You really need to do the math. If the orbit of such a planet went from 1 Astronomical Unit (AU) (Earth's orbit) to ~10 AU (Saturn's orbit), the orbital period would be on the order of 12 years...not 3,600 years---off by a factor of 300. If it were as large as Saturn at that distance, it would be observable with the naked eye. Neither Jupiter nor Saturn have perceptible gravitational effects on Earth.
"Some sources" are full of bullshit. You are better off getting a textbook on orbital mechanics and learning for yourself why it is bullshit.
You really need to do the math. If the orbit of such a planet went from 1 Astronomical Unit (AU) (Earth's orbit) to ~10 AU (Saturn's orbit), the orbital period would be on the order of 12 years...not 3,600 years---off by a factor of 300. If it were as large as Saturn at that distance, it would be observable with the naked eye. Neither Jupiter nor Saturn have perceptible gravitational effects on Earth.
"Some sources" are full of bullshit. You are better off getting a textbook on orbital mechanics and learning for yourself why it is bullshit.