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ✭✷✳✷✾✮ s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ y = (0,0) = y0_{✳ ❙❡ ♦❧❤❛r♠♦s ♣❛r❛ g(y(t)) ❝♦♠♦ ✉♠❛} ❢✉♥çã♦ ❞❡ t ❡♥tã♦✱ ✉s❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❛ ✈❛r✐❛çã♦ ❞❛s ❝♦♥st❛♥t❡s ✭✈❡r ❆♣ê♥❞✐❝❡ ❬❇❪✮ t❡♠♦s✿

y(t) = eAty0(t) +

Z t

0

eA(t−s)_{g(y(s))ds. ✭✷✳✸✶✮} ❊♠❜♦r❛ ❛ ❢✉♥çã♦y(t)❛♣❛r❡ç❛ ❡♠ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ❡q✉❛çã♦ ✭✷✳✸✶✮✱ ✈❛♠♦s ✉s❛r

❡ss❛ ❡q✉❛çã♦ ✐♥t❡❣r❛❧ ♣❛r❛ ❡st✐♠❛r||y(t)_||❡♠ t❡r♠♦s ❞❡ _||y0_{|| ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❞❡ t✳} ❙✉♣♦♥❤❛ q✉❡ ❛s ❝♦♥st❛♥t❡s k ❡ α sã♦ ❞❛❞❛s ❝♦♠♦ ♥♦ ❚❡♦r❡♠❛ ✷✳✷✱ m > 0 t❛❧ q✉❡

mk < α ❡ Bε(0,0) = {y∈R2;||y|| ≤ε} t❛❧ q✉❡ ❛ ❡q✉❛çã♦ ✭✷✳✸✵✮ é s❛t✐s❢❡✐t❛✳ ❙❡❣✉❡

❞♦ ❚❡♦r❡♠❛ ✷✳✷ q✉❡

||y(t)_{|| ≤}ke−αt

||y0_(t)||+

Z t

0

k e−α(t−s)_{m||y(s)||ds. ✭✷✳✸✷✮} ❝♦♠ ||y(s)_|| < ε ❡ 0 _≤ s _≤ t✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✸✷✮ ♣♦reαt_{✱ t❡♠♦s✿}

eαt_||y(t)_{|| ≤}k_||y0(t)_||+

Z t

0

km eαs_||y(s)_||ds. ✭✷✳✸✸✮ ❙❡ ❛♣❧✐❝❛r♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ●r♦♥✇❛❧❧ ✭✈❡r ❆♣ê♥❞✐❝❡ ❬❈❪✮ ❡♠ ✭✷✳✸✸✮✱ ♦❜t❡♠♦s✿

eαt_||y(t)_{|| ≤}k_||y0(t)_||ekmt. ✭✷✳✸✹✮ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ♣♦re−_αt

s❡❣✉❡ q✉❡ ❛ ❡st✐♠❛t✐✈❛ q✉❡ ❡st❛♠♦s ♣r♦❝✉r❛♥❞♦ é✿

||y(t)_{|| ≤}k_||y0(t)_||e−(α−km)t_{✱ ♣❛r❛}

||y(t)_{|| ≤}ε. ✭✷✳✸✺✮ P❛r❛ ❝♦♥❝❧✉✐r ❛ ❞❡♠♦♥str❛çã♦✱ s❡❥❛ δ > 0 t❛❧ q✉❡ kδ < ε✳ ❆ss✐♠✱ _||y(t)_||< ε s❡♠♣r❡ q✉❡ α ₋km > 0✱ ♦ q✉❡ ♣r♦✈❛ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ y(t)✳ ❆✐♥❞❛ ♣❡❧❛ ❡q✉❛çã♦ ✭✷✳✸✺✮✱

❝♦♥❝❧✉í♠♦s q✉❡ ||y(t)_{|| →} 0 q✉❛♥❞♦ t _→ +_∞✳ ▲♦❣♦✱ ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ y= (0,0)

é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧✳

❆❧❣✉♠❛s ❞❡✜♥✐çõ❡s ✶✾

❖ ❚❡♦r❡♠❛ ❞❡ ▲✐♥❡❛r✐③❛çã♦ é t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♣r✐♠❡✐r♦ ♠ét♦❞♦ ❞❡ ▲②❛♣✉♥♦✈ ♦✉ ♠ét♦❞♦ ✐♥❞✐r❡t♦✳

❊①❡♠♣❧♦ ✷✳✷✳ ❆ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠ ♣ê♥❞✉❧♦ s❡♠ ❛♠♦rt❡❝✐♠❡♥t♦ é

d2_θ/dt+ω2_{senθ = 0✱ ❝♦♠ω}2 _{=g/L✱ ♦♥❞❡gé ❛ ❝♦♥st❛♥t❡ ❣r❛✈✐t❛❝✐♦♥❛❧ ❡L♦ ❝♦♠♣r✐♠❡♥t♦} ❞♦ ♣ê♥❞✉❧♦✳ ❋❛③❡♥❞♦x=θ✱ y=dθ/dt ♦❜t❡♠♦s ♦ s✐st❡♠❛

(

dx/dt = y

dy/dt = ₋ω2_senx ✭✷✳✸✻✮ ▼♦str❡♠♦s q✉❡ ♦ s✐st❡♠❛ ♥ã♦ ❧✐♥❡❛r ❛❝✐♠❛ é q✉❛s❡ ❧✐♥❡❛r✱ ♦❜t❡♥❞♦ ♦ s✐st❡♠❛ ❧✐♥❡❛r ❝♦rr❡s♣♦♥❞❡♥t❡ ♣❛r❛ ❝❛❞❛ ✉♠ ❞♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦✳

Pr✐♠❡✐r❛♠❡♥t❡ ✈❛♠♦s ❡♥❝♦♥tr❛r ♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ s✐st❡♠❛ ✭✷✳✸✻✮✳

(

y = 0

−ω2_{senx = 0}

▼❛s senx= 0 ♣❛r❛ x= _±nπ ❝♦♠ n = 0,1,2, . . .✱ ♦✉ s❡❥❛✱ ♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ s✐st❡♠❛ ✭✷✳✸✻✮ sã♦(_±nπ,0)✱n = 0,1,2, . . .✳

❖ s✐st❡♠❛ ✭✷✳✸✻✮ s❡rá q✉❛s❡ ❧✐♥❡❛r ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ (x0, y0) s❡♠♣r❡ q✉❡ ❛s ❢✉♥çõ❡sF(x, y)❡G(x, y)t✐✈❡r❡♠ ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s ❝♦♥tí♥✉❛s ❛té ❛ s❡❣✉♥❞❛

♦r❞❡♠✳

❈♦♠♦F(x, y) =y✱ G(x, y) = ₋ω2_{senxsã♦ ❞✐❢❡r❡♥❝✐á✈❡✐s✱ ♦ s✐st❡♠❛ é q✉❛s❡ ❧✐♥❡❛r ❡♠} ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ❝❛❞❛ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✳

❆s ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s sã♦ ❞❛❞❛s ♣♦r

Fx(x, y) = 0✱ Fy(x, y) = 1✱ Gx(x, y) = −ω2cosx✱Gy(x, y) = 0,

❡ ♦ s✐st❡♠❛ ❧✐♥❡❛r ❝♦rr❡s♣♦♥❞❡♥t❡

u′

= 0 1

−ω2_{cos(±nπ) 0}

!

u1 u2

!

, n= 0,1,2, . . . ✭✷✳✸✼✮ ❊①❡♠♣❧♦ ✷✳✸✳ ❯♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ❡q✉❛çã♦ ❞♦ ♣ê♥❞✉❧♦ ❛♠♦rt❡❝✐❞♦✱ ♦✉ ❞❡ ✉♠ s✐st❡♠❛ ♠❛ss❛✲♠♦❧❛ é ❛ ❡q✉❛çã♦ ❞❡ ▲✐é♥❛r❞

d2_x

dt2 +c(x) dx

dt +g(x) = 0. ✭✷✳✸✽✮ ❙❡c(x) ❢♦r ❝♦♥st❛♥t❡ ❡g(x) =kx✱ ❡♥tã♦ ❡st❛ ❡q✉❛çã♦ t❡♠ ❛ ❢♦r♠❛ ❞❛ ❡q✉❛çã♦ ❧✐♥❡❛r ❞♦ ♣ê♥❞✉❧♦✳ ❙✉♣♦♥❤❛♠♦s ❛❣♦r❛ q✉❡cé ❞❡ ❝❧❛ss❡ C1_{✱ g é ❞❡ ❝❧❛ss❡ C}2 _{❡ g(0) = 0✳}

❆❧❣✉♠❛s ❞❡✜♥✐çõ❡s ✷✵

❡q✉✐❧í❜r✐♦(0,0)✳ ❆ss✐♠✱ ❢❛③❡♥❞♦ ❛ s✉❜st✐t✉✐çã♦ y =dx/dt✱ t❡♠♦s✿

( _dy

dt +c(x)y+g(x) = 0 dx

dt −y = 0

✭✷✳✸✾✮

❆❣♦r❛✱ ✈❛♠♦s ♠♦str❛r q✉❡(0,0)é ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ s✐st❡♠❛ ✭✷✳✸✾✮ r❡s♦❧✈❡♥❞♦

(

−c(x)y₋g(x) = 0 (_∗)

y = 0 ✭✷✳✹✵✮

❙✉❜st✐t✉✐♥❞♦ x = 0 ❡ y = 0 ❡♠ (_∗) ❡ ✉s❛♥❞♦ ♦ ❢❛t♦ q✉❡ g(0) = 0 t❡♠♦s q✉❡ (0,0)

s❛t✐s❢❛③ ✭✷✳✸✾✮ ❡ ♣♦rt❛♥t♦ é ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✳

❆s ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s ❞❡F(x, y) =y ❡G(x, y) =₋c(x)y₋g(x)✱ sã♦

Fx(x, y) = 0, Fy(x, y) = 1, Gx(x, y) = −c

′

(x)y₋g′

(x), Gy(x, y) = −c(x),

q✉❡ sã♦ ❝♦♥tí♥✉❛s✳

▲♦❣♦ ♦ s✐st❡♠❛ é q✉❛s❡ ❧✐♥❡❛r ❡ ♦ s✐st❡♠❛ ❧✐♥❡❛r ❝♦rr❡s♣♦♥❞❡♥t❡ ♣ró①✐♠♦ à ♦r✐❣❡♠ é d

dt u1 u2

!

= 0 1

−g′

(0) ₋c(0)

!

u1 u2

!

. ✭✷✳✹✶✮

❖s ❛✉t♦✈❛❧♦r❡s ❞❡st❡ s✐st❡♠❛ ❧✐♥❡❛r sã♦✿

r1,2 = −

c(0)_±p

[c(0)]2_−4g′₍₀₎

2 .

❆ ♣❛rt❡ r❡❛❧ ♣❛r❛ ❛♠❜♦s ❛✉t♦✈❛❧♦r❡s é ♥❡❣❛t✐✈❛ s❡c(0)>0 ❡g′

(0) >0✳ P❛r❛ ✈❡r✐✜❝❛r

❡ss❡ ❢❛t♦✱ ✈❛♠♦s s✉♣♦r ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡[c(0)]2 _>4g′

(0)✳ ❆ss✐♠✱

r1 =− c(0)

2 −

p

[c(0)]2_−4g′₍₀₎

2 <0

❡ r2 =− c(0)

2 +

p

[c(0)]2_−4g′₍₀₎

2 <−

c(0) 2 +

p

[c(0)]2

2 = 0.

❆♥❛❧♦❣❛♠❡♥t❡✱ s✉♣♦♥❞♦ ❛❣♦r❛ q✉❡[c(0)]2 _<4g′

(0)✱ t❡♠♦s

r1 =− c(0)

2 −

λi

2 ❡ r2 =−

c(0) 2 +

λi

2 ✱ ♦♥❞❡ λ=

p

4g′₍₀₎

−[c(0)]2✳ ❙❡[c(0)]2 _{= 4g}′

(0) ♦ ❛✉t♦✈❛❧♦r ❞❡ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞♦✐s é ♥❡❣❛t✐✈♦✳

❊♥tã♦ ♣❡❧♦ ❚❡♦r❡♠❛ ✭✷✳✸✮✱ ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ (0,0) ❞❛ ❡q✉❛çã♦ ♥ã♦ ❧✐♥❡❛r ✭✷✳✸✾✮ é

❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧✳

❱❛♠♦s s✉♣♦r ❛❣♦r❛ q✉❡c(0)<0✳ ❈♦♠♦

r1 =− c(0)

2 +

p

[c(0)]2_−4g′₍₀₎

2 ❡ r2 =−

c(0) 2 −

p

[c(0)]2_−4g′₍₀₎

❖ s❡❣✉♥❞♦ ♠ét♦❞♦ ❞❡ ▲②❛♣✉♥♦✈ ✷✶

✭✐✮ ❙❡[c(0)]2 _>4g′

(0)t❡♠♦sr1 >0✳ ❙❡ t❛♠❜é♠g′(0) <0t❡♠♦s[c(0)]2−4g′(0)>[c(0)]2 ❡ ♣♦rt❛♥t♦ r2 <0✳ ◆♦ ❡♥t❛♥t♦✱ s❡ g′(0)>0t❡♠♦s r2 >0✳

✭✐✐✮ ❙❡ [c(0)]2 _<4g′

(0) ❡♥tã♦ Re(r1,2)>0✱ ♦♥❞❡ Re(r1,2) ✐♥❞✐❝❛ ❛ ♣❛rt❡ r❡❛❧ ❞♦s ❛✉t♦✈❛✲ ❧♦r❡s✳

✭✐✐✐✮ ❙❡[c(0)]2_−4g′

(0) = 0 ❡♥tã♦ r1,2 =− c(0)

2 >0✱ q✉❡ t❡♠ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞♦✐s✳

❈♦♠♦ ❡♠ t♦❞♦s ♦s ❝❛s♦s ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ❛✉t♦✈❛❧♦r❡s t❡♠ ♣❛rt❡ r❡❛❧ ♣♦s✐t✐✈❛✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✷✳✸✱ ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ (0,0) ❞❛ ❡q✉❛çã♦ ♥ã♦ ❧✐♥❡❛r ✭✷✳✸✾✮ é ✐♥stá✈❡❧✳

✷✳✷ ❖ s❡❣✉♥❞♦ ♠ét♦❞♦ ❞❡ ▲②❛♣✉♥♦✈

❖ s❡❣✉♥❞♦ ♠ét♦❞♦ ❞❡ ▲②❛♣✉♥♦✈✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠ét♦❞♦ ❞✐r❡t♦✱ ♣♦r ♥ã♦ s❡r ♥❡❝❡ssár✐♦ ❝♦♥❤❡❝❡r ❛❧❣♦ s♦❜r❡ ❛s s♦❧✉çõ❡s ❞♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✱ é ✉t✐❧✐✲ ③❛❞♦ ♣❛r❛ ❝❤❡❣❛r♠♦s à ❝♦♥❝❧✉sõ❡s s♦❜r❡ ❛ ❡st❛❜✐❧✐❞❛❞❡ ♦✉ ✐♥st❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❛tr❛✈és ❞❛ ❝❤❛♠❛❞❛ ❢✉♥çã♦ ❞❡ ▲②❛♣✉♥♦✈✱ q✉❡ é ✉♠❛ ❢✉♥çã♦ ❛✉①✐❧✐❛r ❛♣r♦♣r✐❛❞❛✳ ❖ s❡❣✉♥❞♦ ♠ét♦❞♦ ❞❡ ▲②❛♣✉♥♦✈ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ t❡♦r✐❛ ❞❡ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s ❞❡ ❞♦✐s ♣r✐♥❝í♣✐♦s ❢ís✐❝♦s ❜ás✐❝♦s ❬✶❪✿

• ❯♠ ♣♦♥t♦ ❞❡ ✉♠ s✐st❡♠❛ ❝♦♥s❡r✈❛t✐✈♦ é ❡stá✈❡❧ s❡ ❡ s♦♠❡♥t❡ s❡ s✉❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧

t❡♠ ✉♠ ♠í♥✐♠♦ ❧♦❝❛❧ ♥❡st❡ ♣♦♥t♦❀

• ❆ ❡♥❡r❣✐❛ t♦t❛❧ ❞❡ ✉♠ s✐st❡♠❛ ❝♦♥s❡r✈❛t✐✈♦ é ❝♦♥st❛♥t❡ ❞✉r❛♥t❡ ❛ ❡✈♦❧✉çã♦ ❞♦ s✐s✲

t❡♠❛✳

❊♠ ❣❡r❛❧ ♦ ♠ét♦❞♦ ❞❡ ▲②❛♣✉♥♦✈ é ✉t✐❧✐③❛❞♦ q✉❛♥❞♦ ♥ã♦ ❝♦♥s❡❣✉✐♠♦s ✉s❛r ♦ t❡♦r❡♠❛ ❞❡ ❧✐♥❡❛r✐③❛çã♦✱ ♥♦ ❝❛s♦ ❡♠ q✉❡ ♦s ❛✉t♦✈❛❧♦r❡s sã♦ ✐♠❛❣✐♥ár✐♦s ♣✉r♦s✳

❈♦♥s✐❞❡r❡♠♦s ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ q✉❡ ❣♦✈❡r♥❛ ♦ ♣ê♥❞✉❧♦ ♥ã♦ ❛♠♦rt❡❝✐❞♦ ✭❛♣r❡s❡♥✲ t❛❞❛ ♥♦ ❡①❡♠♣❧♦ ✷✳✷✮

d2_θ dt2 +

g

Lsenθ = 0, ❝♦♠ ω 2 ₌ g

L. ❯s❛♥❞♦ ❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧

x=θ, y= dθ

dt, ♦❜t❡♠♦s ♦ s✐st❡♠❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠

dx dt =y dy

dt =− g Lsenx.

❖ s❡❣✉♥❞♦ ♠ét♦❞♦ ❞❡ ▲②❛♣✉♥♦✈ ✷✷

❖s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ♣❛r❛ ❡st❡ s✐st❡♠❛ sã♦

y= 0

−_Lg senx= 0

⇒y= 0, x= 0,_±π,_±2π,_±3π, . . . . P❛r❛ ❛ ♦r✐❣❡♠✱ ♦ s✐st❡♠❛ ❧✐♥❡❛r ❝♦rr❡s♣♦♥❞❡♥t❡ é

d dt

u1 u2

!

= 0 1

−Lg 0

!

u1 u2

!

,

♦♥❞❡ s❡✉s ❛✉t♦✈❛❧♦r❡s sã♦

r=_±i

r

g L.

❈♦♠♦ ♦s ❛✉t♦✈❛❧♦r❡s sã♦ ✐♠❛❣✐♥ár✐♦s ♣✉r♦s✱ ♥ã♦ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r s♦❜r❡ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❛ ♦r✐❣❡♠ ✉s❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❛ ❧✐♥❡❛r✐③❛çã♦✳

◆♦ ❡♥t❛♥t♦✱ ❝♦♠♦ ♥ã♦ ❡①✐st❡ ❛tr✐t♦ ❛t✉❛♥❞♦ ♥♦ s✐st❡♠❛✱ s❛❜❡♠♦s q✉❡ ❛ ❡♥❡r❣✐❛ t♦t❛❧ é ❝♦♥st❛♥t❡✳ ❚❡♠♦s ❡♥tã♦

❊♥❡r❣✐❛ t♦t❛❧ = ✭❊♥❡r❣✐❛ ❝✐♥ét✐❝❛✮ ✰ ✭❊♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧✮=EC+EP

= 1 2mv

2_+mgh

❆ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ♣❡s♦ ♥♦ ✜♥❛❧ ❞♦ ♣ê♥❞✉❧♦ éLdθ

dt ❡ ♣♦rt❛♥t♦ EC = 1

2mL

2_y2_.

❈♦♠♦ ❛ ❛❧t✉r❛ ❞♦ ♣❡s♦ ❞♦ ♣ê♥❞✉❧♦ é ❞❛❞❛ ♣♦rh=L(1₋cosθ)t❡♠♦s

EP =mgL(1₋cosθ). ❊♥tã♦✱ ❛ ❡♥❡r❣✐❛ t♦t❛❧ é ❞❛❞❛ ♣♦r

E = 1 2mL

2_y2_+mgL(1

−cosθ). ❈♦♠♦ ❛ ❡♥❡r❣✐❛ é ❝♦♥s❡r✈❛❞❛

0 = dE

dt =mL 2_ydy

dt +mgLsenθ dx

❖ s❡❣✉♥❞♦ ♠ét♦❞♦ ❞❡ ▲②❛♣✉♥♦✈ ✷✸

❙✉❜st✐t✉✐♥❞♦ ✭✷✳✹✷✮ ❡♠ ✭✷✳✹✸✮ ♦❜t❡♠♦s dE

dt =mL 2_y

−_Lg senx+mgLsenθ(y) = 0. Pró①✐♠♦ ❛♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦(0,0)♦♥❞❡ x✱ y sã♦ ♣❡q✉❡♥♦s t❡♠♦s

E = 1 2mL

2_y2_+mgL(1

−cosx)_≈ 1 2mL

2_y2_+mgL

1₋

1₋ x

2

2 +. . .

≈ 1

2mL

2_y2 ₊1

2mgLx

2_.

❆ ❝♦♥❞✐çã♦ q✉❡ E é ❝♦♥st❛♥t❡ ❡♥tã♦ r❡q✉❡r q✉❡x ❡y s❛t✐s❢❛③❡♠ ❛ ❡q✉❛çã♦ ❞❛ ❡❧✐♣s❡ x2

L + y2

g =

2E mgL2.

P♦❞❡♠♦s ❞❡❞✉③✐r q✉❡ ❛s tr❛❥❡tór✐❛s q✉❡ ♣❛ss❛♠ ♣ró①✐♠❛s ❛♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦(0,0)

♥ã♦ ✐rã♦ s❡ ❛❢❛st❛r ❞♦ ♠❡s♠♦✳ ❊♥tã♦✱ ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡♠(0,0) é ❡stá✈❡❧ ✭♠❛s ♥ã♦

♥❡❝❡ss❛r✐❛♠❡♥t❡ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧✮✳

❆ ✐❞❡✐❛ ♣r✐♥❝✐♣❛❧ ♣♦r trás ❞♦ ♠ét♦❞♦ ❞❡ ▲②❛♣✉♥♦✈ é ❞❡t❡r♠✐♥❛r ❝♦♠♦ ❝❡rt❛s ❢✉♥çõ❡s ❡s♣❡❝✐❛✐s ✭❋✉♥çõ❡s ❞❡ ▲②❛♣✉♥♦✈✮ ✈❛r✐❛♠ ❛♦ ❧♦♥❣♦ ❞❛s s♦❧✉çõ❡s ❞♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s X′

=f(X)✳ ❱❛♠♦s ❝♦♠❡ç❛r ❞❡✜♥✐♥❞♦ ❡ss❛s ❢✉♥çõ❡s✳

❉❡✜♥✐çã♦ ✷✳✶✸✳ ❙❡❥❛ U ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❞❡ R2 ❝♦♥t❡♥❞♦ ❛ ♦r✐❣❡♠✳ ❯♠❛ ❢✉♥çã♦

V ❞❡ ❝❧❛ss❡ C1

V :U _→R t❛❧ q✉❡ _{X 7→V(X);}

é ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛ ❡♠ U s❡ ✭✐✮ V(0) = 0❀

✭✐✐✮ V(X)>0 ♣❛r❛ t♦❞♦ X _∈U ❝♦♠ X ₆= 0✳

❯♠❛ ❢✉♥çã♦ V r❡❛❧ ❡ ❞❡ ❝❧❛ss❡ C1 _{é ♥❡❣❛t✐✈❛ ❞❡✜♥✐❞❛ s❡ −V é ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛✳} ❊①❡♠♣❧♦ ✷✳✹✳ ❆ ❢✉♥çã♦V(x, y) =x2_+y2 _{é ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛✱ ♣♦✐sx}2_+y2 _{= 0 ⇔(x, y) =}

(0,0) ❡ x2 +y2 >0 _∀(x, y) _∈ U ❝♦♠ (x, y) ₆= (0,0)✳ ❏á ❛ ❢✉♥çã♦ V(x, y) = x+y2 ♥ã♦ é ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛ ❡♠ q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ ❞❛ ♦r✐❣❡♠ ♣♦✐s V(x, y) > 0 ♥ã♦ ♦❝♦rr❡

q✉❛♥❞♦y2 _{=−x (x <0)❝♦♠ (x, y)6= (0,0)✳}

P❛r❛ ❛♥❛❧✐s❛r ❛ ❡st❛❜✐❧✐❞❛❞❡ ♦✉ ♥ã♦ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ▲②❛♣✉♥♦✈ ♥♦s ❜❛s❡❛♠♦s ♥♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❢✉♥çã♦ V ❛♦ ❧♦♥❣♦ ❞❛s s♦❧✉çõ❡s ❞♦ s✐st❡♠❛✳ ❙❡❥❛ ϕ(t) = (x(t), y(t))✉♠❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛

X′

❖ s❡❣✉♥❞♦ ♠ét♦❞♦ ❞❡ ▲②❛♣✉♥♦✈ ✷✹

♦♥❞❡ X = x

y

!

❡ f(X) = F(x, y)

G(x, y)

!

.

❊♥tã♦✱ ♣❡❧❛ r❡❣r❛ ❞❛ ❝❛❞❡✐❛ t❡♠♦s✱

V′

(ϕ(t)) = ∂V

∂x(ϕ(t))x

′

(t) + ∂V

∂y(ϕ(t))y

′

(t)

♦✉ ❛✐♥❞❛✱

V′

(ϕ(t)) =_∇V(ϕ(t))_·f(ϕ(t))

♦♥❞❡ f(x, y) = (F(x, y), G(x, y))✱ ♦✉ s❡❥❛✱ V′ é ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞♦ ✈❡t♦r

f(x, y) ❝♦♠ ♦

✈❡t♦r ❣r❛❞✐❡♥t❡∇V(ϕ(t))❞❡V ❡♠ ϕ(t)✿

V′

(ϕ(t)) =f(x, y)_{· ∇}V(X) = _||f(x, y)_||._||∇V(X)_||cosθ, ✭✷✳✹✺✮ ♦♥❞❡θ é ♦ â♥❣✉❧♦ ❡♥tr❡ f(x, y)❡ _∇V(ϕ(t))✳

❈♦♠ ♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r ♣♦❞❡♠♦s ❛♥❛❧✐s❛r ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ♣❛r❛ ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦x= (0,0)✳

❚❡♦r❡♠❛ ✷✳✹✳ ✭▲②❛♣✉♥♦✈✮ ❙❡❥❛x= (0,0)✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ X′

=f(X)❡ V ✉♠❛ ❢✉♥çã♦ ❞❡ ❝❧❛ss❡ C1_{✱ ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛ ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ❞❡ (0,0)✳}

✭✐✮ ❙❡ V′

(X)_≤0 ♣❛r❛X _∈U _{− {}(0,0)_}✱ ❡♥tã♦ (0,0) é ❡stá✈❡❧✳

✭✐✐✮ ❙❡ V′

(X)<0 ♣❛r❛ X _∈U _{− {}(0,0)_}✱ ❡♥tã♦ (0,0) é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧✳

✭✐✐✐✮ ❙❡ V′

(X)>0 ♣❛r❛ X _∈U _{− {}(0,0)_}✱ ❡♥tã♦ (0,0) é ✐♥stá✈❡❧✳

❉❡♠♦♥str❛çã♦✳ ✭✐✮ ❙❡❥❛ V ✉♠❛ ❢✉♥çã♦ ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛✳ ❱❛♠♦s t♦♠❛r ε > 0 s✉✜✲

❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ ❞❡ ♠♦❞♦ q✉❡ B = _{(x, y)_∈R2_{;||(x, y)||< ε} ⊂ U} ❡ s❡❥❛

k = min_{V(x, y); _||(x, y)_||=ε_}✱ q✉❡ é ♣♦s✐t✐✈♦ ♣♦✐s V é ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛✳ P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ V✱ ❡①✐st❡ δ ❝♦♠ 0 < δ < ε t❛❧ q✉❡ Bδ(x) ⊂ B ❡ V(x, y) < k✱

∀(x, y)_∈ Bδ(x)✳ ❱❛♠♦s ♠♦str❛r q✉❡ ❛ s♦❧✉çã♦ ❝♦♥st❛♥t❡ x(t) = x ✐♥✐❝✐❛❞❛ ♥❛ ❜♦❧❛

❞❡ r❛✐♦ δ é ❡stá✈❡❧✱ ✐st♦ é✱

||ϕ(t0)−x||< δ ⇒ ||ϕ(t)−x||< ε, ∀t >0. ✭✷✳✹✻✮ ❙❡❥❛ t = min_{s_∈(0, t]; _||ϕ(s)₋x_{|| ≥}ε_}✱ ❛ss✐♠ t❡♠♦s V(x(t)) _≥ k✳ P♦r ❤✐♣ót❡s❡ V′

≤0✱ ♦✉ s❡❥❛✱V é ♥ã♦ ❝r❡s❝❡♥t❡ ❛♦ ❧♦♥❣♦ ❞❛s s♦❧✉çõ❡s✱ ❧♦❣♦✱V(x(t))_≤V(x(t0))≤ k _⇒V(x(t))< k✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❆ss✐♠✱ ♦ ♣♦♥t♦x é ❡stá✈❡❧✳

❖s ❞❡♠❛✐s ✐t❡♥s ❞❡❝♦rr❡♠ ❞❛ ❣❡♦♠❡tr✐❛ ❞♦ ✈❡t♦r ∇V(X)✭❬✻❪✮✳

❉❡✜♥✐çã♦ ✷✳✶✹✳ ❯♠❛ ❢✉♥çã♦ ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛V ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛U ❞❛ ♦r✐❣❡♠ é ❞✐t❛ ✉♠❛ ❢✉♥çã♦ ❞❡ ▲②❛♣✉♥♦✈ ♣❛r❛X′

=f(X)s❡ V′

❖ s❡❣✉♥❞♦ ♠ét♦❞♦ ❞❡ ▲②❛♣✉♥♦✈ ✷✺

◗✉❛♥❞♦ V′

(X) < 0 ♣❛r❛ t♦❞♦ x _∈ U _{− {}(0,0)_}✱ ❛ ❢✉♥çã♦ V é ❝❤❛♠❛❞❛ ✉♠❛ ❢✉♥çã♦ ❞❡ ▲②❛♣✉♥♦✈ ❡str✐t❛✳

❊①❡♠♣❧♦ ✷✳✺✳ ❯♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❛ ❡q✉❛çã♦ ❞❡ ▲✐é♥❛r❞ ❞♦ ❊①❡♠♣❧♦ ✭✷✳✸✮ é d2_u

dt2 + du

dt +g(u) = 0,

♦♥❞❡ g(0) = 0✱ g(u)> 0 ♣❛r❛ 0< u < k ❡ g(u) <0 ♣❛r❛ ₋k < u <0✱ ✐st♦ é✱ ug(u) >0

♣❛r❛ x₆= 0✱ ₋k < u < k✳

❊ss❛ ❡q✉❛çã♦ ♣♦❞❡ s❡r ✐♥t❡r♣r❡t❛❞❛ ❝♦♠♦ ❞❡s❝r❡✈❡♥❞♦ ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠ s✐st❡♠❛ ♠❛ss❛✲♠♦❧❛ ❝♦♠ ❛♠♦rt❡❝✐♠❡♥t♦ ♣r♦♣♦r❝✐♦♥❛❧ à ✈❡❧♦❝✐❞❛❞❡ ❡ ✉♠❛ ❢♦rç❛ r❡st❛✉r❛❞♦r❛ ♥ã♦ ❧✐♥❡❛r✳

❋❛③❡♥❞♦ x = u✱ y = du/dt✱ ♠♦str❡♠♦s q✉❡ ❛ ♦r✐❣❡♠ é ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦

s✐st❡♠❛ r❡s✉❧t❛♥t❡✳ ₍

dx

dt −y= 0 dy

dt +y+g(x) = 0

✭✷✳✹✼✮ ❉❡t❡r♠✐♥❛♥❞♦ ♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ t❡♠♦s

(

y= 0

−y₋g(x) = 0.

❈♦♠♦g(0) = 0✱ s✉❜st✐t✉✐♥❞♦ x= 0 ❡ y= 0 ♥♦ s✐st❡♠❛ ❛♥t❡r✐♦r t❡♠♦s q✉❡ ❛ ♦r✐❣❡♠ é ✉♠

♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✳

❯s❛♥❞♦ ❛ ❢✉♥çã♦ ❞❡ ▲②❛♣✉♥♦✈ ❛❜❛✐①♦✱ ✈❛♠♦s ♠♦str❛r q✉❡ ❛ ♦r✐❣❡♠ é ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡stá✈❡❧✿

V(x, y) = 1 2y

2₊

Z x

0

g(s)ds✱ ₋k < x < k. ❈❛❧❝✉❧❛♥❞♦V′

(x, y)✱

V′

(x, y) =Vx

dx dt +Vy

dy

dt =y(g(x)−g(0)) +y(−y−g(x)) =

=g(x)y₋y2₋g(x)y=₋y2 _≤0,_∀(x, y)_∈U_\{(0,0)_}. ▲♦❣♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✭✷✳✹✮✱ ❛ ♦r✐❣❡♠ é ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡stá✈❡❧✳

▼❡s♠♦ ❝♦♠ ♦ ❛♠♦rt❡❝✐♠❡♥t♦✱ ♥ã♦ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r ❛ ❡st❛❜✐❧✐❞❛❞❡ ❛ss✐♥tót✐❝❛ ❝♦♠ ❡ss❛ ❢✉♥çã♦ ❞❡ ▲②❛♣✉♥♦✈✳ ❆ ❡st❛❜✐❧✐❞❛❞❡ ❛ss✐♥tót✐❝❛ ❞♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ (0,0) ♣♦❞❡

s❡r ❡st❛❜❡❧❡❝✐❞❛ ❝♦♥str✉✐♥❞♦✲s❡ ✉♠❛ ❢✉♥çã♦ ❞❡ ▲②❛♣✉♥♦✈ ♠❡❧❤♦r✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ❛♥á❧✐s❡ ♣❛r❛ ✉♠❛ ❢✉♥çã♦g ❣❡r❛❧ é ✉♠ ♣♦✉❝♦ ♠❛✐s s♦✜st✐❝❛❞❛ ❡ ✈❛♠♦s ♠❡♥❝✐♦♥❛r ❛♣❡♥❛s q✉❡ ✉♠❛ ❢♦r♠❛ ❛♣r♦♣r✐❛❞❛ ♣❛r❛ V é✿

V(x, y) = 1 2y

2_{+Ayg(x) +}

Z x

0

❖ s❡❣✉♥❞♦ ♠ét♦❞♦ ❞❡ ▲②❛♣✉♥♦✈ ✷✻

♦♥❞❡ A é ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ ❛ s❡r ❡s❝♦❧❤✐❞❛ ❞❡ ♠♦❞♦ q✉❡ V s❡❥❛ ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛ ❡ V′ s❡❥❛ ♥❡❣❛t✐✈❛ ❞❡✜♥✐❞❛✳ P❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞♦ ♣ê♥❞✉❧♦

[g(x) = senx]✱ ✉s❛♠♦s V ❝♦♠♦ ♥❛ ❡q✉❛çã♦ ♣r❡❝❡❞❡♥t❡ ❝♦♠A = 1₂ ♣❛r❛ ♠♦str❛r q✉❡ ❛ ♦r✐❣❡♠ é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧

✭♠❛✐s ❞❡t❛❧❤❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬✶❪✮✳

✸ ▼♦❞❡❧♦s P♦♣✉❧❛❝✐♦♥❛✐s

▼♦❞❡❧♦s ❞❡ ❉✐♥â♠✐❝❛ P♦♣✉❧❛❝✐♦♥❛❧ sã♦ ❜♦♥s ❡①❡♠♣❧♦s ♣❛r❛ ✐❧✉str❛r♠♦s ♦s r❡s✉❧t❛✲ ❞♦s s♦❜r❡ ❛♥á❧✐s❡ ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❛♣r❡s❡♥t❛❞♦s ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✳ ❆♣r❡s❡♥t❛r❡♠♦s ♦s ♠♦❞❡❧♦s ❝❧áss✐❝♦s ❞❡ ❝♦♠♣❡t✐çã♦ ❡♥tr❡ ❡s♣é❝✐❡s ❡ ♣r❡s❛ ♣r❡❞❛❞♦r✳

✸✳✶ ❊s♣é❝✐❡s ❡♠ ❝♦♠♣❡t✐çã♦

❖ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ♣♦♣✉❧❛çõ❡s é ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛ ♣r❡✈✐sã♦ ❞❡ ❡①t✐♥çã♦ ❞❡ ✉♠❛ ♦✉ ♠❛✐s ♣♦♣✉❧❛çõ❡s ♦✉ ❛té ♠❡s♠♦✱ s♦❜ q✉❡ ❝♦♥❞✐çõ❡s ❡ss❛s ♣♦♣✉❧❛çõ❡s ♣♦❞❡♠ ❝♦❡①✐st✐r✳ ❆♣r❡s❡♥t❛♠♦s ♣r✐♠❡✐r❛♠❡♥t❡✱ ♦ ♠♦❞❡❧♦ ❞❡ ❝♦♠♣❡t✐çã♦ ❡♥tr❡ ❞✉❛s ❡s♣é❝✐❡s✱ q✉❡ ❢♦✐ ♣r♦♣♦st♦ ✐♥✐❝✐❛❧♠❡♥t❡ ♣♦r ▲♦t❦❛✲❱♦❧t❡rr❛ ❬✶❪✱ ❬✼❪✱ ✐♥tr♦❞✉③✐♥❞♦ ♠♦❞✐✜❝❛çõ❡s ♥❛ ❡q✉❛çã♦ ❧♦❣íst✐❝❛ q✉❡ ✐♥❝❧✉✐✉ ♦s ❡❢❡✐t♦s ✐♥✐❜✐❞♦r❡s ❞❡ ❝❛❞❛ ❡s♣é❝✐❡ ❡♠ r❡❧❛çã♦ à ♦✉tr❛✳

❈♦♥s✐❞❡r❛♥❞♦ x❡ y❛s ❞✉❛s ♣♦♣✉❧❛çõ❡s q✉❡ ❡stã♦ ❡♠ ❝♦♠♣❡t✐çã♦✱ ♦ ♠♦❞❡❧♦ ❞❡ ▲♦t❦❛✲ ❱♦❧t❡rr❛ é ❞❛❞♦ ♣♦r✿

dx

dt =x(ǫ1−σ1x−α1y) dy

dt =y(ǫ2 −σ2y−α2x).

✭✸✳✶✮

❖s ✈❛❧♦r❡s ❞❛s ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s ǫ1✱ ǫ2✱ σ1✱ σ2✱ α1 ❡ α2 ✐rã♦ ❞❡♣❡♥❞❡r ❞❛s ❡s♣é❝✐❡s ❡♠ q✉❡stã♦ ❡ tê♠ q✉❡ s❡r ❞❡t❡r♠✐♥❛❞♦s✱ ❡♠ ❣❡r❛❧✱ ❛tr❛✈és ❞❡ ♦❜s❡r✈❛çõ❡s✳

❊①❡♠♣❧♦ ✸✳✶✳ ❱❛♠♦s ❞✐s❝✉t✐r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ q✉❛❧✐t❛t✐✈♦ ❞❛s s♦❧✉çõ❡s ❞♦ s✐st❡♠❛ dx

dt =x

3

2−x− 1 2y

dy dt =y

2₋y₋ 3

4x

,

✭✸✳✷✮

q✉❡ é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ s✐st❡♠❛ ✭✸✳✶✮✳

❖s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ sã♦ ♦❜t✐❞♦s r❡s♦❧✈❡♥❞♦ ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s

x

3

2 −x− 1 2y

= 0 ❡ y

2₋y₋ 3

4x

= 0. ✭✸✳✸✮