Great question. I don't have a whole lecture on it, but I give a numerical example in the following, along with MATLAB code: th-cam.com/video/gpXAACF5eOI/w-d-xo.html
Great lecture Professor Ross. I wanted to ask, could you recommend any text book which teaches Floquet theory in nonlinear dynamical systems in details. I am currently working on piece wise-smooth (PWS) systems (like impact oscillator, Filippov systems which show chattering and sliding motion near a discontinuity boundary) which shows route to chaos via period doubling, period adding cascade etc. I want to understand/study the Floquet spectrum for such PWS systems. If you could recommend any book/paper/s; it would be very helpful.
In general, I very much like the book which taught me: 'Introduction to Applied Nonlinear Dynamical Systems and Chaos' by Wiggins (@stephenwiggins1). What you're describing, I think I've heard them called "non-smooth systems" or "hybrid systems", those which have both continuous and discrete dynamics. I don't know where that's treated at the level of detail you're seeking. Perhaps the work of Harry Dankowicz? danko.mechanical.illinois.edu/about.htm
@@ProfessorRoss Thank you for the suggestion. I will look into the book by Wiggins. I follow the book on Applied nonlinear dynamics by Ali H. Nayfeh and B. Balachandran. I am also aware of the work by Prof. Dankowicz. Recently I worked on a paper which calculates the Floquet characteristic multipliers of a hybrid system (impact oscillator; degree of smoothness zero). I worked on deriving the state transition matrix near the non-smooth discontinuity boundary with higher order corrections. Next I would like to study the behavior of such non-smooth systems near grazing and chattering using Floquet theory.
@@deeper1993 My impression is that Nayfeh and Balachandran are focused on mechanical systems, while Wiggins is considering more general mathematical systems, but both are good. The only work I've done on hybrid systems was looking at a simple passive downhill walking mechanism: www.dept.aoe.vt.edu/~sdross/papers/simple_walker.pdf
@@ProfessorRoss Yes indeed. I love the contents of the book by Wiggins which you recommended, its more deeper and covers a broader range of dynamical systems. I will look into this book. I am also interested in the theory of chaos, and it seems Wiggins has discussed a lot about it. Thus this book will help me a lot. So far I have only studied chaos from the book by S. H. Strogatz. Thank you for sharing your work on the walking mechanism; will read it surely.
chaos dynamics is so fascinating... thank you ... great presentation...
Hi , how can i calculate the monodromy matrix if i have the jacobian matrix numerically ?
Great question. I don't have a whole lecture on it, but I give a numerical example in the following, along with MATLAB code: th-cam.com/video/gpXAACF5eOI/w-d-xo.html
Great lecture Professor Ross. I wanted to ask, could you recommend any text book which teaches Floquet theory in nonlinear dynamical systems in details. I am currently working on piece wise-smooth (PWS) systems (like impact oscillator, Filippov systems which show chattering and sliding motion near a discontinuity boundary) which shows route to chaos via period doubling, period adding cascade etc. I want to understand/study the Floquet spectrum for such PWS systems. If you could recommend any book/paper/s; it would be very helpful.
In general, I very much like the book which taught me: 'Introduction to Applied Nonlinear Dynamical Systems and Chaos' by Wiggins (@stephenwiggins1). What you're describing, I think I've heard them called "non-smooth systems" or "hybrid systems", those which have both continuous and discrete dynamics. I don't know where that's treated at the level of detail you're seeking. Perhaps the work of Harry Dankowicz? danko.mechanical.illinois.edu/about.htm
@@ProfessorRoss Thank you for the suggestion. I will look into the book by Wiggins. I follow the book on Applied nonlinear dynamics by Ali H. Nayfeh and B. Balachandran. I am also aware of the work by Prof. Dankowicz.
Recently I worked on a paper which calculates the Floquet characteristic multipliers of a hybrid system (impact oscillator; degree of smoothness zero). I worked on deriving the state transition matrix near the non-smooth discontinuity boundary with higher order corrections. Next I would like to study the behavior of such non-smooth systems near grazing and chattering using Floquet theory.
@@deeper1993 My impression is that Nayfeh and Balachandran are focused on mechanical systems, while Wiggins is considering more general mathematical systems, but both are good. The only work I've done on hybrid systems was looking at a simple passive downhill walking mechanism: www.dept.aoe.vt.edu/~sdross/papers/simple_walker.pdf
@@ProfessorRoss Yes indeed. I love the contents of the book by Wiggins which you recommended, its more deeper and covers a broader range of dynamical systems. I will look into this book.
I am also interested in the theory of chaos, and it seems Wiggins has discussed a lot about it. Thus this book will help me a lot. So far I have only studied chaos from the book by S. H. Strogatz.
Thank you for sharing your work on the walking mechanism; will read it surely.
Thank u so much
Thank you for watching, I'm glad you appreciate it.