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yeah but you make me carios...

Math-dashti 120
June 29 2025
0 0

@mmcdara i can't see it seem something new there is no error just not showing 

> 

restart;
> 

with(plots):
> 

K[1] := 2/`ξ__1`^2; K[2] := (2*`ξ__1`*`γ__1`*`γ__2`+`ξ__2`*`γ__1`^2-`ξ__3`)/(`ξ__1`^2*`ξ__2`)

2/xi__1^2

  

(2*xi__1*gamma__1*gamma__2+xi__2*gamma__1^2-xi__3)/(xi__1^2*xi__2)

 (1)
> 

`ξ__1` := 1; `ξ__2` := 1; `ξ__3` := 1; `γ__1` := 2.2; `γ__2` := 1

1

  

1

  

1

  

2.2

  

1

 (2)
> 

de1 := diff(u(t),t) = v(t);
de2 := diff(v(t),t) = - K[1]*u(t)^3 + K[2]*u(t) + epsilon*sin(theta*t);

diff(u(t), t) = v(t)

  

diff(v(t), t) = -2*u(t)^3+8.24*u(t)+epsilon*sin(theta*t)

 (3)
> 

ic := u(0)=0, v(0)=0;

u(0) = 0, v(0) = 0

 (4)
> 

rt := 0, 200:

dsol := dsolve(
          {de1,de2,ic}
          , numeric
          , parameters=[epsilon, theta]
          , range=rt[1]..rt[2]
          , maxfun=10^5
        );

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := [epsilon = epsilon, theta = theta]; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 28, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([0, 0, 0, Array(1..0, 1..2, {}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..65, {(1) = 2, (2) = 2, (3) = 0, (4) = 0, (5) = 2, (6) = 0, (7) = 0, (8) = 0, (9) = 0, (10) = 1, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 0, (19) = 100000, (20) = 0, (21) = 0, (22) = 1, (23) = 4, (24) = 0, (25) = 1, (26) = 15, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0, (54) = 0, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0, (64) = -1, (65) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = .0, (7) = .0, (8) = 0.10e-5, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = 1.0, (14) = .0, (15) = .49999999999999, (16) = .0, (17) = 1.0, (18) = 1.0, (19) = .0, (20) = .0, (21) = 1.0, (22) = 1.0, (23) = .0, (24) = .0, (25) = 0.10e-14, (26) = .0, (27) = .0, (28) = .0}, datatype = float[8], order = C_order)), ( 6 ) = (Array(1..4, {(1) = 0., (2) = 0., (3) = Float(undefined), (4) = Float(undefined)})), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = .0, (1, 2) = .203125, (1, 3) = .3046875, (1, 4) = .75, (1, 5) = .8125, (1, 6) = .40625, (1, 7) = .8125, (2, 1) = 0.6378173828125e-1, (2, 2) = .0, (2, 3) = .279296875, (2, 4) = .27237892150878906, (2, 5) = -0.9686851501464844e-1, (2, 6) = 0.1956939697265625e-1, (2, 7) = .5381584167480469, (3, 1) = 0.31890869140625e-1, (3, 2) = .0, (3, 3) = -.34375, (3, 4) = -.335235595703125, (3, 5) = .2296142578125, (3, 6) = .41748046875, (3, 7) = 11.480712890625, (4, 1) = 0.9710520505905151e-1, (4, 2) = .0, (4, 3) = .40350341796875, (4, 4) = 0.20297467708587646e-1, (4, 5) = -0.6054282188415527e-2, (4, 6) = -0.4770040512084961e-1, (4, 7) = .77858567237854}, datatype = float[8], order = C_order), Array(1..6, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = 1.0, (2, 1) = .25, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = 1.0, (3, 1) = .1875, (3, 2) = .5625, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = 2.0, (4, 1) = .23583984375, (4, 2) = -.87890625, (4, 3) = .890625, (4, 4) = .0, (4, 5) = .0, (4, 6) = .2681884765625, (5, 1) = .1272735595703125, (5, 2) = -.5009765625, (5, 3) = .44921875, (5, 4) = -0.128936767578125e-1, (5, 5) = .0, (5, 6) = 0.626220703125e-1, (6, 1) = -0.927734375e-1, (6, 2) = .626220703125, (6, 3) = -.4326171875, (6, 4) = .1418304443359375, (6, 5) = -0.861053466796875e-1, (6, 6) = .3131103515625}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .386, (3) = .21, (4) = .63, (5) = 1.0, (6) = 1.0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.544, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = .9466785280815533, (3, 2) = .25570116989825814, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = 3.3148251870684886, (4, 2) = 2.896124015972123, (4, 3) = .9986419139977808, (4, 4) = .0, (4, 5) = .0, (5, 1) = 1.2212245092262748, (5, 2) = 6.019134481287752, (5, 3) = 12.537083329320874, (5, 4) = -.687886036105895, (5, 5) = .0, (6, 1) = 1.2212245092262748, (6, 2) = 6.019134481287752, (6, 3) = 12.537083329320874, (6, 4) = -.687886036105895, (6, 5) = 1.0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = -5.6688, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = -2.4300933568337584, (3, 2) = -.20635991570891224, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = -.10735290581452621, (4, 2) = -9.594562251021896, (4, 3) = -20.470286148096154, (4, 4) = .0, (4, 5) = .0, (5, 1) = 7.496443313968615, (5, 2) = -10.246804314641219, (5, 3) = -33.99990352819906, (5, 4) = 11.708908932061595, (5, 5) = .0, (6, 1) = 8.083246795922411, (6, 2) = -7.981132988062785, (6, 3) = -31.52159432874373, (6, 4) = 16.319305431231363, (6, 5) = -6.0588182388340535}, datatype = float[8], order = C_order), Array(1..3, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 10.126235083446911, (2, 2) = -7.487995877607633, (2, 3) = -34.800918615557414, (2, 4) = -7.9927717075687275, (2, 5) = 1.0251377232956207, (3, 1) = -.6762803392806898, (3, 2) = 6.087714651678606, (3, 3) = 16.43084320892463, (3, 4) = 24.767225114183653, (3, 5) = -6.5943891257167815}, datatype = float[8], order = C_order)]), ( 9 ) = ([Array(1..2, {(1) = .1, (2) = .1}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8]), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8])]), ( 8 ) = ([Array(1..4, {(1) = .0, (2) = .0, (3) = undefined, (4) = undefined}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..2, {(1, 1) = .0, (1, 2) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = .0, (3, 0) = .0, (3, 1) = .0, (3, 2) = .0, (4, 0) = .0, (4, 1) = .0, (4, 2) = .0, (5, 0) = .0, (5, 1) = .0, (5, 2) = .0, (6, 0) = .0, (6, 1) = .0, (6, 2) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = u(t), Y[2] = v(t)]`; YP[2] := -2*Y[1]^3+8.24*Y[1]+Y[3]*sin(Y[4]*X); YP[1] := Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, 0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = u(t), Y[2] = v(t)]`; YP[2] := -2*Y[1]^3+8.24*Y[1]+Y[3]*sin(Y[4]*X); YP[1] := Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 27 ) = (""), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0), ( 28 ) = (0) ] )) ] ); _y0 := Array(0..4, {(1) = 0., (2) = 0., (3) = 0., (4) = undefined}); _vmap := array( 1 .. 2, [( 1 ) = (1), ( 2 ) = (2) ] ); _x0 := _dtbl[1][5][5]; _n := _dtbl[1][4][1]; _ne := _dtbl[1][4][3]; _nd := _dtbl[1][4][4]; _nv := _dtbl[1][4][16]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then if _Env_smart_dsolve_numeric = true or _dtbl[1][4][10] = 1 then if _xout = "left" then if type(_dtbl[2], 'table') then return _dtbl[2][5][1] end if elif _xout = "right" then if type(_dtbl[3], 'table') then return _dtbl[3][5][1] end if end if end if; return _dtbl[1][5][5] elif _xout = "method" then return _dtbl[1][15] elif _xout = "storage" then return evalb(_dtbl[1][4][10] = 1) elif _xout = "leftdata" then if not type(_dtbl[2], 'array') then return NULL else return eval(_dtbl[2]) end if elif _xout = "rightdata" then if not type(_dtbl[3], 'array') then return NULL else return eval(_dtbl[3]) end if elif _xout = "enginedata" then return eval(_dtbl[1]) elif _xout = "enginereset" then _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); return NULL elif _xout = "initial" then return procname(_y0[0]) elif _xout = "laxtol" then return _dtbl[`if`(member(_dtbl[4], {2, 3}), _dtbl[4], 1)][5][18] elif _xout = "numfun" then return `if`(member(_dtbl[4], {2, 3}), _dtbl[_dtbl[4]][4][18], 0) elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return procname(_y0[0]), [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _dtbl[4] <> 2 and _dtbl[4] <> 3 or _x0-_dtbl[_dtbl[4]][5][1] = 0. then error "no information is available on last computed point" else _xout := _dtbl[_dtbl[4]][5][1] end if elif _xout = "function" then if _dtbl[1][4][33]-2. = 0 then return eval(_dtbl[1][10], 1) else return eval(_dtbl[1][10][1], 1) end if elif _xout = "map" then return copy(_vmap) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); _i := false; if _par <> [] then _i := `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then _i := `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) or _i end if; if _i then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars); if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') and _dtbl[1][4][10] <> 1 then procname("right") := _y0[0]; procname("left") := _y0[0] end if end if; if _xout = "initial" then return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)] elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] else return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)], [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] end if elif _xin = "eventstop" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then return 0 end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 <= _dtbl[5-_i][4][9] then _i := 5-_i; _dtbl[4] := _i; _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) elif 100 <= _dtbl[_i][4][9] then _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) else return 0 end if elif _xin = "eventstatus" then if _nv = 0 then error "this solution has no events" end if; _i := [selectremove(proc (a) options operator, arrow; _dtbl[1][3][1][a, 7] = 1 end proc, {seq(_j, _j = 1 .. round(_dtbl[1][3][1][_nv+1, 1]))})]; return ':-enabled' = _i[1], ':-disabled' = _i[2] elif _xin = "eventclear" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then error "no events to clear" end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 100 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-100 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-100; if irem(round(_dtbl[_i][3][1][_j, 4]), 2) = 1 then error "retriggerable events cannot be cleared" end if; _j := round(_dtbl[_i][3][1][_j, 1]); for _k to _nv do if _dtbl[_i][3][1][_k, 1] = _j then if _dtbl[_i][3][1][_k, 2] = 3 then error "range events cannot be cleared" end if; _dtbl[_i][3][1][_k, 8] := _dtbl[_i][3][1][_nv+1, 8] end if end do; _dtbl[_i][4][17] := 0; _dtbl[_i][4][9] := 0; if _dtbl[1][4][10] = 1 then if _i = 2 then try procname(procname("left")) catch: end try else try procname(procname("right")) catch: end try end if end if end if; return elif type(_xin, `=`) and member(lhs(_xin), {"eventdisable", "eventenable"}) then if _nv = 0 then error "this solution has no events" end if; if type(rhs(_xin), {('list')('posint'), ('set')('posint')}) then _i := {op(rhs(_xin))} elif type(rhs(_xin), 'posint') then _i := {rhs(_xin)} else error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; if select(proc (a) options operator, arrow; _nv < a end proc, _i) <> {} then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _k := {}; for _j to _nv do if member(round(_dtbl[1][3][1][_j, 1]), _i) then _k := `union`(_k, {_j}) end if end do; _i := _k; if lhs(_xin) = "eventdisable" then _dtbl[4] := 0; _j := [evalb(assigned(_dtbl[2]) and member(_dtbl[2][4][17], _i)), evalb(assigned(_dtbl[3]) and member(_dtbl[3][4][17], _i))]; for _k in _i do _dtbl[1][3][1][_k, 7] := 0; if assigned(_dtbl[2]) then _dtbl[2][3][1][_k, 7] := 0 end if; if assigned(_dtbl[3]) then _dtbl[3][3][1][_k, 7] := 0 end if end do; if _j[1] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[2][3][4][_k, 1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to defined init `, _dtbl[2][3][4][_k, 1]); _dtbl[2][3][1][_k, 8] := _dtbl[2][3][4][_k, 1] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to rate hysteresis init `, _dtbl[2][5][24]); _dtbl[2][3][1][_k, 8] := _dtbl[2][5][24] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to initial init `, _x0); _dtbl[2][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to fireinitial init `, _x0-1); _dtbl[2][3][1][_k, 8] := _x0-1 end if end do; _dtbl[2][4][17] := 0; _dtbl[2][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("left")) end if end if; if _j[2] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[3][3][4][_k, 2], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to defined init `, _dtbl[3][3][4][_k, 2]); _dtbl[3][3][1][_k, 8] := _dtbl[3][3][4][_k, 2] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to rate hysteresis init `, _dtbl[3][5][24]); _dtbl[3][3][1][_k, 8] := _dtbl[3][5][24] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to initial init `, _x0); _dtbl[3][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to fireinitial init `, _x0+1); _dtbl[3][3][1][_k, 8] := _x0+1 end if end do; _dtbl[3][4][17] := 0; _dtbl[3][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("right")) end if end if else for _k in _i do _dtbl[1][3][1][_k, 7] := 1 end do; _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); _dtbl[4] := 0; if _dtbl[1][4][10] = 1 then if _x0 <= procname("right") then try procname(procname("right")) catch: end try end if; if procname("left") <= _x0 then try procname(procname("left")) catch: end try end if end if end if; return elif type(_xin, `=`) and lhs(_xin) = "eventfired" then if not type(rhs(_xin), 'list') then error "'eventfired' must be specified as a list" end if; if _nv = 0 then error "this solution has no events" end if; if _dtbl[4] <> 2 and _dtbl[4] <> 3 then error "'direction' must be set prior to calling/setting 'eventfired'" end if; _i := _dtbl[4]; _val := NULL; if not assigned(_EnvEventRetriggerWarned) then _EnvEventRetriggerWarned := false end if; for _k in rhs(_xin) do if type(_k, 'integer') then _src := _k elif type(_k, 'integer' = 'anything') and type(evalf(rhs(_k)), 'numeric') then _k := lhs(_k) = evalf[max(Digits, 18)](rhs(_k)); _src := lhs(_k) else error "'eventfired' entry is not valid: %1", _k end if; if _src < 1 or round(_dtbl[1][3][1][_nv+1, 1]) < _src then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _src := {seq(`if`(_dtbl[1][3][1][_j, 1]-_src = 0., _j, NULL), _j = 1 .. _nv)}; if nops(_src) <> 1 then error "'eventfired' can only be set/queried for root-finding events and time/interval events" end if; _src := _src[1]; if _dtbl[1][3][1][_src, 2] <> 0. and _dtbl[1][3][1][_src, 2]-2. <> 0. then error "'eventfired' can only be set/queried for root-finding events and time/interval events" elif irem(round(_dtbl[1][3][1][_src, 4]), 2) = 1 then if _EnvEventRetriggerWarned = false then WARNING(`'eventfired' has no effect on events that retrigger`) end if; _EnvEventRetriggerWarned := true end if; if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then _val := _val, undefined elif type(_dtbl[_i][3][4][_src, _i-1], 'undefined') or _i = 2 and _dtbl[2][3][1][_src, 8] < _dtbl[2][3][4][_src, 1] or _i = 3 and _dtbl[3][3][4][_src, 2] < _dtbl[3][3][1][_src, 8] then _val := _val, _dtbl[_i][3][1][_src, 8] else _val := _val, _dtbl[_i][3][4][_src, _i-1] end if; if type(_k, `=`) then if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then error "cannot set event code for a rate hysteresis event" end if; userinfo(3, {'events', 'eventreset'}, `manual set event code `, _src, ` to value `, rhs(_k)); _dtbl[_i][3][1][_src, 8] := rhs(_k); _dtbl[_i][3][4][_src, _i-1] := rhs(_k) end if end do; return [_val] elif type(_xin, `=`) and lhs(_xin) = "direction" then if not member(rhs(_xin), {-1, 1, ':-left', ':-right'}) then error "'direction' must be specified as either '1' or 'right' (positive) or '-1' or 'left' (negative)" end if; _src := `if`(_dtbl[4] = 2, -1, `if`(_dtbl[4] = 3, 1, undefined)); _i := `if`(member(rhs(_xin), {1, ':-right'}), 3, 2); _dtbl[4] := _i; _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if; return _src elif _xin = "eventcount" then if _dtbl[1][3][1] = 0 or _dtbl[4] <> 2 and _dtbl[4] <> 3 then return 0 else return round(_dtbl[_dtbl[4]][3][1][_nv+1, 12]) end if elif type(_xin, `=`) and lhs(_xin) = "setdatacallback" then if not type(rhs(_xin), 'nonegint') then error "data callback must be a nonnegative integer (address)" end if; _dtbl[1][28] := rhs(_xin) else return "procname" end if end if; if _xout = _x0 then return [_x0, seq(evalf(_dtbl[1][6][_vmap[_i]]), _i = 1 .. _n-_ne)] end if; _i := `if`(_x0 <= _xout, 3, 2); if _xin = "last" and 0 < _dtbl[_i][4][9] and _dtbl[_i][4][9] < 100 then _dat := eval(_dtbl[_i], 2); _j := _dat[4][20]; return [_dat[11][_j, 0], seq(_dat[11][_j, _vmap[_i]], _i = 1 .. _n-_ne-_nd), seq(_dat[8][1][_vmap[_i]], _i = _n-_ne-_nd+1 .. _n-_ne)] end if; if not type(_dtbl[_i], 'array') then _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if end if; if _xin <> "last" then if 0 < 0 then if `dsolve/numeric/checkglobals`(op(_dtbl[1][14]), _pars, _n, _y0) then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars, _i) end if end if; if _dtbl[1][4][7] = 0 then error "parameters must be initialized before solution can be computed" end if end if; _dat := eval(_dtbl[_i], 2); _dtbl[4] := _i; try _src := `dsolve/numeric/SC/IVPrun`(_dat, _xout) catch: userinfo(2, `dsolve/debug`, print(`Exception in solnproc:`, [lastexception][2 .. -1])); error end try; if _dat[17] <> _dtbl[1][17] then _dtbl[1][17] := _dat[17]; _dtbl[1][10] := _dat[10] end if; if _src = 0 and 100 < _dat[4][9] then _val := _dat[3][1][_nv+1, 8] else _val := _dat[11][_dat[4][20], 0] end if; if _src <> 0 or _dat[4][9] <= 0 then _dtbl[1][5][1] := _xout else _dtbl[1][5][1] := _val end if; if _i = 3 and _val < _xout then Rounding := -infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further right of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further right of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further right of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further right of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further right of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further right of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further right of %1", evalf[8](_val) end if elif _i = 2 and _xout < _val then Rounding := infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further left of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further left of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further left of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further left of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further left of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further left of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further left of %1", evalf[8](_val) end if end if; if _EnvInFsolve = true then _dig := _dat[4][26]; if type(_EnvDSNumericSaveDigits, 'posint') then _dat[4][26] := _EnvDSNumericSaveDigits else _dat[4][26] := Digits end if; _Env_dsolve_SC_native := true; if _dat[4][25] = 1 then _i := 1; _dat[4][25] := 2 else _i := _dat[4][25] end if; _val := `dsolve/numeric/SC/IVPval`(_dat, _xout, _src); _dat[4][25] := _i; _dat[4][26] := _dig; [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] else Digits := _dat[4][26]; _val := `dsolve/numeric/SC/IVPval`(eval(_dat, 2), _xout, _src); [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] end if end proc, (2) = Array(0..0, {}), (3) = [t, u(t), v(t)], (4) = [epsilon = epsilon, theta = theta]}); _vars := _dat[3]; _pars := map(lhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_rkf45, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_rkf45, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_rkf45, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_rkf45, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_rkf45), 'string') = rhs(x_rkf45); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_rkf45), 'string') = rhs(x_rkf45)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_rkf45) else _ndsol := 1; _ndsol := _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_rkf45) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error end try end proc

 (5)


A correct Maple 2015 procedure

> 

figures := proc(__theta, __epsilon, col::{name, string}, planes::list)
  local p2, p3, data, ru, rv, i, j, k, t_plane, u_plane, v_plane, black_back:

  dsol(parameters=[__theta, __epsilon]):

  p2 := odeplot(
          dsol
          , [u(t),v(t)]
          , t=rt[1]...rt[2]
          , color=col
          , background=black
          , axes=frame
          , refine=1
        ):  

  # u and v ranges

  data := plottools:-getdata(p2)[3]:
  ru   := (min, max)(data[.., 1]);
  rv   := (min, max)(data[.., 2]);

  # The 3 selected background planes.
  
  i, j, k := op(planes):
  t_plane := [ [rt[i], ru[1], rv[1]], [rt[i], ru[2], rv[1]], [rt[i], ru[2], rv[2]], [rt[i], ru[1], rv[2]] ]:
  u_plane := [ [rt[1], ru[j], rv[1]], [rt[2], ru[j], rv[1]], [rt[2], ru[j], rv[2]], [rt[1], ru[j], rv[2]] ]:
  v_plane := [ [rt[1], ru[1], rv[k]], [rt[2], ru[1], rv[k]], [rt[2], ru[2], rv[k]], [rt[1], ru[2], rv[k]] ]:

  black_back := display(
    PLOT3D(
      POLYGONS(t_plane, COLOR(RGB, 0$3)),
      POLYGONS(u_plane, COLOR(RGB, 0$3)),
      POLYGONS(v_plane, COLOR(RGB, 0$3))
    )
  ):

  p3 := odeplot(
          dsol
          , [t, u(t), v(t)]
          , t=rt[1]...rt[2]
          , color=col
          , axes=frame
          , refine=1
        ):

  return p2, p3, black_back
end proc:


Can you run these commands to see what function behaves differently with you present Maple version?

> 

# Command 1
  
dsol(parameters=[1, 1]):
dsol(Pi);

[t = 3.14159265358979, u(t) = HFloat(-0.12143458408390591), v(t) = HFloat(-1.4703737891342488)]

 (6)
> 

# Command 2
  
p2 := odeplot(
          dsol
          , [u(t),v(t)]
          , t=rt[1]..rt[2]
          , color=red
          , axes=frame
        ):
p2;  

Warning, extending a solution obtained using the range argument with 'maxfun' large or disabled is highly inefficient, and may consume a great deal of memory. If this functionality is desired, it is suggested to call dsolve without the range argument

  

 
> 

# Command 3
  
odeplot(
          dsol
          , [u(t),v(t)]
          , t=rt[1]..rt[2]
          , color=red
          , axes=frame
          , refine=1
        );  

 
> 

# Command 4
  
odeplot(
          dsol
          , [u(t),v(t)]
          , t=rt[1]..rt[2]
          , color=red
          , background=black
          , axes=frame
          , refine=1
        );  

 
> 

# Command 5
  
data := plottools:-getdata(p2)[3]:
ru   := (min, max)(data[.., 1]);
rv   := (min, max)(data[.., 2]);

HFloat(HFloat(undefined)), HFloat(HFloat(undefined))

  

HFloat(HFloat(undefined)), HFloat(HFloat(undefined))

 (7)
> 

# Command 6
  
plot_1 := figures(1, 1, cyan, [1$3]):

plot_1[1];

 


EXAMPLE

argument 1 : value of θ
argument 2 : value of ε
argument 3 : color
argument 4 : a list of three integers in {1, 2}:
                     integer 1 = 1 means "plane [u, v] for t = min(t range)
                     integer 1 = 1 means "plane [u, v] for t = min(t range)
                     integer 2 = 2 means "plane [t, v] for t = max(u range)
                     integer 2 = 1 means "plane [t, v] for t = min(u range)
                     integer 3 = 2 means "plane [t, u] for t = max(v range)
                     integer 3 = 2 means "plane [t, u] for t = max(v range)

> 

plot_1 := figures(2.73, 2.7, cyan, [1$3]):
plot_2 := figures(3.43, 9.17, gold, [1$3]):
> 

display(
  <
    plot_1[1]
  |
    display(plot_1[2..3])
  >
)

 

 
 

 

 

  
> 

display(
  <
    plot_2[1]
  |
    display(plot_1[2..3])
  >
)

 

 
 

 

 

  
> 

ru_1 := max(map2(op, 2, op([1, 1], plot_1[3])));
rv_1 := max(map2(op, 3, op([1, 1], plot_1[3])));
ru_2 := max(map2(op, 2, op([1, 1], plot_2[3])));
rv_2 := max(map2(op, 3, op([1, 1], plot_2[3])));

# What follows might not be robust bur seems sufficient

backgrounds := plot_1[3]:
if ru_2 > ru_1 then
  backgrounds := plot_2[3]
end if:

HFloat(HFloat(undefined))

  

HFloat(HFloat(undefined))

  

HFloat(HFloat(undefined))

  

HFloat(HFloat(undefined))

 (8)
> 

display(
  plot_1[2]
  , plot_2[2]
  , backgrounds
)

 

Download chaotic-2_mmcdara_2.mw

i think something missing...

Math-dashti 120
June 29 2025
0 0

@dharr in this two equation  we have three different  coeficent which lead us to get three different condition  like that
#u^3=$u^3 & @u=%u & #u''=@u''  when we set up we have to get the condition which there is parameter in code i saw that you mark  duble term as one  which i think this is gonna be mistake i have to check it analytically by solution

> 

restart
> 

with(PDEtools)
> 

with(plots)
> 

undeclare(prime, quiet)
> 

with(LinearAlgebra)
> 

declare(u(x, y, t), quiet); declare(U(xi), quiet); declare(V(xi), quiet); declare(W(x, y, t), quiet); declare(V(x, y, t), quiet)
> 

E8 := (2*k[1]*p[2]+4*k[2]*p[1])*U(xi)^3+(-k[1]*p[1]^2*p[2]+k[1]*p[3])*U(xi)+(k[1]^3*p[2]+2*k[1]^2*k[2]*p[1])*(diff(diff(U(xi), xi), xi)) = 0

(2*k[1]*p[2]+4*k[2]*p[1])*U(xi)^3+(-k[1]*p[1]^2*p[2]+k[1]*p[3])*U(xi)+(k[1]^3*p[2]+2*k[1]^2*k[2]*p[1])*(diff(diff(U(xi), xi), xi)) = 0

 (1)
> 

NULL
> 

con1 := k[1] = solve(2*k[1]*p[2]+4*k[2]*p[1], k[1])

k[1] = -2*k[2]*p[1]/p[2]

 (2)
> 

NULL
> 

con2 := p[2] = solve(-k[1]*p[1]^2*p[2]+k[1]*p[3], p[2])

p[2] = p[3]/p[1]^2

 (3)
> 

NULL
> 

con3 := k[2] = solve(k[1]^3*p[2]+2*k[1]^2*k[2]*p[1], k[2])

k[2] = -(1/2)*k[1]*p[2]/p[1]

 (4)
> 

NULL
> 

CC := subs({con2, con3}, con1)

k[1] = k[1]*p[2]*p[1]^2/p[3]

 (5)
> 

NULL
> 

E9 := 2*k[2]*U(xi)^3+(-2*k[1]*p[1]*p[2]-k[2]*p[1]^2+k[3])*U(xi)+k[1]^2*k[2]*(diff(diff(U(xi), xi), xi))

2*k[2]*U(xi)^3+(-2*k[1]*p[1]*p[2]-k[2]*p[1]^2+k[3])*U(xi)+k[1]^2*k[2]*(diff(diff(U(xi), xi), xi))

 (6)
> 

NULL
> 

NULL
> 

W1 := 2*k[1]*p[2]+4*k[2]*p[1] = 2*k[2]

2*k[1]*p[2]+4*k[2]*p[1] = 2*k[2]

 (7)
> 

R := k[1] = solve(W1, k[1])

k[1] = -k[2]*(2*p[1]-1)/p[2]

 (8)
> 

NULL
> 

W2 := -k[1]*p[1]^2*p[2]+k[1]*p[3] = -2*k[1]*p[1]*p[2]-k[2]*p[1]^2+k[3]

-k[1]*p[1]^2*p[2]+k[1]*p[3] = -2*k[1]*p[1]*p[2]-k[2]*p[1]^2+k[3]

 (9)
> 

R2 := k[2] = solve(W2, k[2])

k[2] = (k[1]*p[1]^2*p[2]-2*k[1]*p[1]*p[2]-k[1]*p[3]+k[3])/p[1]^2

 (10)
> 

NULL
> 

W22 := k[2] = solve(simplify(subs(R, R2)), k[2])

k[2] = k[3]*p[2]/(2*p[1]^3*p[2]-4*p[1]^2*p[2]+2*p[1]*p[2]-2*p[1]*p[3]+p[3])

 (11)
> 

NULL
> 

RR := subs(W22, R)

k[1] = -k[3]*(2*p[1]-1)/(2*p[1]^3*p[2]-4*p[1]^2*p[2]+2*p[1]*p[2]-2*p[1]*p[3]+p[3])

 (12)
> 

NULL
> 

W3 := k[1]^3*p[2]+2*k[1]^2*k[2]*p[1] = k[1]^2*k[2]

k[1]^3*p[2]+2*k[1]^2*k[2]*p[1] = k[1]^2*k[2]

 (13)
> 

F1 := p[1] = solve(W3, p[1])

p[1] = -(1/2)*(k[1]*p[2]-k[2])/k[2]

 (14)
> 

F11 := simplify(subs({RR, W22}, F1))

p[1] = p[1]

 (15)
> 

NULL
 

NULL

Download C11.mw

the coding is amazing but ......

Math-dashti 120
June 29 2025
0 0

@mmcdara i can't see any effect in thus plot, i don't know why 

Sorry for that.....

Math-dashti 120
June 28 2025
0 0

@acer i think maybe no one answer i post easy question in begin then go to main problem, and you are right i should post question in begin next time i will , sorry, and your code is good but becuase problem are too much and if do  like i mentioned it will be great 

is beautiful...

Math-dashti 120
June 28 2025
0 0

@janhardo code in simple shape if possible?

this is so perfect......

Math-dashti 120
June 28 2025
0 0

@acer but i need another step if possible, i want the outcome be like this picture, i can arange by hand but for being mistakles say S[1]=# and S[2]=# 

> 

restart
> 

with(PDEtools)
> 

undeclare(prime, quiet)
> 

with(LinearAlgebra)
> 

NULL
> 

ode := 2*k[2]*U(xi)^3+(-2*k[1]*p[1]*p[2]-k[2]*p[1]^2+k[3])*U(xi)+k[1]^2*k[2]*(diff(diff(U(xi), xi), xi))

2*k[2]*U(xi)^3+(-2*k[1]*p[1]*p[2]-k[2]*p[1]^2+k[3])*U(xi)+k[1]^2*k[2]*(diff(diff(U(xi), xi), xi))

 (1)
> 

Typesetting:-Settings(prime=xi):
Typesetting:-Settings(typesetprime=true):

 

> 

de := diff(y(t), t$2) + 3*diff(y(t), t) + 2*y(t) = exp(t);

diff(diff(y(t), t), t)+3*(diff(y(t), t))+2*y(t) = exp(t)

 (2)
> 

raw := DEtools[convertsys]({ode}, {}, {U(xi)}, xi, s)[1..2];

[[YP[1] = s[2], YP[2] = -(2*k[2]*s[1]^3+(-2*k[1]*p[1]*p[2]-k[2]*p[1]^2+k[3])*s[1])/(k[1]^2*k[2])], [s[1] = U(xi), s[2] = diff(U(xi), xi)]]

 (3)
> 

eval(raw, [s[1]=u, s[2]=v]);

[[YP[1] = v, YP[2] = -(2*k[2]*u^3+(-2*k[1]*p[1]*p[2]-k[2]*p[1]^2+k[3])*u)/(k[1]^2*k[2])], [u = U(xi), v = diff(U(xi), xi)]]

 (4)
> 

f := proc (u, v) options operator, arrow; v end proc; g := proc (u, v) options operator, arrow; -S[1]*u^3+S[2]*u end proc

proc (u, v) options operator, arrow; v end proc

  

proc (u, v) options operator, arrow; -S[1]*u^3+S[2]*u end proc

 (5)
> 

NULL

Download S.mw

this is good but i need something simple...

Math-dashti 120
June 28 2025
0 0

@dharr i have a lot example and the lymic cycle topic contain a lot of phenomena which i will study more on it i have to find the best way for a system have periodic solution or not, i will collecting data more, thank you so much

You did something new and intresting...

Math-dashti 120
June 28 2025
0 0

@dharr  this is first time i saw this, which a system have two different consevation  depend on u and v and  really is so importan but becuase the conservation must contionuios and out conservation is not continious becuase of that the pretty option of plot is not work on it but this is very new for me i have to work on it more, thank you so much it is a very big help 

thank you,can you plot it too?...

Math-dashti 120
June 28 2025
0 0

@dharr i don't have idea to plot polar system like that i have a lot example ,even in some of them we have trapping region 

It seems you are really pro in this topi...

Math-dashti 120
June 28 2025
0 0

@Rouben Rostamian  i didn't mention that ODE becuase i say maybe is not related i take  out the ODE  and will upload here,  i am say maybe you are work just with system not PDe-->ODE, i have the whole idea about this topic just i need some help of coding specially in last question about conservation quantity is called first integral i study that, Let me upload program and picture i hope help me more and if i did mistake please modified , please if need anything more tell me, i will work on it more and more

F-ode-Fpaper.mw

Is not about what i beliave...

Math-dashti 120
June 27 2025
0 0

@acer is about why not coming out the result is not that 

this is so good!...

Math-dashti 120
June 27 2025
0 0

@Rouben Rostamian  i watch your is so good, but there is some questions appear for me, i am back the problem and try to figure out he give some constant a number but didn't mentioned all, like k,tau, and delta, and i set them 1 but the figure not show me anything

i have to change the problem, in case they didn't mentioned 

mw.mw

thank a lot...

Math-dashti 120
June 26 2025
0 0

@dharr  i didn't have any idea about that untill you explain it, i didn't notice that which we must have that  Conserved quantity

i have to study about that more to get full notion thank you so much

Good point...

Math-dashti 120
June 26 2025
0 0

@dharr  @Rouben Rostamian  it is first time i face this How we can calculate this even by hand if lecture exist  please mention it? if we can do phase portrait of system by Conserved quantity so we don't need the jacobian even?  always is times by -1/2?

yes this is exactly what i am want ......

Math-dashti 120
June 26 2025
0 0

@Rouben Rostamian  thank you so much , it seems you are a profisional in this topic i am glad to have you here.

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